Le Corbusier modulor. Practical application of Modulor

MODULOR LE CORBUSIER

Proportionation of parts of buildings and structures, corresponding to the natural proportions and proportions of a person, his perception of reality and sensations, is the most important factor in the normal functioning of the human body. Increasingly, in the scientific literature, the fruitful influence on a person of structures proportional to the golden section is noted. It is believed that the most significant contribution to the architectural development of new systems of proportioning in the 20th century. was made by the French architect Le Corbusier, who at the end of the 40s proposed a modulor table with a step equal to the golden number F.

The modulor was based on specific proportions of the human body - the height of a person of one height - one model. Moreover, Le Corbusier had to work out several variants of a model man. And since it was a sample, the size of its growth was determined as average or above average. Le Corbusier writes: “... in the first version of the modulor, he was 175 cm tall, and in the position with a raised arm he had a size of 216 cm. The rest were calculated from these initial data” (Fig. 8).

I will return to this fundamental principle of the modulor, but first I will note the obvious advantages that provided the architectural structures built on its basis with the achievement of aesthetically perfect proportions, the versatility of layouts and their certain proportionality with human proportions.

As already mentioned above, the golden number is obtained mainly either by a geometric method (by dividing the segment in the extreme and average ratios), or by the method of successive approximations along the Fibonacci number series. (I note that there are many such series, Fibonacci was the author of the first fixed series, and all of them, before A.A. Pilecki, seem to have been single. The first double series formed the basis of Le Corbusier's modulor, although he himself probably did not understand this , since his attempts to represent the red and blue lines as a single matrix are not reflected in the publications.)

Rice. 8. Modulor

Le Corbusier's modulor is built as a single series on two shifted Fibonacci series, conventionally called the red and blue lines by the author. Doubling dramatically increased the possibilities of architectural combinatorics. Consider what coefficients are associated with the numbers of the red and blue lines (Table 3):

Table 3

0,806 0,806 0,806 0,806 0,806 0,806

red 0.164 0.266 0.431 0.697 1.128 1.825

blue 0.204 0.330 0.533 0.863 1.397 2.260

1,306 1,306 1,306 1,306 1,306

If we now shift the numbers of the blue line to the red line, we get the full Le Corbusier modulor series: 0.164; 0.204; 0.266; 0.330; 0.431; 0.533; 0.697; 0.863; 1.128; 1.397; 1.825; 2.260. If we divide each number of the red line of the table by the number of the blue line standing diagonally below and to the left of it, then at each division we will get the same coefficient 1.306, and when dividing the numbers of the red line by the numbers of the blue line to the left and below them - coefficient 0.806. This indicates that these shifted lines constitute one numerical matrix having a structure similar to that of the A.A. matrix. Pilecki, only, in contrast to it, the ratio of the number Ф does not run diagonally, but horizontally, and the basic step is not equal to 2. This connection makes the Le Corbusier modulator the possibility of wide compositional combination in the variant linked to the height of a person. The fact that the modulor was limited to only two rows of the matrix A.A. Pilecki and another basic step is its main drawback. This is what limited the possibility of variation in human height options, and in the final version, the modulor was calculated based on a person's height of 6 feet -183 cm (the last rounded number of the red line), and the size in the position with a raised arm - 226 cm (blue line). Consider the variant of constructing the Le Corbusier modulor according to the structure of the matrix A.A. Pilecki (matrix 4):

Matrix 4

1,160 1,319 1,512 2,260

0,819 0,932 1,068 1,397 1,825

0,578 0,659 0,754 0,863 1,128

0,409 0,465 0,533 0,697

0,289 0,330 0,376 0,431

0,204 0,232 0,266

0,144 0,164 0,188

Analyzing matrix 4, we are convinced that its structure completely repeats the structure of the matrix of A. A. Pilecki, including the absence of the basic 1, and this is where the similarity ends. The step of numbers along the vertical, which in the matrix A.A. Pilecki is equal to 2, in the Le Corbusier matrix it is equal to 1.41556... , all cells of the matrix can be filled in (shown in light font on the example of the three left columns), but in this area they do not form a commensurate system of measures, similar to the system of Old Russian sazhens, and therefore, they cannot be recommended for use in proportioning objects.

Modulor Le Corbusier allows, of course, to obtain some common types of proportions of the golden number:

F = 1.618; 2/F = 1.236; F2/2 = 1.309; 2/F2 = 0.472 ...

Without dwelling on their architectural significance, I note that there are quite a lot of them, they determine the conjugation and aesthetics of buildings and structures, and only a small part of them is included in the proportions of Le Corbusier. Moreover, the limitation of the modulor by the initial data of one person (a sample of a certain height) does not automatically measure the proportions of the modulor with the growth of other people, and therefore causes a deviation from proportionality in the construction of parts of objects. Is that why Le Corbusier repeatedly changed the size of the sample, trying to expand the range of applicability of the modulor.

But this drawback should not be considered the most significant. Once again, let's return to its structure and note that the golden number Ф is obtained by successively dividing the numbers of both the red and blue lines into each other. If we carry out a sequential division of each number by each other

2,260/1,829 = 1,236; 1,829/1,397 = 1,309;

1.397/1.130 = 1.236; 1.130 / 0.863 = 1.309, etc., then we get the alternation of two numbers 1.236 and 1.309. Now we define for each of these numbers the one that is a multiple of them:

1,309/1,236 = 1,05492... .

The number that is a multiple of all numbers of the Le Corbusier series is also irrational and is equal to 1.05492... . And this, as will be shown below, means that all structures built on the basis of the Le Corbusier modulor are a multiple of a single factor and therefore, when introduced into the structure of a building object, turn this object into a structure unsuitable for habitation. Consequently, the beauty and aesthetics of a building object created by a modulor are not yet a guarantee of the safety of living in it.

The word is for those who used Modulor. Introduction

Six years of use of Modulor in almost all parts of the world marked the beginning of the first stage of experimental verification.

Six years of using Modulor in the workshop on the street. Sèvres allowed the creation of complete compositions in the development of both large and small projects, providing exceptionally favorable conditions for creativity. It was an undeniable success. We have gained confidence. True, there are still separate and sometimes quite large gaps in the series of Modulor's dimensional quantities, leading, perhaps, to impoverishment of the solutions. Many wrote to us about this, offering to fill these gaps with additional rows of numbers. Some of them spoke about the need to create special rulers with divisions: some on a scale for the design of architectural structures, others on a scale for urban planning projects. It was also proposed to produce a pocket tape with divisions from 0 to 226 cm, corresponding to the main dimensions of the human figure.

The practical application of Modulor has led to a very significant simplification of the table of numerical values, which in printed form takes only half a page; the set of these numbers includes all the necessary for the architect in his work. This is the red and blue rows of the Modulor; people with a good memory do not even need auxiliary tools.

We believe that the ease of use of the Modulor and its success are ultimately due to its construction in accordance with the size of the human figure, which even the "divine proportions" of the Renaissance did not meet. In this sense, Modulor is closer to systems of measures based on proportional relations inherent in the human figure, and the highest achievement of which was the system of the Egyptian cubit.

Wishing to be modest in assessing our own discovery, we will quote the words from the letter of the mathematician Le Lionnet.

“... As you know, I reproach many authors for attributing too much importance, bordering on mysticism, to the use of the golden ratio. I hasten to assure you that this does not apply to you. Speaking about the ratio of numbers in the golden ratio, I always considered it necessary to express a personal judgment on this issue. There is no need for me to repeat myself, since from this point of view our relations coincide. In the field of technology, the ratio of the golden section does not, in my opinion, have any important or significant significance; however, here too it can become a useful condition; as is often the case, the acceptance of any particular condition, even an arbitrary one, may, if consistently observed, lead to success and become the basis for selection and the establishment of order. The order, for example, of the letters of the alphabet has no natural basis; nevertheless, in practice it has proved convenient, and there is no reason to dispute it. Of course, I succumbed in this case to one of the vices inherent in mathematicians, sharpening my thought to the utmost in order to make it more intelligible. Obviously, even if the Modulor does not become the only indispensable tool in the field of plastic arts, it has a number of inherent qualities that, along with other numerical values, can attract the attention of both artists and engineers.

This is the point of view of a mathematician.

Let me express my point of view as an architect, urban planner and artist. It is possible that for modern mathematicians the ratios of numbers in the golden ratio are something very common. With the help of computers, they were able to create sensational combinations of numbers (understandable to them, but inaccessible to the understanding of other people).

However, we should not forget that the numbers in the golden section relations underlie the structure of many objects around us - the structure of the leaf, the structure of the crown of trees and shrub branches, the skeletons of a giraffe or a person that have developed over many thousands and millions of years. They form the environment around us (higher mathematics is not capable of this).

We, the workers who are called upon to create, maintain and modify the human environment, are not upset by the everyday life for mathematicians of the "golden" ratios of numbers. As specialists called to build, sculpt, draw, organize space, we are blinded by the variety of possible combinations of numbers in the golden ratio that we can use in creativity.

The new and this time clear geometric construction of Modulor confirms the hypothesis put forward in 1942:

“Draw the figure of a man with a raised hand 2 m 20 cm high, fit it into two squares placed on top of each other with sides of 1.10 m each; fill in these two squares with a third one that will help you find the solution you are looking for. The inscribed right angle rule will determine the position of the third square. Such a grid, installed on construction sites, will help determine the system of dimensions that link the human figure ... and mathematical relationships (Modulor, 1948). This building was opened in a workshop on the street. Sèvres by Uruguayan Justine Serralta and Frenchman Meissonier. At the Triennale exhibition in 1951 in Milan, in the "Divine Proportion" section, the graphic image of Modulor was exhibited along with the manuscripts and first editions of Vitruvius, Villars de Honnecura, Piero della Francesco, Dürer, Leonardo da Vinci, Alberti, etc.

Andreas Speiser, a mathematician from the University of Basel, who devoted much of his work to the application of mathematics to the visual arts and music, exclaimed before this construction: “How beautiful it is!”

Proof of. Discussion

Here is the final Modulor circuit (Figure 3). Two equal squares with sides FROM cm are placed one on top of the other. The third square is superimposed on them with the ratio of the parts corresponding to the golden ratio; its position is determined by the inscribed right angle rule.

The right angle is strictly (this time) inscribed in a rectangle consisting of two squares, and defines two points at the intersection of the sides of the third square...

Drawing an oblique line through these two points, we get a decreasing series on the left and an increasing series on the right, defining a wonderful spiral of blue and red series of proportional numbers. In May 1950, Dufour de Coderans of the Gironde drew my attention to a mistake made in the first book of Modulor 1.

“You, of course, understand the joy that seized me at the thought that in time the Modulor system would become widespread and give us the opportunity to admire the boundless splendor of proportional relations. We can say in the end that it is amazing ... "Along with this, he points out the error:" ... this mistake can shake the credibility of the universally recognized system; fortunately, this only applies to the theoretical part and will not interfere with the practical implementation of Modulor.

We are talking about a graphical representation of the Modulor series, in which, in my humble opinion, many errors have crept in, and in some cases, uncertainties that prevent finding the right solution.

I propose a different, very simple construction, devoid of these shortcomings and capable of satisfying any captious critic (there are, of course, other constructions that can lead to a similar result).

So, without further ado, allow me to proceed with the presentation of my remarks:

It is impossible to fit the right angle in question into two squares. If it's really two squares, then the angle isn't right. If the angle is right, then one of the two quadrilaterals is not a square.

The position of an inscribed right angle can be determined by a semicircle built on a straight line equal to twice the side of a square. This is the only solution.

Below I will take the liberty of laying out the proposed solution:

Initial square / Its golden ratio

The double square is built using a red dot, the position of which is determined by the golden ratio. Monsieur Dufour's proposal is important, precise, very simple and elegant. But ... after all, I went the other way! My response was:

“In drawing A - I reproduced your construction. In drawing B - I show my construction, (see "Modulor-1", Fig. 14). The initial data was used as the basis of the proportional grid: Person with a raised hand = 2 squares with sides 113 (226).

The position of the third square is determined by the "rule of inscribed right angle".

The position of the third square would have to be determined by dividing the side of this square in relation to the golden ratio, and not by dividing it in half. Hence the noted error (in Modulor, 1948) in fig. 3, 4, 5, 6, 9, leading to ambiguities and uncertainties in fig. 18 and 60.

Such an assumption (Fig. 2) is the result of a natural game of the imagination. That was a preliminary a priori representation, and not the result of a calculation.

This is how the position of point i was determined (Fig. 6, Mod-1). This is repeated (Fig. 9) when determining the position of the point l of length and the straight line g - 1, on which two equal squares with bases gk and ki are constructed by dividing into equal parts. I agree, Monsieur Dufault, that in this construction they sought to express a certain idea and the task was not to ensure the complete accuracy of the drawing. Dufour's construction is very clear and simple, but executed a posteriori, and could not have occurred to anyone: it is, in essence, a verification and clarifying construction.

Now (in 1954) it is necessary to take into account the conditions in which the work was carried out then (1942-1948). We tried to create a proportional grid, a working tool for construction sites. We have come to the definition of mutually agreed numbers.

We set ourselves a practical goal: to help the construction site. In those years we were building a residential building in Marseille (Residential Unit).

Justin Serralta and Meissonier sought to find out the possibility of harmonizing the Modulor with the traditional systems of the past, in particular with the Egyptian cubit.

I was struck by the fact that Modulor provided for the first time harmonious proportion - proportionality based on the size of the human figure. This is truly curious. In the Renaissance, they were passionately engaged in questions of proportions (“divine proportions”). Then they reveled in mathematical calculations, the use of numbers in all sorts of algebraic and geometric constructions. Magnificent polyhedrons were built with axial lines and circles inscribed in them, into which, in turn, human figures and cloaks of structures were inscribed.

The boundless game of digital combinations allowed in each case to create its own system of sizes. The system of "divine proportions" equally subdued the construction of both a building up to 100 meters high and pottery several centimeters high. At that time, they were overly carried away by the very pattern of proportional constructions, giving them a self-contained significance. When drawing numerous polyhedrons and star-shaped diverging axes, they often forgot that a person's eyes are located on the front side of the head and that, depending on his height, the visual perception of objects changes. Thus, the understanding of the true relationship between man and his environment was lost.

Man already at the dawn of his development created devices that provide him with practical help and, more importantly, give him moral satisfaction.

He invented measuring instruments, whose name is: foot (foot), cubit, span, inch, sazhen, etc., etc. ... Using these tools (measurements), he built houses, roads, bridges, palaces and cathedrals. These measures: foot (foot), span, elbow, etc. ... come from parts of the human figure. They contribute to the creation of harmonious relationships and are subject to the same mathematical laws of growth and development, just like living beings.

The Parthenon, pyramids, temples, fishermen's houses and shepherds' huts are built on the basis of these human dimensions.

Subsequently, the metric system was adopted; it was a great invention. Any calculations using the feet and inches system are extremely complex and time consuming. However, dimensional values ​​​​of 10, 20, 30, 40, 50 centimeters or 1, 2, 3, 4, 5 meters are in no way related to the size of our body. Unexpectedly for the inventor himself, Modulor made it possible to create a wide variety of mathematical and geometric combinations, which can be expressed both in meters and feet-inches, etc. ... All dimensions assigned by the Modulor are based on the dimensions of our figure and therefore allow us to create objects adapted to man and his environment, in the field of both architecture and mechanics.

The whole gamut of the numerical dimensions of the Modulor, on the one hand, tends to zero, and on the other, rushes to infinity; within the limits of human height, that is, from 0 to 2 m 26 cm, it is divided into a small and perhaps too limited number of intervals; it is possible, however, that this limitation is its merit!

Some have explored the connection between the old measurement systems and the Modulor. Striking coincidences have been established. The study of Serralt and Meissonier made it possible to include intermediate values ​​common to both systems of measures, borrowed from the system of the Egyptian cubit.

The Egyptian cubit was widely used in ancient times. It is possible that he will enrich the series of Modulor dimensional numbers, which can then be combined with the old measures: inch, palm, foot and cubit.

There are four inches in a palm tree,

in a foot - four palms,

in a cubit, one foot and two palms.

Ancient civilizations originated in certain geographical areas and in different social formations. The units of measure were also different. So, the Egyptian cubit is 45 cm, Greek - 46.3 cm, Roman - 44.4 cm. When erecting places of worship in ancient Egypt, a larger, royal cubit, equal to 52.5 cm, was used, which gave the abodes of the gods an emphatically majestic scale. In Morocco, a cubit length of 51.7 cm and sometimes 53.3 cm is used, while the size of the Tunisian cubit is reduced to 47.3 cm, and in Calcutta to 44.7 cm and in Ceylon to 47 cm. In the Arab countries used the so-called cubit of Omar, equal to 64 cm. the other, the so-called "palm major", was equal to 3D feet. These units of measurement were used until the advent of the metric system, and in different places they had different meanings: in Carrara, the main unit of measurement was a foot, equal to 24.36 cm, in Genoa - 24.7 cm, in Naples - 26.3 cm, in Rome - 22.3 cm, etc.

On fig. 5, made by Serralta and Meissonier, is based on a square with an inscribed "figure of a man with a height of 1.83". But Serralta, as a man with a tender heart, instead of a man depicted a woman with a height (oh horror!) 1 m 83 cm. In two squares placed on top of each other with sides 113 + 113 = 226, a right angle is inscribed, the intersection points of which serve as the basis for construction. .. Height 183 is equal to four cubits of 45.75 cm or six feet of 30.5 cm, and each foot is four palm trees of 7.625 ...

There is only one discrepancy in the marks according to the Modulor based on (183) - 226 and on the basis of the Egyptian cubit (183) - 228.75. Below we will see that such discrepancies, which can be called "additions", do not create significant inconvenience in the construction business when they relate to additional elements. The expression of Modulor in Egyptian cubits harmonizes Modulor with ancient geometric constructions. On fig. 4 values ​​1, 2, 3, 4 and 5 are obtained from the square in which the triangle is inscribed by dividing the sides of the square in half.

On fig. 6 the same construction is given in a more ordered, expressive and clear form. The division of the size 228 cm into 5 cubits and the size 183 cm into 4 cubits is shown; 183 is seen to be equal to 6 feet, 8 half-cubits, or 24 palms. Thus, the resulting construction makes it possible to introduce additional divisions into the intervals between the divisions of the Modulor, corresponding to the values ​​​​of the historical units of measurement: inch, palm, foot and cubit.

These additional dimensional values ​​can be used in construction practice in the secondary parts of the composition to indicate the specific dimensions of some elements of building materials [the thickness of stone slabs (in quarries), the width of sheet iron, the dimensions of standardized materials: bricks, tiles, facing materials, etc. ...]. A discrepancy of 2.75 cm for sizes exceeding five cubits, called "addition", is easily repaid by the thickness of the seams if the number of seams is equal to or more than 6, 8, 11, 18, etc. Serralta and Meisognier argue that the walls , whose height is determined by the Modulor, can be successfully dissected in the most diverse way.

We see that Modulor is successfully combined with beautiful ancient systems of measures. Continuing the traditions, Modulor brings something new and fruitful to contemporary art.

A number of additional constructions performed by Meissonier confirm the possibility of the coexistence of the Modulor and the Egyptian cubit. The human figure can equally well be inscribed in cubes with sides equal to 226 cm (according to the Modulor) or 22.875 cm (Egyptian cubit), complementing each other perfectly if necessary. Further we will see that the volumetric unit 226 x 226 x 226 will be successfully used in the design of apartments, and especially their internal equipment. But let's not get ahead of ourselves!

Below are the statements of a retired mining engineer, Crussard (Paris).

1. Some thoughts on Modulor

The modulor can be expressed by a geometric construction and a system of numerical values. To fully master it, you must master both methods. A small book about Modulor attracts and, one might say, excites precisely because its author is confused between these methods; he mixes them up, giving the impression of a person striving at once, with one glance, to see both the front side of the carpet and its inside, which, of course, he does not succeed. The front side is geometry based on intuition and artistic flair. The wrong side is a numbers game. Often it is recognized as an occupation too rational, not in need of creative imagination; there is no need to say that such an opinion is fundamentally erroneous; both Pythagoras and Plato would have rebelled against him.

I am convinced that for a complete understanding of the Modulor, both geometric constructions, made with a ruler and a compass, and numerical calculations are necessary, provided that they are necessarily performed separately. Geometric constructions should be done as if there were no numbers at all, and calculations as if there were no figures, no space. Only after such studies have been carried out should they be compared and summarized. I have no doubt that only in this way it will be possible to fully understand.

The remarks given below refer only to the numerical values ​​of the Modulor and do not concern geometry at all.

2. Initial numerical values

The basis of the Modulor, the original numerical value on the basis of which it is built, is the number C = 1.618 (exactly (√ 5/2) + ½). Squared, it gives 2.617924, or 2.618 in four digits - in other words, the same number C increased by one. By squaring (√ 5/2) + ½) we get the same value increased by one.

Arithmetic knows no other positive number that has this property. It is this property of the number C that underlies the Modulor. It is the basis of the entire grid.

3. Mesh C.

and the third number is equal to the sum of the previous two. For the development of the grid, we will establish the fourth number of the series - С С С. It, apparently, can be obtained by multiplying all three numbers of the series (1) by the value С:

FROM	C C	S S S
1,618	2,618	4,236

and the latter, of course, will be equal to the sum of the two previous ones.

The Modulor grid can get further and quite obvious development:

1) starting position........ 1

2) the main numerical value ........ 1.618

3) the sum of 1) and 2)........ 2.618

4) the sum of 2) and 3) ........ 4.236

5) the sum of 3) and 4)........ 6.854

6) the sum of 4) and 5)........ 11.090

4. Foundation

In any mesh or fabric, in addition to the weft, there must also be a warp. Modulor determines it by doubling the previous digits. Naturally, the new blue row will have the same properties as the red one. Each number in the series is equal to the sum of the two previous numbers:

1") home position........ 2

2") basic numerical value 2 1.618........ 3.236

3") sum of 1") and 2")........ 5.236

4") sum of 2") and 3")........ 8.472

5") sum of 3") and 4")........ 13.708

b1) the sum of 4") and 5") ........ 22.180

5. Cross-weave mesh

It remains to be seen how the weft and the base of the Modulor mesh are intertwined. The interlacing is quite satisfactory, since in the increasing series of numbers the numbers of both series are consistently repeated (see diagram).

Let us discard for a while the first members of the series, which are, as it were, the edge of the grid; but more on that below.

We see that there is an absolutely correct alternation of numbers in the red, blue, red rows. Note that the values ​​of the intervals between the numerical values ​​are shown by the numbers on the oblique straight lines. These numbers have interesting properties:

1. Each member of the red row determines the exact midpoint between two adjacent members of the blue row, of which one is smaller, the other is larger than it;

2. The interval between a member of the red row and two adjacent members of the blue row is constantly increasing in accordance with the numbers of the row 1 - 1.618 - 2.618 - 4.236, etc. ...

There is nothing mysterious about these properties; they are easy to explain: this is a direct consequence of the properties inherent in the number C (multiplied by 2).

6. Changing the direction of the grid

Let's return to the starting position in a series of numbers from 1 to C, equal to 1.618. Instead of going from left to right and forming numbers by adding them: 1 - C = 2.618, we can go to the left, forming numbers that add up to one to C; this will obviously be C - 1 = 0.618. From here we get three numbers of the series:

C-1 ..... 1 ..... C

Relevant

0,618 ..... 1 ..... 1,618

Taking into account the known properties of the number C, we can count on the fact that by multiplying the first term by C, we will get the second. And indeed, 0.618 1.618 = 0.999924, or practically 1 (since, strictly speaking)

0.618 = (√5/2) + ½).

Thus, a new numerical series is formed, going from right to left, in which each new term (on the left) is the difference of the two previous ones. The new grid of numbers will look like this:

1) starting position..... 1

2) the main numerical unit ..... 0.618

3) difference between 1) and 2)..... 0.382

4) difference between 2) and 3)..... 0.236

5) difference between 3) and 4)..... 0.146

6) difference between 4) and 5)..... 0.090

7) difference between 5) p 6)..... 0.056

7. Changing the directions of the warp when the grids are cross-woven

The numbers in the blue row are double the numbers in the red row; cross-linking is preserved.

The properties listed above have been preserved (see diagram).

8. Combination of ascending and descending series

Now it has become clear how the extreme indicators of the grids are combined (we will restrict ourselves only to numerical values ​​close to the boundary indicators): The conjugation of numerical values ​​is flawless. All patterns are preserved from start to finish; there was not even a trace of that “edge” from which we counted to the right and left.

These are the foundations of "the Modulor theory in its arithmetic expression."

If you want to see the “wrong side of the carpet”, there is nothing better to look for.

Let both the mathematician and the artist study its front side.

A complete understanding of the Modulor is achieved immediately by summarizing both sides.

P.S. In order not to confuse everything in the world, I have highlighted in the postscript a particular issue that could be called the relationship between the grid and the base.

To do this, consider the successive numbers of the red series adjacent to the conjugation of decreasing and increasing numbers of the series

1C	2C	3C	4C
2 - C	C– 1	1	C
0,382	0,618	1	1,618

Each number, of course, is the sum of the previous two; in addition, the sum of the extreme numbers (1C and 4C) is equal to 2, i.e., double 3C, i.e., the beginning of the blue row. Therefore, from the members of the red series alone, it is possible to form a blue series by adding two non-adjacent members of the series, abandoning, in other words, the basic rule set forth in paragraph 3. This is tantamount to skipping intermediate threads in the manufacture of a net. From these purely numerical examples, one can trace the foundations on which two adjacent squares are built.

Taking the square as a basis, we repeat the well-known construction of division in relation to the golden section three times. The sum of the sides of the upper and main squares is equal to twice the side of the middle square. This important property is obvious from the given drawing (Fig. 7).

All attempts at more simplified constructions (Palladio's construction, Maillard's solution) give only approximate results.

The numerical construction of Palladio (√ 5/2) + ½) + (√2 - 1) = 2.032 indicates an error of 1.6%;

The Maillard solution corresponds to the expression (√ 5/2) + (2/√5) = 0.9√5 = 2.0124 with an error of 0.6%. It is 2.5 times more accurate than Palladio's construction. However, the construction shown in Fig. 7.

ABCD is the original square. Using the classical construction, we find the square DEFG and GHJI, then the position of the point K.

We transfer the value of AB to IL by drawing the line AI and BL parallel to it. Obviously, KL= 2GH. If you start building with GH, you must:

1) determine DE, and then AB, building in the reverse order;

2) then find KL in the usual way.

The summation of KI and AB gives the desired solution.

From a letter from Jean Deyre (ASCORAL), State Administration for Economics.

1. On the basis of Modulor you will be able to develop a logarithmic system of measures.

2. This system would make it possible to simplify the numerical expressions of large and small quantities.

3. You can use the properties of logarithms for simplified calculations of areas and volumes.

4. However, it is necessary to check the limits of application of the additive properties of this system.

1. Ability to create a logarithmic system of measures based on Modulor

Ratio Ø = (1+√5) / 2 = 1.6178 ≈ 1.62

as the basis of the Fibonacci series can be taken to build a new logarithmic system that can compete with the natural and decimal system of logarithms.

Let's call them, with your permission, the golden logarithms (based on the golden ratio) or, even more simply, the logor. (S)*.

*From logarithme aural golden logarithm. (Note, ed.)

The golden logarithm of the number N is:

Фx \u003d N or 1.6178x \u003d N. Therefore, loghor. 1.62° or loghor. 1/=0

loghor. 1.62 = 1;

loghor. 1.62kv = 2, etc.

In order to keep to human scale, you accept an auxiliary additional value - 1.83 m, corresponding to the height of an athlete of 6 pounds.

Let's call this unit a megalanthrope, or megan for short (1 megan = 1.83 m).

We obtain the following conversion table of metric measures, which can be extrapolated if necessary. Let's convert to logos.

As a logarithmic unit, we take the golden logarithm of Ø megan = 1.62 megan. Let's call this unit almegan (from the algorithm). Here is a translation table:

2.96 meters = 1.62 meganes = 1 almegan;

0.70"=0.37"=2";

3.66" = 2" = 1.45".

(for the red row we get fractional almegans).

2. Dimensional quantities expressed in almegans (like all logarithmic quantities) are convenient for expressing very small and very large quantities.

They determine the required number of gradations (ascending and descending) of the main red row, separating the starting point, corresponding to the growth of the megalanthropus, from the desired value. Examples (regardless of the possibility of errors in the calculations):

1. Distance from Paris to Marseille

800,000 m = 800,000 meganes / 1.83 = about 28 almeganes.

2. Water drop diameter

5 mm = 0.005 meganes / 1.83 = about 13 almeganes.

3. Diameter of the Milky Way

5000 light years = 10 to 21 meters = 10 to 21 megans / 1.83 = about 100 almegans.

4. The length of the light wave in the void

0.0006 mm = 6 meters / 10 in 7 = 6 / 1.83 x 10 in 7 almegans = about 31 almegans.

Thus, all the largest and smallest; values ​​expressed in almegans give numerical values ​​corresponding to the human scale. Obviously, if we took the meter as the initial unit of length, we would find such a pattern. Whatever dimensions the accepted values ​​correspond to, the number of meters, expressed in logarithms, will correspond to the human scale). In the range between the numerical values ​​of the length of sound waves and the diameter of the Milky Way, only 131 divisions are placed on the Modulor scale.

3. Using Modulor Properties to Calculate Areas and Volumes

This is a simple use of the properties of logarithms.

For example, let's calculate the area of ​​​​a room in square meters and in square megans:

1 square megane \u003d 1.832 \u003d 3.35 m².

Room dimensions: 4.79 x 7.74 m or 2.62 x 4.24 megans

An arithmetic calculation determines the area of ​​\u200b\u200bthe room at 37 m² or 11 square meters. megans.

We use golden logarithms or logora:

loghor. 2.62 meganes = 2 almeganes;

loghor. 4.2 megans = 3 almegans.

Logor. area, expressed in square megans will be 2 + 3 = 5.

Extrapolating the conversion table, we get:

11 sq. meganes or 11 × 3.35 = 37 m², which confirms the result of the arithmetic calculation.

A detailed conversion table would make it possible to quickly determine the values ​​of fractional almeganes.

4. Extending Modulor's Additive Properties

Here we will talk about the most serious difficulties in using the Modulor as a universal system of measures.

The main property of any system of measures is the ability to add dimensional quantities.

Logarithmic systems, as a rule, do not have this property.

By this I mean that the logarithm of the sum of two numbers cannot be directly determined from the logarithms of these numbers. So, for example, in decimal:

However,

log(1000 + 10) = log 1010 = 3.0043.

We determine this result by the table of logarithms. There is no direct relationship between log 1010 (3.0043), log 10 (1) and log 1000 (3).

At the same time, the system of golden logarithms has a number of additive properties, since a number of numbers have a direct relationship between the logarithms of these numbers and the logarithm of their sum.

These are the main properties of Modulor related to the Fibonacci series:

Ø + Ø in n+1 = Ø in n+2

Thus, considering three successive terms of the red series, we see that the golden logarithm of the third term (which is the sum of the previous two) is in a simple relationship with the golden logarithms of the first two terms.

If n is the golden logarithm of the first, n + 1 is the logarithm of the second, then the golden logarithm of the sum is n + 2.

Thus, we can sum some quantities using the properties of golden logarithms. This is the main difficulty, because what has been said does not mean at all that these properties of the Modulor extend to all quantities.

So, for example, taking any two randomly chosen numbers, the golden logarithms of which are equal to (1.83 and 2.67), it is obviously impossible to determine the logarithms of their sum based on the values ​​of 1.83 and 2.67. If it were possible to prove such a possibility, then Modulor would win a complete victory and could become a universal harmonic system not only in essence, but also in practice.

This question is very important, and, in my opinion, mathematicians should study it.

Be that as it may, your discovery is remarkable. Whether it is fully or only partially additive, the Modulor is a tool that has been lacking for those involved in standardization, able to harmoniously combine precision and rigor with artistic qualities.

Jean Deir"

Letter from Doctor of Mathematics Andreas Speiser

Dear friend, thank you for your letter and especially for the wonderful book about Modulor. I read it with great pleasure and took it as a sign of respect for an artist passionate about mathematics. You are in great company, as all great artists have been under the spell of numbers. In your letter you ask: is it true that one can resort to the help of both geometry and numbers at the same time? I answer:

We have two possibilities for knowing the external world:

1. Numbers. With their help, we "know the outside world", i.e. the existence of many other people, order, proportions, beauty, etc.

2. Space. In it we see a lot of inanimate objects, devoid of life, beauty, but "occupying a place" (lying, standing, prostrate, etc...).

Now about Modulor. You know, of course, that Luca Pacioli wrote a wonderful treatise on divine proportion. In it, he talks about 13 wonderful properties of the golden ratio. He gave each of them a magnificent name and told us about the joys they brought to Leonardo da Vinci. Your merit lies in the fact that you discovered the fourteenth property.

You built two Fibonacci series, of which the second is a doubling of the first, and identified a pattern in the properties of four consecutive numbers in such a series. Taking, for example, the numbers 5, 8, 13, 21, we see that the sum of the first and last numbers, that is, 5 + 21, is equal to twice the third number; 5+21=26. At the same time, the difference between the fourth and first numbers is equal to twice the second 21 - 5 = 16 = 2 8.

I want to present this regularity in a general form, understandable for any student. Denote by letters a, b, c, d four consecutive numbers, with c = a + b and d = a + 2b, whence a + d = 2a + 2b = 2c and, finally, d - 2b.

This explains the connection between your red and blue rows.

Jean Deir's letter is also true, but I must say that logarithms are no longer used today. Now all calculations are done on calculating machines (many times faster and more accurate). I understand that you wanted to have a system of measures that is easy to apply in architecture, and that you need whole numbers to achieve harmony. Therefore, it seems to me that your system of measures is really acceptable for artists. Ultimately, when it comes to the worker, you will have to give him all the dimensions in meters, which, by the way, will not present any difficulties. To do this, you just need to multiply your numbers by your own unit, expressed in meters. As for interplanetary distances, I am skeptical about them. For many centuries they have been trying to establish these laws, this was done in their time by Kepler and Titius, who determined some of them; Professor Weizsäcker at Göttingen is currently working hard on these questions. I don't really believe that this riddle can be solved using the golden ratio. Accept, dear friend, the assurances of my best feelings.

A. Speizer.

Our discussion was conducted at a high scientific level, at a very high level.

However, the word is up to those who used the Modulor... After all, there are no trifles at all - neither in painting, nor in architecture, nor in life! Alfred Neumann acknowledges the significance of the Ø relation. A lover of all kinds of calculation tables, numbers, numerical similarities and combinations, he compiled many tables based on the ratio Ø.

These tables made it possible to establish a number of numerical values. So, for example, 0.462 m is close to the value of the Attic cubit, equal to 0.46 m; with the help of the ratio of the golden section, the cubit is converted into the metric system, which explains the origin of the exact metric dimensions in the Parthenon, the height of the columns of which is 10 m. This was established by me. The Egyptian (royal) cubit, equal to 0.524, according to the Neumann table is 0.5236 m (Fig. 8).

Further, Neumann writes: “In order to create a correct system of measures and proportioning, it is necessary to combine the “geometric” unit of measures with the “anthropometric”. The meter is still the basis of scientific measurements and, consequently, of technical civilization. Curiously, the meter is also an "anthropometric" measure. I established that there must be a commonality between the meter, which is the earth's measure of length, and human dimensions. Many criticize the use of the meter as the basis of a system of measures, because they consider it not an anthropometric measure, but a scientific abstraction. Such an opinion is unfounded. The meter is an updated form of the old human measurements. The meter is a double cubit, which was only later divided into three feet, which is still the unit of the English system of measures "... (Yard = 3 feet).

The oldest unit of length known today. is a double cubit of the Babylonian king Gudea; it was installed in the 22nd century BC. e. and is equal to 990–996 mm, i.e., about a meter.

The connection between the measures of time and space was known in the days of ancient civilizations. Measures of weight of the past corresponded to about a kilogram. In ancient Greece, when assigning the diameter of columns, a module close to a meter was often used, for example, in Tezeion in Athens 1.004 m, in the Aegina temple 1.01 m ...

In our time, the English Standards Institute approved the module value of 101.6 mm; in America, the module is set to 10.16 cm.

From here, Neumann concludes: “What we have said confirms the mandatory need to combine the decimal system m with the ratio of the golden section “Ф”. This system can be called "mF" - the system "Em-fi" ... "

Wonderful, I welcome such touching agreement and even closer union. However, I draw your attention to the stumbling block - the American module is 10.16 cm. This value is in the red row: 10.2. But there is a whole gulf between building the entire human environment on an infinite addition of 10 cm (or 10.16) and building on the basis of Modulor.

Neumann recognizes the Modulor as noteworthy, despite the fact that it is based on the "arbitrary" growth (in this he is right!) Of a person of 1 m 83 cm; he is delighted that he was able to establish that the table of values ​​of mФ includes the Modulor series with minor deviations; in this he sees confirmation that "Le Corbusier is not devoid of intuition."

A mechanical engineer from Lille expressed the desire to link the Modulor with the "Renard series" used in mechanics. Here is a letter addressed by him to my deputy Vozhensky, which is a response to a report made by the latter in Lille.

Monsieur, I regret that I was unable to attend the report you gave on January 18 in Lille. I met him through the text and was struck by the image under the name Modulor.

You may say that remarks made by a mechanic are unacceptable to an architect. But why don't you take instead of the exact value of the ratio of the golden ratio

1,618 = 1 / 0,618

close to this value of the Renard series

The relative error, defined by the difference between 1.618 and 1.585, is 2%. Is it essential from the point of view of harmonious proportions? The rows take the following form:

(The decimal places don't matter: you can therefore limit yourself to the second line). The height of a person is determined at 1.80 m, the height of a chair is 0.45, the table is 0.71, the door is 2.20, the low chair is 0.36 m, the size of the brick is 11 x 22 cm .., the facing tile is 11 cm. ..

The red and blue rows have 10 two- and three-digit numbers, which exactly corresponds to the Renard series R - 20, on which standardization in mechanical engineering is based.

Wouldn't this make the job easier, given that architects use prefabricated materials in construction? When in your report you compared the dimensions of the Modulor with the feet and inches of our English friends, I thought about the possibility of even greater mechanical simplification by adding ten intermediate numbers to the R - 10 series:

To make it easier to remember the table, we start from a person's height of 1,800 mm, and then multiply or divide this value by 2, conditionally retaining only the first two characters:

or to the major third according to the tempered scale 1.2589 = 10√10

It is possible that I am knocking on an open door - but I would be happy to know why (I think) you are neglecting this door.

A. Martineau-Lagarde»

Similar proposals - to round up the numerical values ​​​​of the Modulor and bring them into line with other series - have already been made. I believe that the Modulor is a tool that gives confidence in the idea and its development ... What is true today will remain true in six months, in six years and in six days, whether it is made on the drawings of the same or different designers in different workshops of any countries.

The intervals between Modulor values ​​make it possible to nuance the time signatures as desired, similar to the way vibrato on a violin complements the tone with higher and lower tones, ensuring the perception of the correct tone. Of course, there is a lot to think about here, so readers can agree or disagree with it and argue for good.

Playing with numbers can get you far. Here is a short letter sent to Labart on the subject of space exploration (left unanswered).

Two months ago I sent the editors of your Constellation magazine my book Modulor.

Our troubled time, of course, is not very conducive to such activities.

In the eight years since its invention in 1948, I have not once attempted to make a fuss about the Modulor. But your space exploration, which amazed everyone in the Nicolas Vedres film, thrilled me, and here's why.

There are (approximately) 270 intervals between 15/1000 (fifteen thousandths) of a millimeter and the circumference of the earth in Modulor. Therefore, the ordinal numbers in the series are:

No. 1 = 15 thousandths of a millimeter;

No. 270 = 40,000 kilometers,

and number 300 will already be a cosmic value.

Based on this data, it is already possible to draw up schedules, calculate time, solve supply issues, etc., etc. The Earth-Moon distance is (approximately) the number of No. 285 Modulor (a) ~ + No. 41 (b) + No. 9(c)

This means that the number 285 defines colossal distances.

The number 41 corresponds to the order of distances from a meter to a kilometer; number number 9 - leads us to microscopic quantities (the numbering of the numbers of the series given here is arbitrary). They could be written:

MOD 285. 41. 9., which makes it possible to make accurate calculations.

I already thought about this before, but now I used the MOU index for the first time. It will be necessary to think more.

Modulor covers dimensional quantities from infinitesimal to infinitely large. It is, in all its meanings, a return series.

Over time, it will be possible to record dimensions as follows MOD 47.3, etc., etc. .., to abolish feet, inches and meters, extending the decimal system of measures to the whole world.

The research and creation of Modulor did not set itself goals of such a cosmic scale.

Concluding this section, I will say about one more type of intervals. They were reported to me by the Parisian architect M. Rothier, who claims that such intervals are very suitable for determining the modular dimensions of areas and volumes in housing construction. As an architect, he takes into account the difference in the thickness of materials, in the height of people with a height of 1.73, 1.83 and 1.93 m, which leads him to introduce intermediate divisions that divide the intervals of the Modulor scale in half. These considerations of the architect-practitioner are quite fair. In this matter, the situation is the same as in the question of the use of graphic methods of construction in painting, where it is first necessary to determine which part of the picture should be corrected by this method. And in architecture, it should be established which structural elements are subject to adjustment by the method of graphic construction or, if necessary, taking into account Modulor gradations.

The challenge is to take into account what we see. We perceive lengths, areas and volumes with our eyesight, and it is necessary to finely proportion them. What should be given the main attention? The space of the room or the thickness of the partition?

Which window sizes are most important: the size of the glass or the entire window opening? This must be established in each individual case.

So: let's carefully look around us, measure, let there be an interest in proportioning; it is not given immediately. We have to admit the bitter truth that we build spontaneously, regardless of the system of well-balanced and coordinated proportions. Engineers have taken a step forward in this direction, guided in standardization by the requirements of efficiency. In an effort to build bridges across the seas and oceans, and considering the fact that industrial products should be applied everywhere, they developed standards. Their standardization is characterized by a certain simplification and does not provide complete creative freedom. However, the progress of mankind and the established rules should not exclude or even limit the creative imagination. We began to look around with open eyes and study our home.

Often, studying the dwelling of the past, once created by masons, carpenters, plasterers, we find answers to the questions that arise: this is explained by shop rules passed down from generation to generation; they are, however, accumulated, distorted, and over time saturated with all sorts of mysterious secrets. All this wisdom comes to us in a simplified and purely "applied" form.

From time to time (and this is also reflected in the letters we received), heralds appear, extolling the theories of a thousand years ago. Naturally, these theories do not meet and are not connected with the obvious demands of modernity. Such heralds revel in themselves and hint at their learning and knowledge. Sometimes... they begin to "rite sacred" and broadcast in a mysterious language. We are told that 8 by 108 = 864: that 108 and 7 signify the number 108 and the holy spirit; and that 216 is twice 108...

Personally, they occupy me a little now, when I am doing research in the field of numbers: but I know well that twice 54 = 108; eight times 108 = 864 etc.

I have always considered and believe that the dimensional value of 108 centimeters is not at all equivalent to the number 108, the meaning and purpose of which are unknown to me. If I convert 108 cm to feet, I get 26 inches and the number 26 ceases to be a sacred number ascending to the number 108, and so on ... The number 108 in 1945 served as the basis for the first Modulor, built on a human height of 1 m 75 cm. The coincidence of numbers does not mean anything yet. I know that there is a metaphysics associated with thousands of symbols to which a thousand and one meanings are attributed. But I'm just a builder. I consider it necessary to strongly reaffirm the importance of the thought:

“Modulor is a tool that gives confidence when making decisions. What is true today will be true in six months, in six years and in six days, in the drawings of the same or different designers in different workshops of any countries.

What's right is right! We are dealing with the realm of numbers. Do you want to “round up” and agree to compromises? In whose name? In the name of what? The only way to solve is the truth.

Practical application of Modulor

The Parisian architect Andre Siv writes: “I give you my opinion, a person who uses Modulor.

First, it is a working tool. Each of my assistants on the drafting table must have both numerical series of the Modulor attached (I myself know them by heart).

Of course, using the Modulor, we still do not solve artistic problems, but it automatically protects us in the process of working from the “assignment” of approximate proportions, from errors in the architectural composition, in details and in general relationships. If Modulor were put as the basis for standardizing building details, this would eliminate the randomness of proportions and the randomness of scales. They would finally become acceptable.

I consider it necessary to oblige the use of Modulor in school construction. This would contribute to the development in children of a sense of artistic harmony, which is necessary in the future, when architecture becomes a true expression of culture.”

To the letter, he attached a layout plan for the city of Meudon-le-Villages as an example of the use of Modulor in solving urban problems.

Parisian architect Marcel Roux, consultant to the Ministry of Reconstruction and Urban Planning, states:

“I consider it necessary to inform you that after two years of using Modulor, I now force everyone who works with me to use the ratios and proportions you proposed.

Although a number of existing rules and regulations prescribe, unfortunately, some of the indicators you reject, it is always possible, with a certain effort and ingenuity, to observe the wonderful proportions recommended by you.

I am convinced that with the universal use of Modulor, architecture would receive an extremely interesting development.

Van der Mesren designed a detached house of just 167 m3, consisting of five living rooms, a kitchen, a bathroom, a garage and a small shop. According to him, thanks to the consistent use of Modulor, all difficulties were overcome.

Riboulet, Turnauer and Vere sent a design for a typical student dormitory room on a university campus in Fez, designed by architect Ecochard (Morocco).

Kandylis developed a project for a residential building in Casablanca, adapted to the conditions of the Moroccan climate. The use of Modulor allowed him to systematize and coordinate the layout of all living quarters. He wrote:

“You wrote somewhere that whoever once uses Modulor, this well-tuned tool, will no longer be able to part with it. This is absolutely fair.

Woods and I have been working in Africa for two years now. Our activities are very diverse: we design, participate in competitions, build and do research. We are accustomed to the Modulor, it has become an indispensable working tool for us.

Before moving on to this, we experienced indecision and doubt, made erroneous decisions.

Over time, we began to work clearly and confidently. Our thoughts are given full expression in ordinary drawings; each assigned size corresponds exactly to a well-defined purpose, excluding any accident and exaggeration. Everything is subject to harmony and corresponds to the human scale...

We were guided by Modulor when assigning the dimensions of spans of areas and volumes of premises, as well as when designing internal equipment and openings for various purposes: we received accurate and interconnected dimensions.

I quote the statement of the Parisian architect father and son Auger, with whom I developed the project of the gliding club in Lorraine. “Thanks to Modulor,” says the son, “we can work quietly, each in his own office and meet only occasionally. We used one of the typical coatings from Jean Prouvé's catalogue. During the development of the working draft, we did not encounter any difficulties; thanks to Modulor, which was the basis of the project, a complete agreement was established between us, since we all used the same well-tuned tool.

Our friends from Baranquilla (Colombia) on the Caribbean coast are developing one of the most pressing issues - "Housing Units". They adopted our terminology with a slight change, denoting by the term "living volume" what we called "volumetric living cell". With the help of Modulor, they created living cells that can be used in a wide variety of conditions. Just as they did in Marseilles, they built an 18-story concrete shelving unit with 100 cells containing a corresponding number of apartments. Having made such a decision, it remains only to achieve its implementation, select materials, develop a technology for the production of work, types of apartments, etc.

They accompanied their project with the following statement: “It is quite obvious that the implementation of such a project requires a unified system of measures, which would be subject to all linear dimensions and volumes, linked to each other and to the growth of a person. Modulor, which combines both metric and foot-inch units, allows for the factory production of relatively cheap building elements in a wide variety of shapes, proportions and solutions.

Prefabrication of modular building elements will make housing accessible to the public and lead to architectural solutions designed for mass and widespread use while maintaining the originality and characteristics inherent in every people and every region.

My workshop assistant at st. Sevres André Vozhensky is now completing the construction of his house, in the design of which he widely used Modulor. He's writing:

“When designing a house, I consistently applied Modulor not only when developing plans, sections, but also when developing working drawings of individual details, for example, when determining the thickness of some elements (the crown of a building, stairs, etc.). I did this even in cases where the thickness is not perceived visually. I have also used Modulor to design furniture and interior fittings such as custom designed hardware and kitchen electrical equipment.

The use of Modulor has never hampered or limited me in my work. I used it, as a rule, at the end of the work for clarification, or rather, for the final adjustment of the accepted sizes and proportions.

The plan was developed on the basis of the grid shown in fig. 13 left. Shown on the right is a solution based on this ground floor plan grid. It should be noted that this grid was not chosen arbitrarily before the start of the design. It appeared as a result of the work as a result of searching for the internal organization of the building, determining the required dimensions and clarifying the layout. Only after that the grid was gradually defined. The choice of the grid was the final push that allowed us to refine the layout decision and set the final dimensions.

Work on the plan was never interrupted from work on sections and facades. The use of a grid does not mean at all that Modulor is applicable only to two-dimensional projections on a plane (plans, sections, facades). On the contrary, its application is associated with the search for three-dimensional volumetric solutions, the orthogonal projections of which are plans, sections, facades, and, consequently, the grid itself.

Architecture is perceived by the viewer in space, he determines the dimensions of the structure and its parts best when moving and changing points of observation, when it seems to unfold in front of him and around him. On fig. 14 shows the east facade; it shows the division of the building in height: the height of the rooms 2.26 is divided by 86 and 140. It should be noted that almost all dimensions without any exaggeration correspond to the values ​​of the blue row.

At the end of the work, we came to the conclusion that such dimensions ensure the unity of the composition, apparently, more than with a combination of the numerical values ​​of both series.

Disagreements

Around 1940, the French Society for Standardization AFNOR (ARCOC) was formed to study the problems facing modern industry. Leading civil engineers, architects, etc. were invited to participate in the work of this organization.

I wasn't invited. For five years, I have not had a chance to build a single cubic centimeter of buildings or build even a square centimeter of urban territory. In 1942, on my own initiative, I founded ASCORAL and headed the work of its commissions, some of which, by the time of liberation, had prepared a number of useful books for publication, including Thoughts on Urban Development, Three Forms of Settlement, and Modulor. ASCORAL is to publish the following works: “To be able to live”, “Urban planning and medicine”. During this time, I personally published books: "At the Crossroads", "The Fate of Paris", "Athenian Charter" and "Conversations with Students of Architecture Schools".

Back in the twenties, twelve articles were published under my signature in Esprit Nouveau magazine. The article "Typical Dwelling" caused a wave of indignation - a residential building in it was considered as a "machine for housing." To implement my ideas, I appealed to industry. In another article, using the example of the Parthenon and the car, I “demonstrated the virtues of “typing”, showing its effectiveness, its essence, standardization as a prerequisite for the creation of artistic things. In an article on graphic construction methods, I highlighted the importance of proportions in architectural structures.

In 1925, in the Esprit Nouveau pavilion at the International Exhibition of Decorative Arts in Paris, there was an appeal to industrialists with an appeal to "take construction into their own hands."

April 1, 1953 I arrived in London to receive; Gold medal awarded to me by the Queen of England for my work in the field of architecture. A student handed me a rotator-printed Modulor Sossti questionnaire on the establishment of standard foot-inch sizes. A few months later, this issue was raised again in the pages of Prefabrication magazine in connection with a vote taken at the congress of the International Union of Architects in Lisbon and his appeal to UNESCO, which confirmed the proposal to establish a modular system in construction. It was assumed that the main module will be equal to 4 inches, i.e. 10 centimeters, which should be used without limitation of dimensional values.

I'm not going to enter into a discussion about this. It should be noted, however, the progressive desire to establish methods of standardization and the need for international consistency. However, under the pretext of urgency, a poor standardization system is proposed that excludes the manifestation of creative imagination. The task is precisely to establish and approve a carefully thought out, justified and universally applied system of indicators for both the technical and spiritual fields of human activity. Such issues cannot be resolved in a hurry, their discussion cannot be limited by referring to any international organizations.

I will add that the initiators of this event in vain associated it with the term "Modulor". The name of their organization is Modular Society, very close to Modulor. I have always abhorred all confusion and hated ambiguity.

The modern world is in the grip of conditional and arbitrary regulations, adopted on the basis of compromises and ill-founded indicators that simply prevent “doing well”. All this is familiar to me, because in Marseilles I built the Residential Unit, regardless of such regulation. He boldly walked, despite stormy protests, their violation and all sorts of retreats.

This concludes the first part of the book "Modulor-2". In it, the floor was given to those who used the Modulor.

In the second part, I will try, without going into deep mathematical reasoning, to show the main dignity and vitality of the Modulor as a general working tool applicable in architecture, as well as in mechanics, opening up opportunities for the manifestation of creative imagination.

Reflections. No spells

At the ninth exhibition of the "Triennale" in Milan, 27, 28 and 29 September 1951 were devoted to the "divine proportion"; these days the first international meeting was held on the topic of proportions in art. The discussion that took place, figuratively speaking, resembled a railway station, from which two tracks branch off; one of them leads to boundless expanses, and the second leads to a dead end.

In his speech at the meeting, Professor Witkover (London) emphasized that one of the bases of proportioning is the square. Many medieval artists used the double square. In Europe, up to our time, they adhere to the Pythagorean and Platonic traditions. This tradition is based on two principles: firstly, a system of numerical ratios (1st, 2nd, 3rd and 4th harmonic intervals of the Greek musical scale); secondly, regular geometric figures: an equilateral triangle, a rectangle, an isosceles triangle, a square, a pentagon ...

In our time of non-Euclidean geometry, the concepts of time and space inevitably differ from the concepts of past centuries...

Perhaps the discussion at the congress will help to look at the problem of interest to us from a new point of view?

Professor Siegfried Giedion (Zurich-Boston) said:

“... The view of the 19th century: the particular dominates the whole (Nietzsche, 1884).

The golden ratio runs through the entire history of mankind (remember the prehistoric cave painting). The golden ratio has been used in different eras, only the methods of its application have changed.

In contrast to the static proportioning of past eras, in our time there is a tendency towards a more dynamic proportioning. An example is the difference in the image of a person in Vitruvius - "Vitruvian man" and in Le Corbusier - "a man with a raised hand" ...

The experience of the United States of America is a warning that general chaos may ensue if our era is unable to find a specific form of standardization process in which all the various elements would correspond to human scale and provide any combination of them with each other.

Matila Ghika talked about the symmetry of the pentagon; about the pentagon and dodecagon and their articulation in the golden ratio; about an angle of 120° and its multiple angles of 60° and 90° inherent in crystals... “6000 kinds of snowflakes have a hexagonal shape. Pentagon, pentagonal corollas of flowers, golden ratio, pentagonal symmetry, lily flower and asphodel lilies... Geometric series of numbers, Fibonacci series... Fibonacci series in botany... The intuition of Pythagoras, Alaton and Pacioli leads to the same results. The provisions of Einstein, de Broglie, Leonardo da Vinci ...

Some incoherent set of big words and names! The fact that there are many scientists who love abstruse terminology (spell words) in the world is as natural as the fact that masons, concrete workers and locksmiths build houses under the guidance of an architect.

Dr. Hans Keyser's speech was devoted to his theory of sound "Harmony". The Milan Triennale exhibition in 1954, dedicated to the "divine proportion", was a celebration of the golden ratio - the ancient path of humanity indicated by Pythagoras.

In the Finnish magazine "Arkitekti Arkitekten" No. 1 for 1954, a message was published on cubic residential cells.

I do not know Finnish, but the drawings in the article are very convincing. We are talking about residential three-dimensional elements built on a specific module, from which various combinations of apartments can be created. The volumetric unit is a cube with sides of about 2.50 m, which allows creating rooms of sufficient size to accommodate the necessary furnishings: a bed, a table, kitchen equipment, etc.

On fig. 15 shows a system of sequential division of the cube into eight ever-decreasing volumes (or, conversely, the addition of a large cube from smaller ones). This corresponds to the mathematical expression 8n, where "n", the exponent, can be a positive number (with a "+" sign) or a negative number (with a "-" sign). Such a simple method of division can be used as the basis for a system of measures in architecture, provided that 1 cm is taken as the main size and the following series is 2, 4, 8, 16, 32, 64, etc., which could be taken as the initial for any mathematical and technical research. The author of this project was the Finnish architect Aulis Blomstedt, who completed it together with Paul Bernui-Vestere and Keio Peteia. On fig. 16 shows some of the possible combinations. In 1947 -1948. these examples were supplemented with new ones, with the explanatory text saying that:

“The economic advantages of mass production are clear. However, there is an impression that there is a clear contradiction between the prefabrication of houses and the need for an infinite variety of types of residential buildings. It is impossible (and would be disastrous) to standardize human dwellings.

On the other hand, batch production of only unchanging building elements is advantageous.

How, under these conditions, to apply mass production in the construction of dwellings?

Just as in arithmetic they look for a common denominator of two numbers, one must find a common denominator for mass production and type of dwelling. This denominator exists for the simple reason that production is organized by man himself. This study shows that the theoretical foundations of industrial production and housing construction can be successfully combined in construction practice. The geometric and constructive system of "rigid volumetric elements" is applicable in factory production and is able to satisfy all housing requirements. There has been a lot of controversy around the topic of "flexible standardization", but in order to provide it with freedom and flexibility, such a standardization system must have wide adaptability, while maintaining its immutability in accordance with its name.

Aulis Blomstedt.

The same journal soon published an article about the Rock-Rob design proposals (a patent for a cubic element with a size of 226 × 226 × 226 was obtained by us in Paris on December 15, 1950 (Fig. 17). The patent received did not cover equipment and environments that have long been studied and partially solved; he referred to a purely constructive problem: a material was installed (bent profiles made of sheet steel) that provides the most favorable values ​​​​of the moments of inertia during installation (corners, T-shaped and cruciform profiles) with minimal cross-sectional areas; thanks to to this, the actions of compressive, tensile and bending forces seem to be combined, which is facilitated by the use of the most advanced articulation method - by electric welding.

Everything as a whole forms a “voluminous living cell”. Construction according to such a system was carried out on the Cote d'Azur (Fig. 18). The accepted module coincides with the original size of the Modulor and corresponds to the height of "a person with a raised hand - 226 cm".

For the first time, volumetric cells were used in 1950 during the construction of a residential building in Marseilles, where beams from bent sheet steel designed by Jean Prouvé were used; they are extremely light, transportable and easy to install.

“From the city to the bottle; from the bottle to the city"

We are talking about a large report of September 28, 1951, dedicated to the Modulor at the Triennale of Milan, in connection with a meeting on "proportions in art".

After explaining and showing detailed graphic materials about 226x226x226 volumetric cells, I felt it necessary to state the following: “For now, this is only work on a modular structure, a prerequisite for creating a living cell. However, a residential zone can be placed if desired in a Garden City system, where the transport network and management can be freely decided, obeying any rule, not just the Modulor.

Speaking about Chandigarh, I drew a plan for a residential area - a part of an urban area measuring 800 × 1200 meters, designed to accommodate 5, 10 or 20 thousand people, depending on the nature of the development provided for by the task. Within the area of ​​the quarter-sector, I outlined the places reserved for the placement of houses. Architects and developers, entrepreneurs, house-building factories thus got the opportunity to dispose within the allotted areas at their discretion, resorting to Modulor or not. Other activities outside the modular system were envisaged. These include, for example, a network of access roads leading from the main highways to each house in the sector, thus forming a single system in the structure of the entire city. The street network includes streets of seven types, supplemented then by the eighth type. This differentiated road network, including all roads, from out-of-town access roads to each dwelling, I called the law 7v - seven types of roads (but in fact there were eight). The system of the transport network is built according to the biological principle, obeying the terrain, taking into account the speed of movement.

The construction of the secondary structural urban element of Chandigarh - the "sector" in all its internal and peripheral divisions is subject to irrational numerical values, for example, the ratio 0; the construction is based on the simplest, accessible to all, arithmetic relations. An arithmetic series of 1200 m - 800 m - 600 m - 400 m - 200 m is adopted, corresponding to the simple ratios 6-4-3-2-1.

Continuing the report, I digressed from Chandigarh and moved on to the topic of a dwelling, the external dimensions of which (dimensions) do not have to be subject to the Modulor: I began to talk about "Dwelling units of appropriate dimensions." In this case, the dimensions (shell) of the building are derived from the terms of the units (we were talking about a new residential building in Nantes-Réze). I wanted to show that the structural basis of the building, the living cell, is strictly subject to modular relationships, while the overall dimensions of the building as a whole are determined by the number of living cells included in its composition and built-in common utility and service rooms.

They are also a function of the adopted system of internal vertical and horizontal communications, etc. The combination of all these very specific elements determines the final architectural appearance of the building - an erected volume illuminated by the rays of the sun!

All talk of modulation is therefore of secondary importance. Only the search for a geometric construction will reveal the richness or squalor of the accepted ratios, it determines the plasticity and poetry of architecture ... The sculptural expressiveness of a structure does not depend on the constructive solution and the internal equipment of the building. The main thing will be the expressiveness of the division of the main volume. It is important to decide the silhouette perceived from the left, right, top and bottom. Only then should .... turn to “graphic methods of construction, which are then capable or powerless of giving poetry, lyricism to the work. All this is very difficult to explain and even more difficult to do.”

I have titled this section "From the city to the bottle and from the bottle to the city" in order to establish the following: it is quite possible for a very perfect home for the family to exist independently (this is the "bottle"), and the city as a whole does not depend on the decision of the "bottle". ”, since it is not associated with a number of specific urban planning factors. This had to be shown to make it clear that there was no need to modulate everything*.

* The principle of constructive construction of Le Corbusier's "Housing Unit" is based on the use of a monolithic reinforced concrete whatnot, into the cells of which volumetric elements of apartments are inserted, figuratively called "bottles" by Le Corbusier, by analogy with racks for storing wine in bottles. (Note, trans.).

I give the first example of work that came to hand in a workshop on the street. Sèvres, 35.

1. Designers Samper, Perez and Doshi. Street V2 "Capitol" in Chandigarh.

I thought that on one side of the street there should be a shopping arcade two kilometers long. The height of the arcade was taken as 775 cm divided into three parts of 226 cm or two parts of 366 cm with remnants, into two uneven parts 478 + 295 or without division for the entire height of 775 cm. Steps between the supports of the arcade could be taken at will 7 m 75 cm, 4 m 78 cm, 2 m 95 cm, 3 m 66 cm, 5 m 92 cm, etc., without giving preference to any of these sizes. The owners of future stores had a wide choice of room sizes. On fig. 20 shows a fragment of the facade of typical residential buildings with a total length of two kilometers and a cross section along the shopping arcade 775 cm high.

2. Temporary administrative buildings of the city. Subsequently, when institutions leave these houses, caravanserais (hotels for visitors) will be arranged in them.

Pillar verandas 3.66 m deep on the sunny side provided deep shade. The height of the pillars is 226 cm + 295 cm = 521 cm.

Distances between interior partitions can be 226, 295, 366, 525 cm, etc. ...

When designing the buildings of the Supreme Court and the Secretariat (Seven Ministries) in Chandigarh, climatic conditions were taken into account first of all. The buildings are set perpendicular to the direction of the prevailing winter and summer winds. On the sunny side, sun protection devices are provided to shade the windows of the working rooms. Climate grid, developed in the workshop on the street. Sevres, 35, made it possible to correctly take into account the direction of the winds when locating buildings, create the necessary shading and regulate the temperature regime of each room.

In the fifth issue of Jay Dew's London-based Architects Year Book, published in late 1953, Prof. Rudolph Witkover. The collection contains illustrations: five regular polyhedra of Plato; Euclid's pentagon; building on the basis of the Milan Cathedral triangle, 1391; proof of the Pythagorean theorem, taken from the book of Vitruvius, edition of 1521; doubling and dividing the square in half according to the same book of Vitruvius; "Durer's Folding Compasses"...

What are these illustrations? They relate to studies in the field of proportions from the times of Antiquity and the Renaissance. They contain an abyss of wise thoughts. But they have nothing to do with the human figure (pentagon, square, triangle). They can lead to an unbridled play of fantasy and imagination (which threatens with dangerous mistakes). But already in the eras of Pythagoras, Plato, Vitruvius, Dürer, anthropocentric dimensions served as a powerful counterweight: foot, palm tree, cubit, etc.. ., as well as the talent of the authors themselves.

But gradually interest in questions of proportions came to naught.

The final words of the mentioned article by Rudolf Witkover are dedicated to Modulor:

“Many signs speak of the imminent end of an era that abandoned the “system of proportions”. The assertion that the architect expresses the era in which he lives in his work has become a hackneyed truth.

Even if the architect is hostile to this civilization, he still expresses his attitude towards it and its inherent features.

We know that at the end of the last and at the beginning of the present century, non-Euclidean geometry was put as the basis for the ideas about the universe. The gap in the past was as deep, if not deeper, than the gap between the scholastic conception of the universe of the Middle Ages and that of such mathematicians of the Euclidean school as Leonardo, Copernicus, and Newton.

What impact does and what else will it have on the role of proportions in art, the replacement of ideas about the absolute values ​​of time and space with new ideas about the variability of the "space-time" relationship. Modulor Le Corbusier gives us an approach to solving this issue. If you approach it from a historical point of view, we can say that this is a fascinating attempt to harmonize tradition with modern non-Euclidean ideas. The very fact that Le Corbusier took man in his environment as the basis of his system, and not any general provisions, indicates that he decided to move from absolute norms to relative norms. Having taken this position, he seeks to consolidate the achieved result. The old systems of proportions were, one might say, unambiguous and considered them only as a consistent system, expressed by geometric constructions and numerical relations. Modulor Le Corbusier treats them differently. Its main elements are extremely simple: a square, a doubled square, and their division in the extreme and average ratio. These elements are included in a system of geometric and numerical relations: the basic principle of symmetry was expressed in two irrational numbers that differ from each other, located in relation to the golden section. No matter how you feel about Modulor, this is, of course, the first logical generalized system created since the fall of the old systems; it also reflects the modern way of thinking. It is evidence of an inextricable link with inherited cultural values. Like the proportions of medieval planimetry, the arithmetic proportions in the music of the Renaissance, Le Corbusier's double system of irrational quantities is built on ideas that were considered by the adherents of the Pythagorean-Platonic school, inherent in Western civilization.

When, after twelve years of practical activity, you are convinced that everywhere, in all projects or planning work, there is a single, as it were, key, modular unit (I mean the dimensions of 226x226x226), this gives the right to assert the existence of a “volumetric element that meets a person” that can bring in architecture, an order that will help to rework the norms and will contribute to the solution of the most difficult task of modern architecture to create dwellings for people of the era of machine technology.

Rue Sevres, 35

1. The difference between the concepts:

a) arithmetic

b) structural (Modulor)

c) geometric (graphic construction methods)

a) arithmetic. Arithmetic concepts are easy to understand. Two plus two equals four. They are tangible, understandable (I'm not saying that they are obvious).

b) Structural. Dictionary "Larousse" explains: connection, mutual placement of parts of any work, product; location of body parts.

c) Geometric. A phenomenon that is best perceived visually, including rules that in themselves can become the basis of harmony and poetry.

Take a look at the layout of Chandigarh: it refers to the first stage of work, when the city was designed for 150,000 inhabitants.

The city consists of 17 sectors 800x1200 m in size (Fig. 24, left). The invention of the "sector" dates back to the work on the plan of the city of Bogota in 1950 and on the master plan of Buenos Aires in 1929-1939.

The territory of 800 x 1200 m is designed for the resettlement of 5, 10, 15, 20 thousand, etc. people, depending on the density provided for by the task. The territory is easily divided into sections that are in simple arithmetic relationships. The division scheme allows solving the issues of organizing high-speed traffic along the contour of each sector with stops every 400 meters. Stops are located not in the corners of the sectors, but in places that are most convenient for servicing the corresponding sections. Arithmetic led to the creation of the most reasonable and practical layout. A distance of 400 m is not visually perceived; with our consciousness, we reach distances of 400 and 200 m, and multiples of 800, 1200 m, etc., which is automatically connected with the idea of ​​time.

Arithmetic is also the basis of the built plan of the Capitol in Chandigarh. The Chandigarh Capitol is the new administrative center; it is located in a newly created park (to protect against traffic noise, the streets are laid in trenches). The Capitol complex includes the buildings of parliament, ministries, the Palace of Justice and the Palace of the Governor. This park (as, indeed, the whole city) is laid out right among arable land. For reasons of expediency and beauty, it was given an easily perceived, clear, geometrically correct rectangular shape. Using appropriate artistic techniques, the architects managed to make accessible to visual perception what could only be comprehended by consciousness: two squares with sides of 800 m were used as the basis for constructing the layout. A smaller square with sides of 400 m is inscribed in the left square. From the right square with sides of 800 meters they refused, because for the most part it turned out to be within the eroded territories; instead, a second square was created with sides of 400 m, located adjacent to the first one (Fig. 26).

The terrain is a flat plain; from the north, the landscape is closed by a picturesque chain of the Himalayan mountains. Any, even the smallest building makes an amazing impression against the background of this landscape. The complex of palaces is a group of contrasting high and low volumes. In order to enhance the artistic effect, it was decided to emphasize simple arithmetic relations by placing obelisks at characteristic points.

The first group of obelisks will anchor a square of 800 x 800 m; the second - squares 400 x 400 m. The first will be located in an open area; the second, located near the buildings, will participate in their architectural composition.

When deciding on the location of the complex of palaces, the problem of its visual perception was of decisive importance. For this purpose, eight-meter masts, black and white, topped with a white flag, were installed. These masts marked the contours of the proposed development, and the corners of the buildings of the palace complex were marked with black and white striped masts. We have seen that the gaps between them are exaggerated. We were extremely worried and worried that right there, in the midst of an endless expanse, it was necessary to make a final decision. I had conflicting doubts!

I had to evaluate the situation and make a decision alone. I had to be guided not so much by reason as by instinct. Chandigarh is not a medieval city - the residence of governors, princes or kings, with dense buildings within the city walls. It was to be placed on an open plain. In essence, the task was to place the complete deep meaning of the sculptural structure. We did not have clay at our disposal to show our searches in a visual form. We couldn't test our solutions on mock-ups. The question was a deep mathematical calculation, the correctness of which could be verified only after the completion of construction. Determining the optimality of the chosen place of breaks, in order to make a final decision, as if by touch, they began to bring the "masts" together. It was a fight for space. But only a completed construction will reveal everything - arithmetic, structural and geometric relationships. And on the fields scorched by the sun, only herds of cows and sheep with their shepherds were visible ...

The Palace of the Supreme Court was solved taking into account the inclusion in its composition of the premises of eight judicial chambers and the Supreme Court itself. The orientation of the building, as well as the whole city, is dictated by the direction of the prevailing winds, the conditions of insolation and shading. In the sequential arrangement of the chambers, the principle adopted in the very first sketches of the composition of the Capitol is preserved (Fig. 27-29).

Arithmetic ratios were the basis for assigning the dimensions of the premises of the Judicial Chambers and the Supreme Court, and each room was considered as a plastic volume. The main dimensions were set - height, width and depth of the premises: 8 × 8 × 12 m for the court chambers and 12 × 12 × 18 m for the Supreme Court. However, Modulor ratios have been applied in glazing divisions and sun protection devices. Naturally, when structural and purely arithmetic relations were combined, residual dimensions were formed, which were used quite expediently. The cross section (Fig. 30) shows the system for protecting the premises from insolation; here you can see that the use of Modulor gives everything a structural unity. When building a facade, the whole system is based on a combination of structural dimensions according to the blue and red rows of the Modulor with the accepted arithmetic ratios of the dimensions of the supporting frame (Fig. 32).

Consider the building of the Palace of Ministries, 280 m long and 35 m high, designed to employ 3,000 employees (Fig. 33).

First of all, a load-bearing frame breakdown module (reinforced concrete transverse frames) was installed. The longitudinal center step is taken to be 3.66 + 4–0.43 m. The frame of the building consists of 63 frames, in other words, 252 columns extending from the base to the entire height of the building (Fig. 32).

The accepted height of the working premises ensures convenient placement of all channels, pipelines and, if necessary, corridors. The section of the building of the Seven Ministries shows an increase in internal spaces by using rooms of double the height adopted according to the Modulor (Fig. 34).

The layout and silhouette of the Governor's Palace, which occupies a dominant position in the Capitol complex, were determined in accordance with the exact instructions of the assignment; they corresponded directly to the existing initial plan. In a three-year period (1951-1953), the development of the project was completed.

In 1954, the crisis broke out! The construction cost was too high! What's the matter? It turns out that we got carried away and imperceptibly became a victim of series of proportional numbers! Having decided on the layout, we began to assign the dimensions of the height and depth of the premises, proceeding (since it was the Governor's Palace) only from the Modulor ratios.

We've been working hard! And the volume of the building turned out to be twice the volume of the former palace! The scale of the palace was exaggerated! We designed for the scale of giants!

The project had to be completely redesigned. New, more modest dimensions were adopted, and the cubic capacity of the building was halved.

The geometry of individual structures is determined by the very structure of the Modulor. But still, the dimensions of a number of basic elements can be refined using graphic construction methods. For the building of the Supreme Court, the simplest construction was adopted using a square, a doubled square, rectangles with an aspect ratio of Ø and an aspect ratio of √2. Such a construction leads to a harmonious solution, provided, of course, its skillful application (see Fig. 27-30).

We received an unimaginably clear confirmation of the correctness of our plan on March 20, 1955, the day after the grand opening of the Palace of the Supreme Court by Jawaharlal Nehru: in the first and so far the only one of the three planned reservoirs, a new architectural work arose, and with an absolute, it would seem only theoretically possible, clarity. The sketch shown in fig. 37 gives an idea of ​​this. An amazing image of a building washed by the air, as if given to the will of the winds!

2. Architecture, standards, unity

Music continues to sound... From now on, it will accompany all our undertakings.

Museum in Ahmedabad

In 1931, for the magazine "Cahier d'Art", I created a project of a square in plan, capable of continuously expanding museum, "devoid of a facade." At the same time, in a small Parisian bistro, I met with Shchusev. He was on a business trip to get acquainted with the construction of museums in connection with the development of the State Museum project for Moscow entrusted to him. On the back of the menu card, I sketched a diagram of a museum without a façade that could be located somewhere near Paris, among potato fields near one of the state highways, or anywhere else.

Over time, the idea was refined. The creation of the "Museum of Knowledge" could be started by any city from the central column, around which a square spiral with a width of 7 meters will then unfold. Further construction can be carried out as needed; it can go on continuously. The main entrance to the museum is located in the center of the building at low elevations. A passageway, arranged under the supporting frame, leads to it. Over time, it will be possible to place storage facilities between the foundation supports. The museum will thus be deprived of facades. Is it the other way around? Well, let!

In 1939, the project of such a museum was developed for the city of Philippeville in Algeria. But then the war broke out! The museum project was published in the Museum magazine, an organ of the International Organization of Museums, which recognized it as a valuable proposal. All columns are standard. Runs with a span of 7 meters and beams are also unified. The temporary facade was to be made of removable thin reinforced concrete panels. Typical elements of the coating provided natural and artificial lighting of the premises. The adopted proportions gave the whole complex an attractive appearance.

Beautiful models were made, which were exhibited in the Main Pavilion of the exhibition in Paris, dedicated to the possessions of France. Here in June 1940 they were caught by the invasion of enemy troops. In 1954, in Chandigarh, hot from the heat (January in the tropics!), at the foot of the Himalayas, from a letter received from Pierre Jeanneret, I learned that the models are quietly located in the Grenoble Museum. In 1951, I received an order to develop a project for such a museum called the "Museum of Knowledge" from the municipality of the city of Ahmedabad. The task was set for the exposition of the museum to tell the inhabitants of the city about their past, about modern affairs and about the prospects for the future. The climate of Ahmedabad is merciless and obliges to take the necessary defensive exchanges.

When designing the Ahmedabad museum, various means of assigning proportions were simultaneously used: simple arithmetic relations were used to build a square spiral of elements 7 × 7 m in size;

the geometric beginnings were expressed in the system of constructing a spiral; spiral fractures in the corners of the building, as it were, reflect the life of a person, which is characterized by change, and not constancy; geometry - also represented by a square shape of the plan;

structuredness - revealed by the use of Modulor relations and the standardization of elements that contribute to the creation of continuously developing internal spaces and provide the possibility of unlimited expansion of the museum.

The result is a gradual disclosure of a variety of visual impressions and an endless change of architectural images. In general - harmony (Fig. 41,42,43).

Architecture, standardization, unity!

Residential building in Marseille

I will confine myself to mentioning some of the details of the whole complex, their interaction infinitely enriching the charm and poetry of the structure. These are separate reinforced concrete columns and beams, as well as metal structures made of steel or bent aluminum profiles, perforated fences of loggias made of vibrated concrete. During the construction of the building, an atmosphere of complete coordination prevailed at the construction site; from the very foundation to the crowning parts, all lines and surfaces were coordinated with each other. An apartment building in Marseille (the so-called Residential unit) could celebrate the victory; each visit to the construction site was reassuring, as the whole structure had an inner harmony born of a clear structure that was perceived by all; it was inspiring. Despite all the hustle and bustle of construction, we did not find a marriage, there was not a single extra detail, not a single mistake, not a single unjustified part of the building. Everything was handy. Every element was in its place. The exceptions were two unfortunate mistakes made by negligence by one of the engineers: a series of divisions of glazing that did not correspond to the proportions determined by the graphic construction method, and individual concrete tiles made in some alien module (I was in New York at that time and was engrossed in the design of the UN building). Such an unacceptable and gross distortion of dimensions, violating the overall harmony of proportion according to the Modulor, was perceived by me extremely painfully; In desperation, I designed the polychrome decoration of the façades. Moreover, it was adopted very bright in order to divert attention from the mistakes made and completely capture with its violent colorfulness. Without these mistakes, a residential building in Marseille would not have received, perhaps, a polychrome finish of the facades.

The residential building in Nantes Reze replicated many of the new techniques used in Marseille. On fig. 44 shows the solution of the three main facades of the house in Nantes - east, south and west. A solution based on the use of seven different but modulated prefabricated elements made at the construction site. This is true standardization!

3. Always keeping the person in mind

On December 30, 1951, at breakfast on the Cote d'Azur, I sketched out a design for a cabin that I decided to give to my wife for her birthday. The following year, it was built on a rock, on which the waves of the sea break. The design of this hut was completed in 3/4 hours. He was final; the hut was built in full accordance with the drawings without any changes. Thanks to Modulor, the experiment was crowned with success (Fig. 46-48).

By reviewing these sketches, the reader will be convinced that sizing using Modulor ratios provided confidence in the work, leaving room for creative imagination.

On August 29, 1954, a similar experiment was repeated: within half an hour, at the request of the owner of the diner, I completed five projects of tourist houses measuring 226 × 366; in terms of the optimality of their layout and volumetric solution, they are not inferior to the cabin of an ocean liner. And that's in half an hour! Back in 1949, dealing with the issues of the appropriate use of the territory on the Cote d'Azur.

at that time it was built up with buildings of very dubious architecture, I proposed projects for houses based on a three-dimensional residential cell measuring 226 × 226 × 226.

So, we have come to the very essence of the problem: the creation of a three-dimensional residential cell. The condition of physical and moral comfort in this case is the accuracy and clarity of the decision. It goes without saying that all dimensions of such a living cell must correspond to a human scale.

On February 8, 1954, in record time, I gave the dimensions and architectural design for the large gilded bronze front door of the Supreme Court Palace at Chandigarh. Without completing the drawing, I dictated all the dimensions over the phone. On fig. 51 shows this doorway 3.66 wide, 3.66 high; handles are located in the most convenient place; the door rotates on a central vertical axis.

Height - 366 - corresponds to the Modulor; width - also 366 - is the sum of the Modulor dimensional values.

4. Unleashed art

Sculptural emblem of Modulor in concrete

The well-known sculptural emblem of Modulor in monolithic concrete at the Residential Building in Marseille was preceded by a series of preliminary proposals given in the book "Modulor, 1948". The attached photo shows the Modulor cast in concrete (fig. 51). A similar solution has been implemented in Nantes-Réze. When performed in kind, this image has undergone some changes. On the outer side of the wall of the elevator shaft is a diagram of proportional relationships. For public viewing, on the wall, in full size, a section of the apartment is shown so that the residents can be convinced that even with such a small size it is possible to live freely and comfortably.

I repeat my thought: the use of such dimensions will solve the problem of housing with a truly unprecedented reduction in the volume of housing construction.

On fig. 54 shows the through concrete fence of the Residential Building in Marseille. Box-shaped blocks were cast from concrete, the dimensions of which corresponded to the five dimensions of the Modulor. The walls were built from these blocks, and the gaps formed between them in several places were filled with concrete. Pieces of colored or white glass were manually inserted into the blocks on plaster. In this way, in the entrance lobby of the kindergarten on the 16th floor, two original and quite modern concrete stained-glass windows were created, which excluded the fastening of glass on lead and undoubtedly enriched the architecture of these interiors.

The same technique is currently used by us in the construction of villas in Ahmedabad.

Chapel in Ronchamp

In principle, I am against any modules if they fetter the creative imagination, claiming to be indisputable and limiting ingenuity. But I believe in perfect proportions (poetic). Proportions are essentially manifold, variable, innumerable. My mind cannot accept the use of AFNOR or Vignola modular systems in construction.

I reject the canons. I insist on establishing harmony in relation between things. When the construction of the chapel in Ronchamp is completed in the spring of 1955, it will become clear that architecture is determined not by columns, but by a plastic image. Plastic images are not subject to schoolboy or academic proportions; they are free and infinitely varied. The chapel in Ronchamp is a place of pilgrimage. It rules over the plain of the Saône in the west, over the Vosges range in the east, and over two small hills in the south and north. The surrounding landscapes on all four sides serve as both a background and a leading environment for the chapel. It is oriented to all four cardinal directions and creates the effect of "acoustic phenomena manifested in the field of forms." Every thing capable of revealing the effulgence of inexpressible space must have a certain intimacy. The chapel will be white inside and out; her decision will be truly free and unconstrained, the only thing that will determine it is the short duration of the service. Everything in it is interconnected. Poetry in the lyricism of the image is generated by free creativity, the brilliance of strictly mathematically justified proportions, the perfect combination of all elements. It gave me great satisfaction to be able to use in my work all the richness of combinations provided by Modulor; all you had to do was watch furtively to avoid any mistake, which always lies in wait for you in any part of your work and is capable of ruining it.

Open Hand at Chandigarh

In 1951, the idea arose to install an “Open Hand” at the entrance to the state capital against the backdrop of the Himalayan mountains (Fig. 56).

The idea of ​​the "Open Hand" was born in 1948. For a number of subsequent years, I worked on this idea, first embodied in Chandigarh. On a sketch in a travel album made in 1952, it appears on a free site, above a foundation pit excavated in the clay soil of the plain. On March 27, 1952, in Chandigarh, right at the construction site, I proposed the first dimensions of this complex.

On April 6, 1952, while still in Chandigarh, I checked the composition of the complex using the Serralt-Meisognier construction. It was only an attempt - perhaps even that I succumbed to the temptation!

April 12, 1952 the composition was refined. On February 27, 1954, at night, on the plane from Bombay to Cairo, I continued to look for a solution, relying on my ability (though doubtful) to remember numbers.

At the end of July 1954, Varma, the head of the works, came from Chandigarh to me at Cape Marten with a request to think about the possibility of the immediate construction of this monument. Not having my archival materials at hand, I tried to recreate the project using the Modulor ratios. During the period from August 1 to 12, I completed 27 drawings, which seemed to lead me to the final decision. Modulor played the main role in this work. this skillful and obedient assistant. However, quite unexpectedly, while trying a freshened reed stick on August 28 in Bogota, I immediately managed to sketch out the second solution of the “Open Hand” (Fig. 60, 61), clarifying the previous solution adopted in Bogota in 1951. The solution that satisfied me was the fortieth in a row and generalized options numbered from 19 to 27, corresponding to the Modulor. And here freedom was given to the flight of the imagination. However, it was based on reliable numerical ratios.

Consistently and gradually, starting from 1948 (Fig. 62), work was underway on this complex work of architecture, sculpture, technology, acoustics and ethics, going all the way from the original idea to the working drawings.

The reorganization of the hall of superhuman proportions

As already mentioned, the main advantage of Modulor is its proportionality to a person. For the purpose of organizing an exhibition of paintings, for the period from November 1953 to January 1954, I was given a room of superhuman proportions in the National Museum of Modern Art in Paris. The works of the great masters: Matisse, Braque, Picasso, Léger and the sculptors Laurence, Moore and others... lost a lot because of the inconsistency in the size of the hall. I tried to overcome these troubles by... returning to the human scale. Some approved of me, others condemned me. I will leave it to the reader to form his own opinion on this matter.

After all, there are obviously unsuccessful sizes ... how did it happen? Only sometimes it can be explained, but you can always feel it. There are architectural structures designed either for fleas or for giraffes, it’s impossible to determine for sure. But at least not per person. For example, St. Peter's Basilica in Rome* or the desperate hall of the National Museum of Modern Art in question.

* In March 1955, during a stop in Rome on my way from New Delhi, I stopped for a moment at St. Peter's Basilica. I told the person who met me at the Nervi airport: “I did not like this cathedral during his visits in 1910, 1921, 1934 and 1936. There is something wrong in Peter's Cathedral; Michelangelo's successors are to blame for this. Now, March 15, 1955, nothing has changed and my opinion has been confirmed.

The works of art exhibited in such spaces become distorted and incomprehensible to us, the people for whom they are ultimately intended.

The task is thus to restore the necessary contact between the visitors of the exhibition and the exhibits (paintings, sculptures, photographs) with the help of effective means.

We decided on this and created a system of volumes 226 cm high in this disproportionately high hall, combining them in such a way as to make the most advantageous use of their external and internal surfaces to accommodate paintings, sculptures and other exhibits.

On the opening day of the exhibition, my friend Fernand Léger said: "What a pity that you mutilated such a magnificent room." But I am an architect, called to operate with volumes! Perhaps I have ruined this hall; but that's what I was striving for... After the end of my exhibition, everything was restored to its original form.

The photo (Fig. 65) shows the layout of the reorganization of the hall. In such a room, the exhibited works - sculpture and painting - were perceived in their true scale and had an emotional impact.

Carpets with a total area of ​​576 square meters for Chandigarh

The carpets are designed to improve acoustics in the Supreme Court and the eight Small Chambers of Justice in the Palace of Justice (Chandigarh Capitol).

Carpets consist of component elements, the dimensions of which correspond to the proportions according to the Modulor.

For the Supreme Court

8 elements, 1.40 × 4.19 m each (3.66 + 0.53) = 5.866 m²

(4"-7")×(13"-9") = 63 square meters f.;

8 elements, 1.40 × 2.26 m each = 3.164 m²

(4"-7")×(7"-5") = 34 sq. m;

5 elements, each 1.40 x 3.33 m = 4.662 m²

(4"-7")×(10"-11") = 50 sq. f.;

5 elements, 1.40 × 2.26 m in size = 3.164 sq. f.

(4"-7")×(7"-5") = 34 sq. f.;

For Small court chambers

5 elements, 1.40 x 2.26 m each = 3.164 sq. ft.

(4"-7")×(7"-5") = 34 sq. f.;

2 elements of 1.40 × 3.33 m = 4.662 m²

(41 - 7") × (11" - 11") = 50 sq. ft.;

2 elements of 1.40 × 2.26 m = 3.164 sq. f.

(4"-7")-(7"-5") = 34 sq. f.

As a result, the entire order involves the manufacture of carpets: For the Supreme Court - 144 m² (1550 sq. ft.); For the Small Judicial Chambers - 54 X8 = 432 m²

(581 sq. ft. x 8 = 4650 sq. ft./2)

Total: 576 m² (6,200 sq. ft.)

Five hundred seventy-six square meters of carpets. Carpets will consist of:

a) typical elements;

b) individual elements;

c) additional elements.

For the Supreme Court

8 elements 1.40×4.19 m in size + 1 additional element 1.33×4.19 m

(4"– 7")×(13"– 9");

(4"-4.5")×(13"-9");

8 elements 1.40 × 2.26 m each + 1 additional element 1.33 × 2.26 m

(4"– 7")×(7"– 5"); (4"– 5")×(7"– 5");

5 elements 1.40×3.33 m each + 3 individual elements and 1 additional element 1.33×3.33 m

(4"– 7")×(10"– 11"); (4"-4.5")×(10"-11");

5 elements 1.40×2.26 m each + 1 individual element 1.13×2.26 m

(4"– 7")×(7"– 5"); (3"–8.5")×(7"-5") + 1 extension piece 1.33×2.26 m

(4"– 4.5")×(7"– 5")

For Small court chambers

5 elements of 1.40 × 2.26 m each + 1 additional element (0.72 × 2.26 m)

(4"– 7")×(7"– 5"); (2"-4.5")×(7"-5");

2 elements 1.40x3.33 m each + 3 individual elements and 1 additional element 0.72x3.33 m

(4"–7")×(10"-11"); (2"– 4.5")×(10"– 11")

2 elements 1.40X2.26 m each + 1 individual element 1.13×2.26 m

(4"– 7")×(7"– 5"); (3"-8.5")×(7"-5") and 1 extension piece 0.72×2.26 m (2"-4.5")×(7"-5").

At the last moment, we introduced an additional table of four combinations of square or rectangular spots, called "points" and denoted by the letters PA, PB, PC, RO. They are designed to enliven individual monochromatic parts of carpets; black and white dots are provided. Carpet designs such as the sun, clouds, lightning, meanders, arms, legs, etc... are made on specific drawings at a scale of 1:5.

Ordinal numbering

"It's a system of proportions that gets in the way of doing bad and helps to do good."

Einstein. Princeton, 1946

In 1949, the France-Soir newspaper published under the heading “In a quarter of an hour you will know everything: .. The architect Le Corbusier took up arms against the meter ... Down with the metric system! .. further, a number of provisions confirmed this statement. But this is journalism! Even with the best of intentions, she is capable of making noise, often simply unbearable. She goes to the scandal! I never thought of abolishing the metric system (read Modulor, 1948). The metric system is a means of measurement based on the decimal system; it is this circumstance that has turned it into a modern working tool.

Until now, the dimensional values ​​​​of the Modulor scale were expressed both in the metric (decimal) system of measures and in feet-inches (in a non-decimal system). This helps those who use feet and inches to do all the counting and calculations in the decimal system.

In an article published in the Cahiers du Sud magazine, Andre Vozhensky noted a number of inaccuracies in the terminology adopted in the Modulor of 1948, in particular, in the title “Experience of a universal harmonious system of measures” ... I think it would be correct to title: “ The experience of a harmonious system of measures on a human scale, which has universal application, etc...” This question remained open. He notes that the intervals between the divisions of the Modulor, tending on the one hand to the bullet, and on the other to infinity, are not numbered by using simple ordinal numbers both for microscopically small and for astronomical intervals ... I believe that this is not for whom it was not associated with serious complications and did not interfere with anyone. In any case, from a purely theoretical point of view, it can be argued that the Modulor proportional scale is, as it were, a ladder of dimensional values, not supported by anything, since these values ​​never reach zero. On the other hand, it is not suspended from some hypothetical sky, as it tends to infinity. This is all pure sophistry! He, however, had every right to quote. If we want to establish an ordinal numbering for the Modulor, we should start with some real value, taking it as the first ordinal value (the number "1"). From this point you can go up and down. Finding such an initial value is not easy. The persons to whom I addressed this question did not dignify me with an answer, and sometimes expressed the idea that this question was of no interest. True, one of the interviewees frivolously said: "Consider the sole of the foot of a standing person as the starting point." In Modulor's graphic emblem, the feet are indeed on the ground; man stood on the ground, in other words, sank to zero. We have, however, repeatedly pointed out that zero is an unattainable goal. He shows only a general trend: however, he is inaccessible. In June 1951, I suggested to Crussard that he accept the starting point for the numbering in accordance with fig. 67. This point is at 113; then the lower divisions towards zero would be indicated by serial numbers 1, 2, 3, 4, ..., 20, ..., 100, ..., 200 with an index of at least A. They would have styles 1A, 2A , ZA, 4A, 100A, 200A and would quickly reach the designation of microscopic sizes.

Serial numbers of divisions above the mark 113 would receive the index B; the numbering of divisions would have no limits - 1, 2, 3, 4, 5, 9, 27, 99, 205, etc. and would have the inscriptions 1B, 2B, 3B, 4B, 5B, 9B, 27B, 99B, 205B etc.

Such a method of ordinal numbering seems to me disgusting, it is devoid of any expressiveness, it is colorless. I left it to scientists to define a clear and usable system. I emphasize: and convenient, since calculations will be carried out on the basis of this numbering: addition, subtraction, multiplication, division, etc., you may even have to compose algebraic equations. In these cases, it seems to me, indexes A and B will create a number of inconveniences; nevertheless, in my opinion, it is necessary to come up with indices that would mark the “lower” and “upper” numbers in the rows.

Mark 113 marks the most significant point of the Modulor: it corresponds to half of the dimensional value of 226 (blue row) and passes through the solar plexus of a person with a raised hand, etc., and corresponds to the articulation in the golden ratio of value 183, i.e., the growth of a standing person ( red line). The question of the ordinal numbering of the Modulor remains open. Perhaps the reader will be able to answer this question?

Epilogue

It so happened that at the age of more than sixty years, quite unexpectedly, without premeditated intention, I proposed three working tools:

1. Modulor;

2. Urban planning grid CIAM (ASCORAL);

3. Climate grid (workshop at 35 Sevres street).

These tools should ensure unity and coherence.

Painting lessons led me to these discoveries. From childhood, my father took us for walks in the mountains and valleys, showed us things that aroused his admiration: he told us about their diversity, contrasts, about their striking originality, despite the commonality of patterns.

By the age of thirteen, I received elementary knowledge in the field of physics, chemistry, cosmography, and algebra at school. This knowledge opened the doors to the future for me. Then I began to study drawing with the first teacher (Leplatenier), whom I idolized. He took us to the fields and forests and encouraged us to make discoveries. Opening is a big word. Start discovering. Start making discoveries and then commit yourself to it. Discoveries must be made at every step.

At the age of 31, I painted my first painting (it was quite clear, since the creation of a painting involves the imposition of colors, and this is not a difficult task; it is much more difficult to know what to paint). My painting was creative, not imitative. My paintings have always been constructive, organic and clearly constructed due to the fact that they have always been subordinated to the most important human qualities, striving to establish a constant coordinated and balanced interaction between design and implementation.

For this, it was necessary to be able to design, to have a sense of balance and time, endurance and understand what exactly is essential; you also need to be imaginative.

I mastered the skill of painting, realizing that in order for a thing to be poetic, it is necessary to achieve sharpness and originality in the choice of exact ratios.

Accuracy is the springboard to the creation of lyrical works.

Architecture at that time only revealed its secrets to me*.

* I started building at the age of 17 (the first building I built dates back to 1905). Only later, after a series of vicissitudes in life, in 1919, at the age of 32, did I correctly understand the task of architecture.

I was able to apply my knowledge in the field of architecture and construction only after reaching a certain level of intellectual development. The next stage was urban planning activities, which included a wide range of issues: the social sphere, the problem of the relationship between man and society, love for man, human scale, natural laws, mastery of space...

That is why, one day, passing next to the wall behind which the games of the gods were going on, I began to listen. I have always been irremediably inquisitive.

On Monday, August 9, 1954, at Cape Marten, I finished reading the final text of this book. In June I dictated it to my secretary, Zhanna. The reader will understand the reasons for the individual roughness of the text and, I hope, will not be angry with me. Let us hope that his attention will be focused on the essence of the problem presented in this work.

Monologue in a good mood

The main task is to excite, excite, using any opportunity that illuminates, generates, overflows, excites and awakens the soul.

On fig. 67 shows a good, if crudely made wooden model, which makes me think of Ahmedabad, India. It's hot, terribly hot, we conceived a snail shell-shaped dwelling, provided with sun protection devices that keep you cool even in hot summer. In winter, the rays of the sun can penetrate deep into the premises. Comfortable conditions are created thanks to through ventilation. The solution of the covering and facades provides shading. The layout is convenient. Air circulates freely in the premises, as the location of the house takes into account the direction of the prevailing winds.

Together with Trouen, for a number of years, we worked to restore the glory of the architectural and iconographic monuments of Sainte-Baume, to create an underground, mysterious and gloomy basilica .., and above it, on the earth's surface, the life of ordinary people would flow on the scale of the surrounding landscape and meet their external and internal needs. That would be great! Those would be the fruits of persistent labor that elevates us. But the archbishops and cardinals of France imposed a ban.

I was then completely absorbed in the struggle in Marseille: the years 1946-1952 went on. Comrades in the profession (architects and their organizations) stood across the road.

The construction of an apartment building in Marseille was a battlefield. What a cruel test! You had to have a lot of patience! Here's Marseille! Look at the residential building in Marseille! I agree that this architecture is unusual for professional circles.

This is a bridge thrown in our time from the Middle Ages. This is not architecture for kings or princes, this is architecture for ordinary people: men, women, children. And in the summer, under the Mediterranean sun, the apartment is cool. The house is located in the heart of Marseille, and the expanse of the sea penetrates through the windows, and the mountains are located on the opposite side. It is a landscape worthy of Homer, similar to the Delphic landscape in the Ionian Islands, which the Marseillais, who live in their houses and huts, behind closed shutters, do not suspect.

Walk through the floors, interview 1600 residents of our residential building in Marseille. Was not a new life revealed before them?

Now, in the spring of 1955, the second "vertical residential complex" is being settled in Nantes Reze. Marseille is six years of struggle; but the majestic Ship daily pleases the inhabitants. This is the reward for forty years of searching; it is the result of a lifetime's work and the selfless help of an army of dedicated, enthusiastic young architects, French and international. Patience, perseverance and modesty in searches and actions. Work without big words. That was an experiment. Seven successive ministers authorized this construction; some only put up with him, others actively helped him. Today tourist buses arrive directly from Malmö, Calais and Cologne. In terms of the number of visitors, this building is second only to the famous castles in the Loire Valley...

The house in Nantes-Reze, built in eighteen months, at the price of ordinary houses being built in France, crowns the hard work of young people working in the workshop on the street. Sevr.

Reader, look for yourself at the photographs of these structures, which Modulor gave a joyful look. Modulor, which "helps to do well."

Rice. 68 - a clear silhouette of the building against the sky (Marseille). Rice. 70 - facade of the house with a shopping street on the eighth floor; there is a bakery, a butcher's shop, a greengrocer's, a pastry shop, a laundry, etc. ... Everywhere, from top to bottom, the concrete is left untreated; reinforced concrete is ranked among the noble materials.

Rice. 71 - supporting pillars - the basis of the urban planning solution adopted in the project of the "Radiant City"; the entire surface of the earth is at the complete disposal of pedestrians.

A kind of fencing of a shopping street made of glass, wood, concrete ... centuries-old trees below, on the one hand - mountains, on the other - the sea, Modulor gave everything here a "Greek", "Ionic" joyful look; it is the gracious gift of the mathematical designation of human proportions.

At a height of fifty-six meters above the ground, children in kindergarten can enjoy the water and the sun, admire the natural landscape ... Go upstairs and ask them: are they happy? Rice. 74 - this is Chandigarh, the colonnade of the gallery of the Palace of Justice, solemnly opened on March 19, 1955 in the presence of D. Nehru. Wait a little more! Currently, large reservoirs are being built in front of the palace. And then the photographer will be able to capture the symphony of Nature and Architecture against the backdrop of a wonderful landscape.

Rice. 73 - view of a factory building in Saint-Dieu. The premises of the directorate in the superstructure on the flat roof of the factory. Le Corbusier's only implemented proposal from the urbanization project for the city of Saint-Dieu, rejected in 1946. Rice. 74 - hall in the workshop of finished products. It would be necessary to show the color scheme of the interior. The intense and bright tones of the ceiling painting gave the workrooms the character of medieval grandeur (of course, only in spirit).

I delivered this monologue in a good mood, because it tells about a work entirely devoted to matters of great importance to people: modern housing and modern public and industrial buildings.

And one day arose from a dream, From this soul praying, Like grass, like water, like birches, A marvelous marvel in the Russian wilderness.

N. Rubtsov

It's time to look for proportions. The spirit of architecture is affirmed.

Le Corbusier

In 1784, the humble father of the Bogolyubov monastic brethren asked His Eminence Victor, Archpastor of Vladimir, for permission to dismantle the dilapidated and half-abandoned church for the monastic needs. The permission was graciously granted, but, as they say, life had its own way: customers and contractors did not agree on the price. The work did not begin, and there they were completely forgotten. So, by the will of fate, the monument remained alive, which was bypassed by the hordes of Batu and Mamai, spared centuries and conflagrations of endless wars, a masterpiece of ancient Russian architecture, the Church of the Intercession of the Virgin on the Nerl.

On clear summer days, among the greenery of flood meadows, its slender whiteness, reflected by the smooth surface of the old Klyazma, breathes the poetry of a fairy tale. Only in the short minutes of sunset the white candle of the church lights up with an alarming crimson flame. In severe winters, an endless snow veil, like a caring mother, wraps and hides her frozen child. "In all Russian poetry, which has given the world so many unsurpassed masterpieces, there is perhaps no monument more lyrical than the Church of the Intercession on the Nerl, for this architectural monument is perceived as a poem etched in stone. A poem of Russian nature, quiet sadness and contemplation" (L . Lyubimov).

Before approaching the mystery of the charm of ancient Russian architecture, we need to get acquainted with the system of measures that existed in Ancient Russia. We have already noted (p. 198) that in different places on the globe, at different times and among different peoples, the standards of length were in principle the same: they somehow came from the human body. These so-called anthropometric measures possessed the most valuable quality for architecture, which was forgotten with the introduction of the metric system of measures, but to which Le Corbusier returned in the 20th century. The fact is that anthropometric measures due to their origin, they are commensurate with a person and therefore convenient for constructing an artificial human habitat - architectural structures. Moreover, in "human" measures there are proportions selected by nature itself, such as halving, the golden section, the function of the golden section. Consequently, the harmony of nature is naturally embedded in anthropometric measures.

The main building measure in ancient Russia was a sazhen, equal to the span of the arms to the sides. The sazhen was divided into 2 half fathoms, half a sazhen - by 2 elbow- distance from fingertips to elbow, elbow - by 2 spans- the distance between the thumb and little finger extended in opposite directions. Everything is clear and logical. However, the more closely historians studied the ancient Russian chronicles, the more fathoms became, and when their number exceeded ten, the historians' head went round. It became necessary to restore mathematical order in the old Russian system of measures. This was done by the historian, academician B. A. Rybakov and the architect I. Sh. Shevelev. The beginning of anthropometric measures is given by the height of a person a. The main of all types of fathoms is measured, or flyweight, sazhen C m, which is equal to the span of a person's arms to the sides. The study of the proportions of the human body shows that C m = 1.03a. Another important measure among all peoples was the double step, which is equal to the height of the body from the feet to the base of the neck. The last distance, as we know (p. 220), is 5/6 AU. In this way, double step, or small(Tmutarakan) sazhen, C t \u003d 5/6 a \u003d 0.833a. But the main surprise lies in relation to these two main dimensions:

(17.1)

Therefore, the small sazhen C t refers to the measured C m as the side of a double square to its diagonal without a small side:

From (17.1) it is clear that the ratio of the measured half-sazhen C m / 2 to the small sazhen C t is equal to the golden ratio:

(17.2)

So, in the ratio of the half-span of the arms (RS) to the height of the body (LQ), established by nature itself, that is, in relation to the two main measures of Ancient Russia, the golden section is concluded, which is so common in ancient Russian architecture.

Man's height: a = AB

measuring sazhen: C n \u003d AC \u003d CN \u003d 1.03a

Small (Tmutarakan) sazhen:

Fathom without a quarter:

Oblique Novgorod sazhen:

Oblique great fathom:

Relations between fathoms:

golden ratio

golden ratio function

Having built squares on small C t and measured C m fathoms and drawing diagonals in them, we get two more types of fathoms: oblique Novgorod sazhenand great oblique fathom. In contrast to the first two sazhens (small and measured), expressing natural measures, oblique sazhens were obtained in a purely geometric way. It's clear that

(17.3)

Finally, there was another sazhen obtained geometrically. This so-called fathom without a quarter C h, equal to the diagonal AM of half the square built on the measured sazhen C m. This sazhen did not have a corresponding oblique pair, and therefore it was called a sazhen without a pair, without a couple, or without a four. It follows from the ACM triangle that , where

(17.4)

i.e. the ratio of a sazhen without a quarter C h to a measured sazhen C m is equal to the function of the golden section (see p. 219).

These are just the main types of sazhens that existed in ancient Russian metrology. The Novgorod measuring stick, found in 1970 (see p. 219), made it possible to clarify their dimensions. Novgorod measures of the 12th century correspond to the height of a person: a = 170.5 cm. Then C m = 175.6 cm, C t = 142.1 cm, K n = 200.9 cm, K v = 248.3 cm, C h \u003d 196.3 cm. If the height of a person is taken equal to 6 Greek feet: a \u003d 6 * 30.87 \u003d 185.22 cm, then for the main fathoms (measured and small) we get the values: C m \u003d 190.8 cm and C m = 154.3 cm. It is these measures that are most often found in ancient Russian churches of the 11th century, the construction of which, apparently, was carried out by Byzantine masters. So, together with Christianity, Russia inherited the Byzantine system of measures, which, in turn, grew up on the ancient Mediterranean culture. The absolute sizes of sazhens in Russia fluctuated greatly over time until the introduction of the metric system of measures in 1918. But it is important that proportional relationships between paired sazhens were preserved. These proportions became the proportions of architectural structures.

The fact that the measures were taken by ancient Russian builders in pairs is evidenced, for example, by a Novgorod letter of the 16th century, which describes the size of the St. Sophia Church in Novgorod in this way: "and inside the chapter, where the windows are, there are 12 sazhens, and from the Spasov image from the forehead to the church bridge - 15 sazhens measured". (Measurements show that the mentioned sazhens correlate as: 2.) The Novgorod measured cane also speaks of the use of paired measures, in which a small sazhen С t was used either in tandem with a measured sazhen С m (С t: С m = 1:( - 1 )), or with oblique Novgorod K n (C t: K n \u003d 1: √ 2). If measured half-sazhens were taken on a Novgorod cane paired with a small sazhen, then this pair gave the golden ratio (C m / 2: C t \u003d φ). So, the beauty of the proportions of ancient Russian architecture lies in the very system of ancient Russian measures, which gives such important proportions as the golden section, the function of the golden section, the ratio of the double square.

But in addition to all these proportions, which from nature itself passed into the system of measures, and then into architectural monuments, the ancient Russian masters had one more secret. It was this secret that made it possible to give each ancient building a unique charm, "nuance", as the architects say. This secret is revealed in the in-line record of the carpenter Fyodor for the construction of the wooden church of the Ust-Kuluysky churchyard (end of the 17th century), where it is said: beauty says...

"As measure and beauty say..." This wonderful formula of an obscure Russian carpenter expresses the essence of the dialectic of the interaction of rational (measure) and sensual (beauty) principles in achieving beauty, the union of mathematics (measure) and art (beauty) in the creation of architectural monuments.

Let us finally turn to an analysis of the proportions of the Church of the Intercession on the Nerl. This architectural masterpiece means as much to a Russian person as the Parthenon does to a Greek. Therefore, it is not surprising that the proportional structure of a small church was analyzed by many researchers and each of them tried to give his own "final" clue to the mystery of her charm. Let us briefly consider the proportions of the Church of the Intercession on the Nerl from two points of view.

According to the architect Shevelev, the proportional structure of the Church of the Intercession is based on the ratio of a sazhen without a quarter to a measured sazhen, which is a function of the golden section (C h: C m = √5: 2), and the plan of the church itself was built as follows. First, a rectangle 3 fathoms long without a quarter and 3 measured fathoms wide was marked out, which outlined the pillars carrying the drum and vaults. Since 3C h: 3C m = √5:2 = 1.118, then the sides of this rectangle refer to the function of the golden section, and the rectangle itself is almost a square, or, in Zholtovsky's terminology, a "living square". Having drawn diagonals in the original rectangle, the architect received the center of the temple, and having set aside 1 measured sazhen on the diagonals from the tops to the center, the domed rectangle and the dimensions of the supporting pillars. So the core of the plan was built, which determined all further horizontal and vertical dimensions of the structure. The measured sazhen of the builders of the Church of the Intercession was C m = 1.79 m.

Having measured from the Center of the temple to the east 3C m and to the west 3C h, the master received the length of the outer

rectangle equal to:

And putting this size in measured sazhens, its width is 5 3/4 cm. Thus, the outer rectangle of the plan of the church is similar to the core of the plan and is also a "living square". The diagonal of the rectangle under the dome determined the diameter of the central apse (under the dome of the altar ledge) and the diameter of the drum of the temple. The short side of the domed rectangle set the diameters of the side apsides.

Finally, the height of the base of the temple - the quadrangle, read by the height of thin columns - is equal to twice the length of the core of the plan, i.e. 2 * 3C h \u003d 6C h, and the height of the drum with a helmet-shaped dome * is twice the width of the core, i.e. 2 *3С m = 6С m. Thus, the main vertical dimensions of the temple - the height of the base and the height of the completion - also relate to the function of the golden section. The quadrangle itself is "almost a cube", the base of which is "almost a square", and the height is almost equal to the sides of the base. So, in the construction of the quadrangle of the temple, the principle of approximate symmetry is clearly visible, which is so often found in nature and art (see Chapter 4). You can also point to smaller divisions of the temple, related to the function of the golden section, that is, in relation to a sazhen without a quarter to a measured sazhen. For example, the stone belt crowning the columnar frieze, which covers the entire church and is its important architectural detail, divides the height of the quadrangle as a function of the golden section.

* (Initially, the Church of the Intercession had a helmet-shaped dome, characteristic of ancient Russian churches, resembling a warrior's helmet. In the 17th century, the helmet-shaped dome was converted into a bulbous one, which we see today.)

Let us now consider the ichnography of the Church of the Intercession on the Nerl, as seen by the connoisseur of ancient Russian architecture K. N. Afanasyev. According to Vitruvius, "ichnography is the proper and consistent use of a compass and straightedge to obtain the outlines of a plan." According to Afanasiev, the initial size of the Church of the Intercession is the smaller side of the domed rectangle, equal to 10 Greek feet: a = 10 Greek. foot. \u003d 308.7 cm. Then the large side of the domed rectangle is obtained as the diagonal of a double square with side a / 2. Thus, the rectangle under the dome is a "living square", the sides of which are related in the golden section function. The thickness of the pillars is determined by the ratio of the golden section to the modulus a/2. Further constructions are clear from the figure. This is how the core of the plan is built. The remaining dimensions of the plan are obtained by similar constructions, relying mainly on the module a/2.

Note that together with the function of the golden section, the law of the golden section also determines the proportional structure of the Church of the Intercession. This is not surprising, because these relations are connected by the geometry of a double square. As Afanasiev established, the main verticals of the temple, which determine its silhouette, are primarily subject to the law of the golden section: the height of the base, equal to the height of the thin columns of the quadrangle, and the height of the drum. The diameter of the drum is related to its height also in the golden ratio. These proportions are visible from any point of view. Moving on to the western facade, the series of the golden section can be continued: the shoulders of the temple refer to the diameter of the drum in the golden ratio. So, taking the height of the white stone part of the church (from the base to the dome) as a unit, we get a number of the golden section: 1, φ, φ 2, φ 3, φ 4, which determines the silhouette of an architectural structure. This series can be continued in smaller details. (Of course, the western facade from the point of view of the golden ratio is no exception and is taken by us only as an example.)

Let's sum up some results. We see that the seemingly incomprehensible harmony of the Church of the Intercession is subject to mathematically strict laws of proportionality. The plan of the church is built on the proportions of the golden section function - "living squares", and its silhouette is determined by the number of the golden section. This chain of mathematical patterns becomes a magical melody of interconnected architectural forms. Of course, the laws of proportionality determine only the "skeleton" of the structure, which must be correct and proportionate, like the skeleton of a healthy person. But besides the mathematical laws of measure, in the bowels of an architectural masterpiece, there are also unfamiliar laws of beauty: "as measure and beauty say ..."! It is the dialectics of the interaction between the laws of measure and the laws of beauty, which often manifest themselves in deviations from the laws of measure, that creates a unique image of an architectural masterpiece.

Note that from the point of view of geometry, the reconstructions of the proportional structure of the Church of the Intercession considered by us are similar. They are consistent with each other and give in the plan three "living squares" inscribed in each other, the ratio of the sides of which is √5:2 determines the entire proportional structure of the temple. However, from the point of view of the history of architecture, these reconstructions differ fundamentally. The first of them is based on the old Russian system of measures and, therefore, suggests that the Church of the Intercession was built by Russian architects. The second one has a Greek measure as the main size and therefore gives reason to believe that the church was built by craftsmen invited from Byzantium ... Who and how created the pearl of Russian architecture? Perhaps we will find out the answer to this question...

The Church of the Intercession was built in 1165. And 73 years later, she witnessed an unprecedented misfortune in the history of Russia: the hordes of Batu, having turned Ryazan, Kolomna and Moscow into ashes, laid siege to Vladimir. The Russian state, tormented by princely strife, was dealt a mortal blow, from which Russia was able to fully recover only 200 years later, by the end of the 15th century.

In 1530, in the royal estate - the village of Kolomenskoye near Moscow - the future tsar of the awakening Russia, Ivan the Terrible, was born. And two years later, here, in Kolomenskoye, on the steep bank of the Moskva River, the construction of a church was completed, erected in memory of this event. The architects seemed to foresee the birth of an unprecedentedly formidable king: the church was also unprecedented. Everything in it ", and the height (almost 62 m), and the stone tent, and the shape directed upwards - was unprecedented. The new temple seemed to symbolize Russia's breakthrough into a future free from the Tatar yoke. "... But that church is very wonderful in height and beauty and lordship, such has not happened before in Russia, "the chronicler wrote about her. The entire proportional structure of the church, all its unrestrained aspiration upwards corresponded to the name - the temple of the Ascension.

But for us, the Ascension Church is also interesting because it is not only the anthem of Russia spreading its wings, but also the architectural anthem of geometry.

None of the considered architectural masterpieces, including the Parthenon, is so permeated with geometry, so simple and concise in its dimensional structure, as the Church of the Ascension in Kolomenskoye. The proportions of the temple with the utmost clarity are determined by two pair measures: horizontal - small (Tmutarakan) sazhen C t and oblique Novgorod sazhen K n (C t: K n \u003d 1: √ 2), vertical - small sazhen C t and measured sazhen C m ( C t: C m = 1: (√5 - 1)) and their combination C m: 2C t = (√ 5 - 1): 2 = φ, giving the golden ratio. Thus, the Church of the Ascension is also an excellent example of the use by Moscow masters of a measuring tool such as the Novgorod measuring stick, created, as we remember, to work with these two pairs of measures (see p. 220). Consider the proportional analysis of the temple, made by the architect Shevelev.

The plan of the Church of the Ascension is based on the square ABCD with a side of 10 small sazhens: a = AB = 10С t. It is clear that the diagonals of the square are 10 oblique Novgorod sazhens: AC = BD = 10√2ST = 10K n. So, with the help of paired measures C t and K n, the correctness of the construction of the initial square was monitored. The circle of radius R = 5K n, describing the square, determines the position of all 12 outer corners of the plan of the temple. By inscribing a new square through the midpoints of the sides in the square ABCD and making constructions, we get the outer contour of the plan - 20- square. The parts protruding above the original square are called vestibules, their width is equal to a / 2 = 5С m. Expressing the radius of the circumscribed circle R in measured fathoms and putting this value in small fathoms, the builders received the side of the square b, which determines the internal space of the temple:

Of course, the Kolomna craftsmen did not calculate any radicals! They simply applied the measuring stick on different sides and automatically switched from one measure to another. The plan of the church is built. And we will also express the side of the square c, covering the vestibules: c \u003d √7 / 2 a (the triangle from which c / 2 is located is not shown in the drawing so as not to spoil the beauty of the central symmetry of the plan; find it). Knowing a, b, c, it is easy to express all other dimensions of the plan and the relationships between them.

Let's move on to the volumes and vertical divisions of the temple. The Church of the Ascension is surrounded on all sides by a covered gallery, raised above ground level and called promenade. The ambush was made at the level of the ceiling basement- semi-basement used for business purposes. The entrance to the church was arranged from the graveyard, to which three porches lead in the Ascension Church, and, thus, the vertical dimensions of the church with the graveyard are perceived from the level of the latter.

The main volume of the temple is a 20-sided prism placed on the basement. Its height is equal to the side of the original square a. Thus, the core of the main volume is a cube - a quadrangle a × a × a (a = 10С t), decorated with narthex faces. Together with the basement, the height of the 20-sided prism is equal to the diagonal of the original square a√2 = 10√2C m = 10K n. So, the side and diagonal of the original square (the core of the plan) completely determine the vertical dimensions of the main volume (the core of the base).

The twenty-sided prism of the main volume passes through the intricate belt of kokoshniks into an octahedral prism - octagon. The octagon is also inscribed in the cube d×d×d(d = 9C t). Then the octagon passes into an octahedral tent, the height of which is h = d√2 = 9√2С t = 9K n, i.e. the tent is inscribed in a rectangular parallelepiped 9С t × 9С t × 9К n. The area of ​​the upper section of the tent is reduced by 16 times, and its linear dimensions - by 4 times. Since 1/4 sazhen is equal to a cubit, therefore, the upper section is inscribed in a square where L t is a small (Tmutarakan) cubit (4L t \u003d C t). Finally, through the crowning cornice, the tent ends with an octagonal drum, the section of which exceeds the upper section of the tent by a small half-cubit. The drum hangs slightly over the tent and is inscribed in a cube f × f × f (f = 9.5L t), and together with the dome, taken without an apple (see the figure on p. 242), the drum is inscribed in a rectangular parallelepiped f × f ×√2f.

So, we see how the side of the core of plan a, measured either by a small sazhen or oblique Novgorod, gives rise to all the main verticals of the temple. Note that the total height of the church from the top of the plinth to the apple on which the cross stands is 4a = 40C m, i.e., it is also expressed in the simplest way in terms of the original size a. And one more important relation. The belt of kokoshniks, through which the quadrangle of the base passes the octagon of the tent, divides the temple into two parts - the base and the completion. The height of the base h 1 ≈14C t, and the height of the completion h 2 ≈14K n, whence h 1:h 2 = C t:K n = 1:√2, i.e. the main vertical divisions of the temple are also referred to as small and oblique Novgorod fathoms.

But the proportions of the Temple of the Ascension are determined not by one, but by two mathematical laws. In addition to the proportion C t: K n \u003d 1: √ 2, which determines the foundation, the static beginning of the temple, there is another theme in it - the theme of upward development, ascension, which is determined by the proportional chain: C t: C m = 1: (√ 5 - 1), as well as the proportion of the golden section: C m: 2C t \u003d φ. In carrying out this theme, the principle of the oncoming movement of proportions, familiar to us from the Parthenon, is observed. Two different proportional circuits are superimposed on each other, collide and oppose. This clash of two opposing principles - horizontal and vertical - is the architectural image of the Church of the Ascension. Without dwelling on the mathematical analysis of these two systems, let's give the floor to the author of an excellent aesthetic analysis of the Church of the Ascension, art critic A. Cires. “In the image of this church,” writes Tsires, “two main leitmotifs are intertwined: the motif of sharp, full of clashes and dissonances of dynamism and the motif of harmoniously calm beauty ... The complex rhythm of the arches of the lower galleries ... goes, becoming more frequent from the edges to the center, .. . pushes the arches from the edges to the corners of the main body of the church and to its middle, ... suggests a change in horizontal movement with an upward movement ... So from the bottom up there is a gradual softening of crystallism and an increase in the compactness of the volume, up to its tightness into a strong knot crowning the entire voluminous composition of the head.

But we would like to end the conversation about the proportions of the Church of the Ascension in Kolomenskoye with the words of the author of the mathematical analysis of its proportions, Shevelev. “Let us emphasize the most expressive detail of the dimensional structure, which most clearly shows the peculiarity of the logic of the ancient master, who strives to express the main thing in metrology with particular precision. cross (10С t Х10С t Х10С t - quadruple; 10С t Х10С t Х10К n - prism of the quadrangle; 10L t Х10Л t - the proportionality of the cross, because for the architect it contains both the semantic symbol of union, and the symbol of the triumph of the vertical, and the symbol of the temple, and the symbol of the proportion that built this image)".

Modulor Le Corbusier. Drawing by Le Corbusier. "The modulor is a measuring device based on human height and mathematics" (Le Corbusier)

We can only add that the village of Kolomenskoye has long been a part of modern Moscow, and for those who do not know this, we recommend getting off at the metro station of the same name and seeing for yourself the genius of unknown Russian masters. Well, those who are familiar with the Temple of the Ascension may now want to look at it with different eyes, to see in it not only the bizarre play of the artist's imagination, but also the wise calculation of the master's sophisticated mind.

Since we are talking about the subway, we will finally move to the modern XX century. The time for the search for proportions has not sunk into oblivion today, on the contrary, according to Le Corbusier, it has only just arrived.

We have already noted (p. 220) that anthropometric measures, due to their origin, turned out to be the best suited for constructing an architectural environment. We have just seen that anthropometric measures contained remarkable proportions, which allowed the ancient masters to create beautiful architectural monuments.

On April 7, 1795, the metric system of measures was introduced in France, in the development of which such prominent scientists as Laplace, Monge, Condorcet participated. Per unit length - meter- 1/10,000,000 part of 1/4 of the length of the Paris geographic meridian was adopted. The metric system had undeniable advantages and was increasingly pushing the boundaries of its existence. However, the meter was in no way associated with man, and, according to Le Corbusier, this had the most serious consequences for architecture^ "Taking part in the construction of huts, houses, temples designed for human needs, the meter, apparently, introduced into they are strange and alien units of measurement and, if we look at it closer, it can be accused of disorienting modern architecture and distorting it ... Architecture built on metric measurements has gone astray.

But the main reason that pushed the architects of the 20th century to search for new measurement systems in architecture was, after all, not in the shortcomings of the metric system of measures. English architecture continued to use feet and inches consistently, but it also had the same problems. The fact was that along with the 20th century, unprecedented volumes and rates of construction came to architecture. The design of the architectural environment has become predominantly typical, and the architecture itself has become industrial. Under these conditions, building elements had to be standardized and unified. In addition, architects would like to reconcile the irreconcilable: beauty and standard. It was necessary to find such proportioning methods that would have maximum flexibility, simplicity and versatility. "If there were some kind of linear meter like music notation systems, wouldn't a number of building problems be alleviated?" Le Corbusier asked. And in 1949, he himself answers this question by proposing a system of modular unification, modulor, as such a meter.

The idea of ​​building a modulor is ingeniously simple. Modulor is a golden ratio series (15.2):

(17.1)

multiplied by two factors. The first coefficient k 1 is equal to the height of a person; multiplying (17.1) by k 1 , Corbusier obtains the so-called red series. The second coefficient k 2 is equal to the distance from the ground to the end of a person's raised arm (this is a large sazhen in the old Russian system of measures) - When (17.1) is multiplied by k 2, a blue row is obtained. It remains only to choose the numerical values ​​of the coefficients. Wishing to reconcile the English and French systems of measures in a fashion, and also following the ancient tradition, according to which a person's height is 6 feet, Corbusier took 6 English feet as k 1, i.e. k 1 \u003d 6 * 30.48 \u003d 182, 88 cm. The value of k 2 is taken equal to 226.0 cm. This is how the red row was obtained:

(17.2)

and the blue row:

(17.3)

The value of k 2 was also chosen so that there is a simple relationship between the red and blue rows:

(17.4)

Therefore, the blue row is actually a doubling of the red row.

Being geometric progressions, the members of both rows of the modulor form a chain of equal relations: a n + 1: a n = b n + 1: b n = Φ, i.e., the principle of harmony is embodied in the modulor: "from everything - one, from one - everything ". Thanks to the additive property of the golden section, the "parts" of the modulor converge into a "whole". Finally, the absolute values ​​of the modulor scales come from humans and are therefore well adapted to the design of the architectural environment. So, according to the author, the modulor brings order, a standard to production and at the same time binds all its elements with the laws of harmony.

Le Corbusier. "Radiant House" in Marseille. 1947-1952 (a). These two antipodes in the work of the great architect, two different philosophies in architecture are linked together by a range of architectural proportions - modulor

However, "chasing two hares" (the desire to have good numbers in both meters and feet) resulted in a serious drawback: the size of the modulor turned out to be disproportionate to the average height of a person. Modulor has not received wide distribution. But the ideas of standard and harmony inherent in the modulor do not cease to excite architects. The eternal search for perfect harmony continues. Recently, the Soviet architect Ya. D. Glikin developed universal system of proportionality, which, as the author shows, incorporates all the proportioning systems known so far: the systems of triangulation on the Egyptian and on the equilateral triangle; systems of Vitruvius, Alberti, Hambridge, Messel, Shevelev; system of ancient Russian measures and modulor Le Corbusier.

What unites all systems of proportionality? The fact is that any proportional system is the basis, the skeleton of an architectural structure, this is the scale, or rather, the mode in which architectural music will sound. It was this property of the modulor that Le Corbusier had in mind Albert Einstein, giving him an enthusiastic assessment: "Modulor is a scale of proportions that makes bad things difficult and good things easy." But the gamma is not yet a melody, not music. Corbusier himself was well aware of this: "Modulor is a scale. A musician has a scale and creates music according to his abilities - banal or beautiful." Indeed, just as the scale has been enabling the composer for the third millennium to create an infinite variety of melodies, so the system of proportioning - modulor - does not in the least constrain the architect's work. Myself

Corbusier brilliantly proved this by building with the help of his modulor both the famous "Radiant House" in Marseilles, and the no less famous chapel in Ronchamps. These two works of the great architect are two antipodes, two different philosophies in architecture. On the one hand, the embodiment of common sense, clear, straightforward and rational. On the other - something irrational, plastic, sculptural, fabulous. The only thing that unites these two outstanding monuments of architecture is the modulor, an architectural scale of proportions common to both works by Le Corbusier.

But why did the great Einstein compare the system of proportioning in architecture - the modulor - with the musical scale? Why does his great compatriot Goethe call architecture music that has ceased to sound? What do architecture and music have in common? This will be the last question we will try to answer in this part of the book.

Ancient Greek temples, like the buildings of Le Corbusier, were built according to the proportions of the human body. However, in both cases, harmony was understood only as mathematical variations on the theme of square roots (Parthenon) and the golden section.

Modulor Le Corbusier.

This is a measuring scale (a system of harmonic quantities) created by Le Corbusier in the 1940s as a tool for the proportional construction of architectural forms.

The modulor scale is based on the proportions of the human body and mathematical calculations. They are the initial dimensions for construction, allowing you to place architectural elements in proportion to the human figure. On the one hand, according to a person with a raised hand, the points of the occupied space are determined: the leg - the solar plexus, the solar plexus - the head, the head - the tip of the fingers of the raised hand - three intervals (triad), which determine the golden section series, called the Fibonacci series. On the other hand, a simple square is created, its doubling and two golden ratios.

The objects of construction are very different receptacles of a person or an extension of his gestures (for example, a car, furniture, a book). Modulor helps to choose the most optimal dimensions of the object and its components, corresponding to the height and proportions of a person. The modulor is built on the base of the tallest person at 6 feet (182.88 cm), as new construction projects measured with the modulor are designed to accommodate people of varying heights.

Modulor components include: a 226 cm (89 inch) ruler, a measuring chart with two series (red and blue) for calculating buildings up to 400 m in height, and a manual for its use.

Description of the modulator:

1) A scale of three intervals: 113, 70, 43 (cm), which are consistent with φ (golden section) and next

Fibonacci: 43+70=113, or 113-70=43. In sum, they give 113+70=183; 113+70+43=226. Due to the equality of the larger element of the triad to the sum of the other two - and this is its meaning - it restores dualism (duality of meaning) and symmetrical division, which it contradicted.

2) Three points of the human figure plus the fourth point - the fulcrum of the lowered hand equal to 86 cm (ratio 140-86) determine the space occupied by him.

Modulor forms a double series of numbers - red and blue. The elements of the triad are the solar plexus, the head, the end of the fingers of the raised hand. The elements of dualism are the solar plexus, the end of the fingers of a raised hand, that is, in both cases, an unlimited possibility of measurements: according to the principle of the triad in the red series of the modulor and dualism in the blue. Size 113 defines the golden ratio of 70, showing the beginning of the first, red series. Size 226 (113x2 - doubling) determines the golden ratio 140-86, showing the beginning of the blue series.
Having improved his modulor in 1950, Le Corbusier used it in the design of his buildings, building them taking into account the proportions of the human body.

Fibonacci numbers (Fibonacci sequence) 1, 1, 2, 3, 5, 8,...(a0 = 1, a1 = 1,..., an+2 = an+1 + an) are defined by recurrence relations its main property is that each subsequent term is equal to the sum of the previous two. If we try to calculate the ratios of neighboring numbers, then each time we will get an infinite fraction, tending to the golden number in the limit (the larger the value, the closer to the desired 1.618 ... or 0.618 ... depending on whether we divide the larger to less or less to more). Later, Kepler and Newton proved that the ratios of the Fibonacci number series determine the radii and periods of revolution of the planets around the Sun, the laws of celestial and terrestrial mechanics.

The Five Points of Architecture by Le Corbusier.

Le Corbusier's "Five Starting Points of Architecture" were published in the journal "L" Esprit Nouveau "in the twenties. In these seemingly simple rules, Corbusier tried to formulate his concept of modern architecture.

5 rules of Le Corbusier:
1. Free-standing support, "pilotis" (pylon);
2. Free interior layout;
3. Walls independent of the frame;
4. Hinged facade, wide windows;
5. Flat roof garden.

And one day arose from a dream, From this soul praying, Like grass, like water, like birches, A marvelous marvel in the Russian wilderness.

N. Rubtsov

It's time to look for proportions. The spirit of architecture is affirmed.

Le Corbusier

In 1784, the humble father of the Bogolyubov monastic brethren asked His Eminence Victor, Archpastor of Vladimir, for permission to dismantle the dilapidated and half-abandoned church for the monastic needs. The permission was graciously granted, but, as they say, life had its own way: customers and contractors did not agree on the price. The work did not begin, and there they were completely forgotten. So, by the will of fate, the monument remained alive, which was bypassed by the hordes of Batu and Mamai, spared centuries and conflagrations of endless wars, a masterpiece of ancient Russian architecture, the Church of the Intercession of the Virgin on the Nerl.

On clear summer days, among the greenery of flood meadows, its slender whiteness, reflected by the smooth surface of the old Klyazma, breathes the poetry of a fairy tale. Only in the short minutes of sunset the white candle of the church lights up with an alarming crimson flame. In severe winters, an endless snow veil, like a caring mother, wraps and hides her frozen child. "In all Russian poetry, which has given the world so many unsurpassed masterpieces, there is perhaps no monument more lyrical than the Church of the Intercession on the Nerl, for this architectural monument is perceived as a poem etched in stone. A poem of Russian nature, quiet sadness and contemplation" (L . Lyubimov).

Before approaching the mystery of the charm of ancient Russian architecture, we need to get acquainted with the system of measures that existed in Ancient Russia. We have already noted (p. 198) that in different places on the globe, at different times and among different peoples, the standards of length were in principle the same: they somehow came from the human body. These so-called anthropometric measures possessed the most valuable quality for architecture, which was forgotten with the introduction of the metric system of measures, but to which Le Corbusier returned in the 20th century. The fact is that anthropometric measures due to their origin, they are commensurate with a person and therefore convenient for constructing an artificial human habitat - architectural structures. Moreover, in "human" measures there are proportions selected by nature itself, such as halving, the golden section, the function of the golden section. Consequently, the harmony of nature is naturally embedded in anthropometric measures.

The main building measure in ancient Russia was a sazhen, equal to the span of the arms to the sides. The sazhen was divided into 2 half fathoms, half a sazhen - by 2 elbow- distance from fingertips to elbow, elbow - by 2 spans- the distance between the thumb and little finger extended in opposite directions. Everything is clear and logical. However, the more closely historians studied the ancient Russian chronicles, the more fathoms became, and when their number exceeded ten, the historians' head went round. It became necessary to restore mathematical order in the old Russian system of measures. This was done by the historian, academician B. A. Rybakov and the architect I. Sh. Shevelev. The beginning of anthropometric measures is given by the height of a person a. The main of all types of fathoms is measured, or flyweight, sazhen C m, which is equal to the span of a person's arms to the sides. The study of the proportions of the human body shows that C m = 1.03a. Another important measure among all peoples was the double step, which is equal to the height of the body from the feet to the base of the neck. The last distance, as we know (p. 220), is 5/6 AU. In this way, double step, or small(Tmutarakan) sazhen, C t \u003d 5/6 a \u003d 0.833a. But the main surprise lies in relation to these two main dimensions:

Therefore, the small sazhen C t refers to the measured C m as the side of a double square to its diagonal without a small side:

From (17.1) it is clear that the ratio of the measured half-sazhen C m / 2 to the small sazhen C t is equal to the golden ratio:

(17.2)

So, in the ratio of the half-span of the arms (RS) to the height of the body (LQ), established by nature itself, that is, in relation to the two main measures of Ancient Russia, the golden section is concluded, which is so common in ancient Russian architecture.

Man's height: a = AB

measuring sazhen: C n \u003d AC \u003d CN \u003d 1.03a

Small (Tmutarakan) sazhen:

Fathom without a quarter:

Oblique Novgorod sazhen:

Oblique great fathom:

Relations between fathoms:

golden ratio

golden ratio function

Having built squares on small C t and measured C m fathoms and drawing diagonals in them, we get two more types of fathoms: oblique Novgorod sazhenand great oblique fathom. In contrast to the first two sazhens (small and measured), expressing natural measures, oblique sazhens were obtained in a purely geometric way. It's clear that

(17.3)

Finally, there was another sazhen obtained geometrically. This so-called fathom without a quarter C h, equal to the diagonal AM of half the square built on the measured sazhen C m. This sazhen did not have a corresponding oblique pair, and therefore it was called a sazhen without a pair, without a couple, or without a four. It follows from the ACM triangle that , where

(17.4)

i.e. the ratio of a sazhen without a quarter C h to a measured sazhen C m is equal to the function of the golden section (see p. 219).

These are just the main types of sazhens that existed in ancient Russian metrology. The Novgorod measuring stick, found in 1970 (see p. 219), made it possible to clarify their dimensions. Novgorod measures of the 12th century correspond to the height of a person: a = 170.5 cm. Then C m = 175.6 cm, C t = 142.1 cm, K n = 200.9 cm, K v = 248.3 cm, C h \u003d 196.3 cm. If the height of a person is taken equal to 6 Greek feet: a \u003d 6 * 30.87 \u003d 185.22 cm, then for the main fathoms (measured and small) we get the values: C m \u003d 190.8 cm and C m = 154.3 cm. It is these measures that are most often found in ancient Russian churches of the 11th century, the construction of which, apparently, was carried out by Byzantine masters. So, together with Christianity, Russia inherited the Byzantine system of measures, which, in turn, grew up on the ancient Mediterranean culture. The absolute sizes of sazhens in Russia fluctuated greatly over time until the introduction of the metric system of measures in 1918. But it is important that proportional relationships between paired sazhens were preserved. These proportions became the proportions of architectural structures.

The fact that the measures were taken by ancient Russian builders in pairs is evidenced, for example, by a Novgorod letter of the 16th century, which describes the size of the St. Sophia Church in Novgorod in this way: "and inside the chapter, where the windows are, there are 12 sazhens, and from the Spasov image from the forehead to the church bridge - 15 sazhens measured". (Measurements show that the mentioned sazhens correlate as: 2.) The Novgorod measured cane also speaks of the use of paired measures, in which a small sazhen С t was used either in tandem with a measured sazhen С m (С t: С m = 1:( - 1 )), or with oblique Novgorod K n (C t: K n \u003d 1: √ 2). If measured half-sazhens were taken on a Novgorod cane paired with a small sazhen, then this pair gave the golden ratio (C m / 2: C t \u003d φ). So, the beauty of the proportions of ancient Russian architecture lies in the very system of ancient Russian measures, which gives such important proportions as the golden section, the function of the golden section, the ratio of the double square.

But in addition to all these proportions, which from nature itself passed into the system of measures, and then into architectural monuments, the ancient Russian masters had one more secret. It was this secret that made it possible to give each ancient building a unique charm, "nuance", as the architects say. This secret is revealed in the in-line record of the carpenter Fyodor for the construction of the wooden church of the Ust-Kuluysky churchyard (end of the 17th century), where it is said: beauty says...

"As measure and beauty say..." This wonderful formula of an obscure Russian carpenter expresses the essence of the dialectic of the interaction of rational (measure) and sensual (beauty) principles in achieving beauty, the union of mathematics (measure) and art (beauty) in the creation of architectural monuments.

Let us finally turn to an analysis of the proportions of the Church of the Intercession on the Nerl. This architectural masterpiece means as much to a Russian person as the Parthenon does to a Greek. Therefore, it is not surprising that the proportional structure of a small church was analyzed by many researchers and each of them tried to give his own "final" clue to the mystery of her charm. Let us briefly consider the proportions of the Church of the Intercession on the Nerl from two points of view.

According to the architect Shevelev, the proportional structure of the Church of the Intercession is based on the ratio of a sazhen without a quarter to a measured sazhen, which is a function of the golden section (C h: C m = √5: 2), and the plan of the church itself was built as follows. First, a rectangle 3 fathoms long without a quarter and 3 measured fathoms wide was marked out, which outlined the pillars carrying the drum and vaults. Since 3C h: 3C m = √5:2 = 1.118, then the sides of this rectangle refer to the function of the golden section, and the rectangle itself is almost a square, or, in Zholtovsky's terminology, a "living square". Having drawn diagonals in the original rectangle, the architect received the center of the temple, and having set aside 1 measured sazhen on the diagonals from the tops to the center, the domed rectangle and the dimensions of the supporting pillars. So the core of the plan was built, which determined all further horizontal and vertical dimensions of the structure. The measured sazhen of the builders of the Church of the Intercession was C m = 1.79 m.

Having measured from the Center of the temple to the east 3C m and to the west 3C h, the master received the length of the outer rectangle, equal to . And putting this size in measured sazhens, its width is 5 3/4 cm. Thus, the outer rectangle of the plan of the church is similar to the core of the plan and is also a "living square". The diagonal of the rectangle under the dome determined the diameter of the central apse (under the dome of the altar ledge) and the diameter of the drum of the temple. The short side of the domed rectangle set the diameters of the side apsides.

Finally, the height of the base of the temple - the quadrangle, read by the height of thin columns - is equal to twice the length of the core of the plan, i.e. 2 * 3C h \u003d 6C h, and the height of the drum with a helmet-shaped dome * is twice the width of the core, i.e. 2 *3С m = 6С m. Thus, the main vertical dimensions of the temple - the height of the base and the height of the completion - also relate to the function of the golden section. The quadrangle itself is "almost a cube", the base of which is "almost a square", and the height is almost equal to the sides of the base. So, in the construction of the quadrangle of the temple, the principle of approximate symmetry is clearly visible, which is so often found in nature and art (see Chapter 4). You can also point to smaller divisions of the temple, related to the function of the golden section, that is, in relation to a sazhen without a quarter to a measured sazhen. For example, the stone belt crowning the columnar frieze, which covers the entire church and is its important architectural detail, divides the height of the quadrangle as a function of the golden section.

* (Initially, the Church of the Intercession had a helmet-shaped dome, characteristic of ancient Russian churches, resembling a warrior's helmet. In the 17th century, the helmet-shaped dome was converted into a bulbous one, which we see today.)

Let us now consider the ichnography of the Church of the Intercession on the Nerl, as seen by the connoisseur of ancient Russian architecture K. N. Afanasyev. According to Vitruvius, "ichnography is the proper and consistent use of a compass and straightedge to obtain the outlines of a plan." According to Afanasiev, the initial size of the Church of the Intercession is the smaller side of the domed rectangle, equal to 10 Greek feet: a = 10 Greek. foot. \u003d 308.7 cm. Then the large side of the domed rectangle is obtained as the diagonal of a double square with side a / 2. Thus, the rectangle under the dome is a "living square", the sides of which are related in the golden section function. The thickness of the pillars is determined by the ratio of the golden section to the modulus a/2. Further constructions are clear from the figure. This is how the core of the plan is built. The remaining dimensions of the plan are obtained by similar constructions, relying mainly on the module a/2.

Note that together with the function of the golden section, the law of the golden section also determines the proportional structure of the Church of the Intercession. This is not surprising, because these relations are connected by the geometry of a double square. As Afanasiev established, the main verticals of the temple, which determine its silhouette, are primarily subject to the law of the golden section: the height of the base, equal to the height of the thin columns of the quadrangle, and the height of the drum. The diameter of the drum is related to its height also in the golden ratio. These proportions are visible from any point of view. Moving on to the western facade, the series of the golden section can be continued: the shoulders of the temple refer to the diameter of the drum in the golden ratio. So, taking the height of the white stone part of the church (from the base to the dome) as a unit, we get a number of the golden section: 1, φ, φ 2, φ 3, φ 4, which determines the silhouette of an architectural structure. This series can be continued in smaller details. (Of course, the western facade from the point of view of the golden ratio is no exception and is taken by us only as an example.)

Let's sum up some results. We see that the seemingly incomprehensible harmony of the Church of the Intercession is subject to mathematically strict laws of proportionality. The plan of the church is built on the proportions of the golden section function - "living squares", and its silhouette is determined by the number of the golden section. This chain of mathematical patterns becomes a magical melody of interconnected architectural forms. Of course, the laws of proportionality determine only the "skeleton" of the structure, which must be correct and proportionate, like the skeleton of a healthy person. But besides the mathematical laws of measure, in the bowels of an architectural masterpiece, there are also unfamiliar laws of beauty: "as measure and beauty say ..."! It is the dialectics of the interaction between the laws of measure and the laws of beauty, which often manifest themselves in deviations from the laws of measure, that creates a unique image of an architectural masterpiece.

Note that from the point of view of geometry, the reconstructions of the proportional structure of the Church of the Intercession considered by us are similar. They are consistent with each other and give in the plan three "living squares" inscribed in each other, the ratio of the sides of which is √5:2 determines the entire proportional structure of the temple. However, from the point of view of the history of architecture, these reconstructions differ fundamentally. The first of them is based on the old Russian system of measures and, therefore, suggests that the Church of the Intercession was built by Russian architects. The second one has a Greek measure as the main size and therefore gives reason to believe that the church was built by craftsmen invited from Byzantium ... Who and how created the pearl of Russian architecture? Perhaps we will find out the answer to this question...

The Church of the Intercession was built in 1165. And 73 years later, she witnessed an unprecedented misfortune in the history of Russia: the hordes of Batu, having turned Ryazan, Kolomna and Moscow into ashes, laid siege to Vladimir. The Russian state, tormented by princely strife, was dealt a mortal blow, from which Russia was able to fully recover only 200 years later, by the end of the 15th century.

In 1530, in the royal estate - the village of Kolomenskoye near Moscow - the future tsar of the awakening Russia, Ivan the Terrible, was born. And two years later, here, in Kolomenskoye, on the steep bank of the Moskva River, the construction of a church was completed, erected in memory of this event. The architects seemed to foresee the birth of an unprecedentedly formidable king: the church was also unprecedented. Everything in it ", and the height (almost 62 m), and the stone tent, and the shape directed upwards - was unprecedented. The new temple seemed to symbolize Russia's breakthrough into a future free from the Tatar yoke. "... But that church is very wonderful in height and beauty and lordship, such has not happened before in Russia, "the chronicler wrote about her. The entire proportional structure of the church, all its unrestrained aspiration upwards corresponded to the name - the temple of the Ascension.

But for us, the Ascension Church is also interesting because it is not only the anthem of Russia spreading its wings, but also the architectural anthem of geometry.

None of the considered architectural masterpieces, including the Parthenon, is so permeated with geometry, so simple and concise in its dimensional structure, as the Church of the Ascension in Kolomenskoye. The proportions of the temple with the utmost clarity are determined by two pair measures: horizontal - small (Tmutarakan) sazhen C t and oblique Novgorod sazhen K n (C t: K n \u003d 1: √ 2), vertical - small sazhen C t and measured sazhen C m ( C t: C m = 1: (√5 - 1)) and their combination C m: 2C t = (√ 5 - 1): 2 = φ, giving the golden ratio. Thus, the Church of the Ascension is also an excellent example of the use by Moscow masters of a measuring tool such as the Novgorod measuring stick, created, as we remember, to work with these two pairs of measures (see p. 220). Consider the proportional analysis of the temple, made by the architect Shevelev.

The plan of the Church of the Ascension is based on the square ABCD with a side of 10 small sazhens: a = AB = 10С t. It is clear that the diagonals of the square are 10 oblique Novgorod sazhens: AC = BD = 10√2ST = 10K n. So, with the help of paired measures C t and K n, the correctness of the construction of the initial square was monitored. The circle of radius R = 5K n, describing the square, determines the position of all 12 outer corners of the plan of the temple. By inscribing a new square through the midpoints of the sides in the square ABCD and making constructions, we get the outer contour of the plan - 20- square. The parts protruding above the original square are called vestibules, their width is equal to a / 2 = 5С m. Expressing the radius of the circumscribed circle R in measured fathoms and putting this value in small fathoms, the builders received the side of the square b, which determines the internal space of the temple:

Of course, the Kolomna craftsmen did not calculate any radicals! They simply applied the measuring stick on different sides and automatically switched from one measure to another. The plan of the church is built. And we will also express the side of the square c, covering the vestibules: c \u003d √7 / 2 a (the triangle from which c / 2 is located is not shown in the drawing so as not to spoil the beauty of the central symmetry of the plan; find it). Knowing a, b, c, it is easy to express all other dimensions of the plan and the relationships between them.

Let's move on to the volumes and vertical divisions of the temple. The Church of the Ascension is surrounded on all sides by a covered gallery, raised above ground level and called promenade. The ambush was made at the level of the ceiling basement- semi-basement used for business purposes. The entrance to the church was arranged from the graveyard, to which three porches lead in the Ascension Church, and, thus, the vertical dimensions of the church with the graveyard are perceived from the level of the latter.

The main volume of the temple is a 20-sided prism placed on the basement. Its height is equal to the side of the original square a. Thus, the core of the main volume is a cube - a quadrangle a × a × a (a = 10С t), decorated with narthex faces. Together with the basement, the height of the 20-sided prism is equal to the diagonal of the original square a√2 = 10√2C m = 10K n. So, the side and diagonal of the original square (the core of the plan) completely determine the vertical dimensions of the main volume (the core of the base).

The twenty-sided prism of the main volume passes through the intricate belt of kokoshniks into an octahedral prism - octagon. The octagon is also inscribed in the cube d×d×d(d = 9C t). Then the octagon passes into an octahedral tent, the height of which is h = d√2 = 9√2С t = 9K n, i.e. the tent is inscribed in a rectangular parallelepiped 9С t × 9С t × 9К n. The area of ​​the upper section of the tent is reduced by 16 times, and its linear dimensions - by 4 times. Since 1/4 sazhen is equal to a cubit, therefore, the upper section is inscribed in a square where L t is a small (Tmutarakan) cubit (4L t \u003d C t). Finally, through the crowning cornice, the tent ends with an octagonal drum, the section of which exceeds the upper section of the tent by a small half-cubit. The drum hangs slightly over the tent and is inscribed in a cube f × f × f (f = 9.5L t), and together with the dome, taken without an apple (see the figure on p. 242), the drum is inscribed in a rectangular parallelepiped f × f ×√2f.

So, we see how the side of the core of plan a, measured either by a small sazhen or oblique Novgorod, gives rise to all the main verticals of the temple. Note that the total height of the church from the top of the plinth to the apple on which the cross stands is 4a = 40C m, i.e., it is also expressed in the simplest way in terms of the original size a. And one more important relation. The belt of kokoshniks, through which the quadrangle of the base passes the octagon of the tent, divides the temple into two parts - the base and the completion. The height of the base h 1 ≈14C t, and the height of the completion h 2 ≈14K n, whence h 1:h 2 = C t:K n = 1:√2, i.e. the main vertical divisions of the temple are also referred to as small and oblique Novgorod fathoms.

But the proportions of the Temple of the Ascension are determined not by one, but by two mathematical laws. In addition to the proportion C t: K n \u003d 1: √ 2, which determines the foundation, the static beginning of the temple, there is another theme in it - the theme of upward development, ascension, which is determined by the proportional chain: C t: C m = 1: (√ 5 - 1), as well as the proportion of the golden section: C m: 2C t \u003d φ. In carrying out this theme, the principle of the oncoming movement of proportions, familiar to us from the Parthenon, is observed. Two different proportional circuits are superimposed on each other, collide and oppose. This clash of two opposing principles - horizontal and vertical - is the architectural image of the Church of the Ascension. Without dwelling on the mathematical analysis of these two systems, let's give the floor to the author of an excellent aesthetic analysis of the Church of the Ascension, art critic A. Cires. “In the image of this church,” writes Tsires, “two main leitmotifs are intertwined: the motif of sharp, full of clashes and dissonances of dynamism and the motif of harmoniously calm beauty ... The complex rhythm of the arches of the lower galleries ... goes, becoming more frequent from the edges to the center, .. . pushes the arches from the edges to the corners of the main body of the church and to its middle, ... suggests a change in horizontal movement with an upward movement ... So from the bottom up there is a gradual softening of crystallism and an increase in the compactness of the volume, up to its tightness into a strong knot crowning the entire voluminous composition of the head.

But we would like to end the conversation about the proportions of the Church of the Ascension in Kolomenskoye with the words of the author of the mathematical analysis of its proportions, Shevelev. “Let us emphasize the most expressive detail of the dimensional structure, which most clearly shows the peculiarity of the logic of the ancient master, who strives to express the main thing in metrology with particular precision. cross (10С t Х10С t Х10С t - quadruple; 10С t Х10С t Х10К n - prism of the quadrangle; 10L t Х10Л t - the proportionality of the cross, because for the architect it contains both the semantic symbol of union, and the symbol of the triumph of the vertical, and the symbol of the temple, and the symbol of the proportion that built this image)".

Modulor Le Corbusier. Drawing by Le Corbusier. "The modulor is a measuring device based on human height and mathematics" (Le Corbusier)

We can only add that the village of Kolomenskoye has long been a part of modern Moscow, and for those who do not know this, we recommend getting off at the metro station of the same name and seeing for yourself the genius of unknown Russian masters. Well, those who are familiar with the Temple of the Ascension may now want to look at it with different eyes, to see in it not only the bizarre play of the artist's imagination, but also the wise calculation of the master's sophisticated mind.

Since we are talking about the subway, we will finally move to the modern XX century. The time for the search for proportions has not sunk into oblivion today, on the contrary, according to Le Corbusier, it has only just arrived.

We have already noted (p. 220) that anthropometric measures, due to their origin, turned out to be the best suited for constructing an architectural environment. We have just seen that anthropometric measures contained remarkable proportions, which allowed the ancient masters to create beautiful architectural monuments.

On April 7, 1795, the metric system of measures was introduced in France, in the development of which such prominent scientists as Laplace, Monge, Condorcet participated. Per unit length - meter- 1/10,000,000 part of 1/4 of the length of the Paris geographic meridian was adopted. The metric system had undeniable advantages and was increasingly pushing the boundaries of its existence. However, the meter was in no way associated with man, and, according to Le Corbusier, this had the most serious consequences for architecture^ "Taking part in the construction of huts, houses, temples designed for human needs, the meter, apparently, introduced into they are strange and alien units of measurement and, if we look at it closer, it can be accused of disorienting modern architecture and distorting it ... Architecture built on metric measurements has gone astray.

But the main reason that pushed the architects of the 20th century to search for new measurement systems in architecture was, after all, not in the shortcomings of the metric system of measures. English architecture continued to use feet and inches consistently, but it also had the same problems. The fact was that along with the 20th century, unprecedented volumes and rates of construction came to architecture. The design of the architectural environment has become predominantly typical, and the architecture itself has become industrial. Under these conditions, building elements had to be standardized and unified. In addition, architects would like to reconcile the irreconcilable: beauty and standard. It was necessary to find such proportioning methods that would have maximum flexibility, simplicity and versatility. "If there were some kind of linear meter like music notation systems, wouldn't a number of building problems be alleviated?" Le Corbusier asked. And in 1949, he himself answers this question by proposing a system of modular unification, modulor, as such a meter.

The idea of ​​building a modulor is ingeniously simple. Modulor is a golden ratio series (15.2):

multiplied by two factors. The first coefficient k 1 is equal to the height of a person; multiplying (17.1) by k 1 , Corbusier obtains the so-called red series. The second coefficient k 2 is equal to the distance from the ground to the end of a person's raised arm (this is a large sazhen in the old Russian system of measures) - When (17.1) is multiplied by k 2, a blue row is obtained. It remains only to choose the numerical values ​​of the coefficients. Wishing to reconcile the English and French systems of measures in a fashion, and also following the ancient tradition, according to which a person's height is 6 feet, Corbusier took 6 English feet as k 1, i.e. k 1 \u003d 6 * 30.48 \u003d 182, 88 cm. The value of k 2 is taken equal to 226.0 cm. This is how the red row was obtained:

and the blue row:

The value of k 2 was also chosen so that there is a simple relationship between the red and blue rows:

Therefore, the blue row is actually a doubling of the red row.

Being geometric progressions, the members of both rows of the modulor form a chain of equal relations: a n + 1: a n = b n + 1: b n = Φ, i.e., the principle of harmony is embodied in the modulor: "from everything - one, from one - everything ". Thanks to the additive property of the golden section, the "parts" of the modulor converge into a "whole". Finally, the absolute values ​​of the modulor scales come from humans and are therefore well adapted to the design of the architectural environment. So, according to the author, the modulor brings order, a standard to production and at the same time binds all its elements with the laws of harmony.

Le Corbusier. "Radiant House" in Marseille. 1947-1952 (a). These two antipodes in the work of the great architect, two different philosophies in architecture are linked together by a range of architectural proportions - modulor

However, "chasing two hares" (the desire to have good numbers in both meters and feet) resulted in a serious drawback: the size of the modulor turned out to be disproportionate to the average height of a person. Modulor has not received wide distribution. But the ideas of standard and harmony inherent in the modulor do not cease to excite architects. The eternal search for perfect harmony continues. Recently, the Soviet architect Ya. D. Glikin developed universal system of proportionality, which, as the author shows, incorporates all the proportioning systems known so far: the systems of triangulation on the Egyptian and on the equilateral triangle; systems of Vitruvius, Alberti, Hambridge, Messel, Shevelev; system of ancient Russian measures and modulor Le Corbusier.

What unites all systems of proportionality? The fact is that any proportional system is the basis, the skeleton of an architectural structure, this is the scale, or rather, the mode in which architectural music will sound. It was this property of the modulor that Le Corbusier had in mind Albert Einstein, giving him an enthusiastic assessment: "Modulor is a scale of proportions that makes bad things difficult and good things easy." But the gamma is not yet a melody, not music. Corbusier himself was well aware of this: "Modulor is a scale. A musician has a scale and creates music according to his abilities - banal or beautiful." Indeed, just as the scale has been enabling the composer for the third millennium to create an infinite variety of melodies, so the system of proportioning - modulor - does not in the least constrain the architect's work. Myself

Corbusier brilliantly proved this by building with the help of his modulor both the famous "Radiant House" in Marseilles, and the no less famous chapel in Ronchamps. These two works of the great architect are two antipodes, two different philosophies in architecture. On the one hand, the embodiment of common sense, clear, straightforward and rational. On the other - something irrational, plastic, sculptural, fabulous. The only thing that unites these two outstanding monuments of architecture is the modulor, an architectural scale of proportions common to both works by Le Corbusier.

But why did the great Einstein compare the system of proportioning in architecture - the modulor - with the musical scale? Why does his great compatriot Goethe call architecture music that has ceased to sound? What do architecture and music have in common? This will be the last question we will try to answer in this part of the book.