Modulor le Corbusier practical application. Proportioning systems golden ratio modulor le corbusier sequence

The significance of the work of Le Corbusier, the greatest theorist and master, who embodied his innovative ideas in the language of an architect, artist and writer, the clarity of his formulations, catchy like propaganda posters, the sharpness of his compositional ideas, have long been recognized by Soviet architects. The creative path of Corbusier as a thinker and artist is marked by the transition from the slogan of constructivism - a villa in Garsha - to a complex combination of the ideas of the Marseille residential unit and the Chandigarh complex. The range of his searches covers town-planning ideas and new types of dwellings, free plans and facades of frame buildings and, finally, free plasticity of the volume of the chapel in Ronchamps - antitheses of many other works of the master. He is looking for a new interpretation of the principles of tectonics, rhythm, proportions and other patterns. architectural composition. The ideas of proportioning dimensions in architecture were not the main theme that occupied Le Corbusier, but their development accompanied the entire work of the master.

Twenty-three years old in 1910, Le Corbusier (then still a young self-taught artist Charles Edouard Jeanneret) “... painted the facade of the house that was going to be built. Before him stood a painful question that plunged him into confusion: what is the pattern that determines everything, binds everything together? ... "

Starting with this kind of doubt, known to every architect and student of architecture, Corbusier embarks on a search, the results and history of which are set forth in a published book. To understand the peculiarities of working on the Modulor, it should be emphasized that Corbusier the innovator was by no means a subversive architectural values of the past. His very formulation of the question of architecture as “the art of building houses, palaces and temples, building ships, cars, railway cars and airplanes”, as well as creating equipment for buildings, designing books and magazines (printing art) echoes Vitruvius’s broad definition of architecture. The book "Modulor" is replete with references to works of the past, measurement data architectural monuments. With regard to the system of proportions, Corbusier still somewhat underestimates history. He talks about the existence of certain rules that governed the construction of the Parthenon, temples, Gothic cathedrals, but mentions only the rules for applying measures related to the size of a person - cubit, foot, span.

It is known, however, that in the past there were developed systems of proportions in architecture. Vitruvius recorded a clear system for constructing modular proportions of ancient temples, residential buildings and even livestock buildings, geometric constructions of theaters and other structures; the masters of the Middle Ages created a proportional system of Gothic cathedrals, the theorists of the Renaissance and classicism - the canons of orders.

Historical canons have lost their significance, and therefore, according to R. Witkover, no matter how one relates to Modulor, this, of course, is the first logically generalized system created since the fall of the old systems; it also reflects a modern way of thinking and is evidence of an inseparable connection with inherited cultural values.

The Modulor book is by no means scientific treatise. It's more like a memoir by the author. fascinating story his work on proportions, intertwined with thoughts about architecture, conversations with friends and disputes with opponents. Therefore, in order to understand and appreciate main idea Modulor, it is necessary first of all to trace the main stages of its development. Search first in 1909-1910. is carried out almost by touch. Corbusier's attention "... was attracted by a picture of Michelangelo's Capitol in Rome ... Suddenly the thought dawned on him: perhaps the whole composition is subordinated to a right angle and the inscribed right angles determine the construction?" He finds confirmation of the use of geometry in art, analyzing the painting of Cezanne, studying the "History of Architecture" by Choisy. From now on, and especially since 1918, geometric construction, drawing-regulator (Le trace regulareur) accompanies all the work of the master, appearing on the facades of villas and paintings.

At the same time, the idea of ​​introducing a human scale into an abstract geometric construction is ripening - a person with a raised hand determines the height of living quarters of 2.10-2.20 m, adopted "... in all harmonious works of both folk architects and professional architects", the height of comfortable express cabins and ocean packet boats.

The techniques of geometric construction and the human scale are combined in 1943 in the task given by Corbusier to one of his assistants: “Take the figure of a man with a raised arm, 2 m 20 cm high; place it in two squares stacked on top of each other; write in these two squares the third one, which should give you the solution. The place of the top of the inscribed right angle will help you position the third square.

The first schemes made in accordance with this working hypothesis of the future Modulor by Hanning and Eliza Maillard did not yet give an exact solution. The authors place the third square along the axis of the vertex of the inscribed right angle, but shift it from the axis of the original rectangle. In fact, as Corbusier himself later admitted (in a letter to Dufour de Coderans in 1950), the vertex of a right angle divides the sides of a rectangle composed of two squares exactly in half.

The first geometrical constructions nevertheless acquired essential significance for the further development of the idea. In 1945, the dean of the faculty at the Sorbonne drew Le Corbusier's attention to the fact that these constructions lead to the widespread use of the golden section. Based on the golden ratio and the ratios of the Fibonacci series * approaching it, Corbusier and his assistants build a linear scale of proportional sizes.

* A series of numbers 1, 1, 2, 3, 5, 8, 13, 21, 34. 55, 89, 144, 233, 377 ..., each of which, starting from 2, is the sum of the two previous ones, and the ratio of the two adjacent terms gradually approaches the ratio of the golden section (named after Fibonacci, by which the 13th-century Italian mathematician Leonardo from Pisa was known).

This is how a system of proportional values ​​was born - Modulor, whose name, found in 1945, merged with the emblem - an image of an exaggerated muscular male figure with a raised hand; the human figure is accompanied by intertwining spirals of "red" and "blue" series of sizes, increasing in proportion to the golden section.

The basis of the "red row" is the conditional growth of a person. The first division, which reduces the original value in the golden section, determines the side of the square, the doubling of which corresponds to the height of a person with a raised hand and gives rise to the "blue row" of sizes.

The conditional height of a person, originally taken at 175 cm, was then increased to 182.8 cm = 6 feet, which made it possible to express all divisions of the Modulor both in centimeters and in inches. The height of the figure with the arm raised was 226 cm (89 inches).

The final results are summarized in a table (p. 66), which shows that the values ​​of the blue series, for example ... 3.66; 2.26; 1.40; 0.86; 0.53 m ..., are by construction a doubling of the corresponding values ​​of the red series: . .. 1.83; 1.13; 0.70; 0.43; 0.27 m...

Some other patterns inherent in the Modulor numbers were identified by the mathematician Andreas Speizer and the engineer Krussar, who indicated that the value of each member of the red series is the average between two adjacent members of the blue series, of which one is greater and the second is less than it. So, Modulor, the development of which began with a geometric study, received an exact numerical expression.

The system found corrects the initial approximate geometric constructions, a new, elegant and this time accurate interpretation of which was created in 1948 by the young assistants of Corbusier, the architects Serralta and Meissonier. The task formulated in 1943 was completed, but with some amendments. The initial size, equal to the height of a person with a raised arm, is taken to be 2.26 m instead of 2.20 m. It corresponds to a rectangle made up of two equal squares with sides of 1.13 m. An inscribed right angle divides the rectangle in half. The third square is not located along the axis of the right angle, but is shifted down so that its height is articulated by the top of the right angle in the golden section.

The same right angle cuts off on the sides of the third square the points through which the oblique straight line is drawn, which determines by constructing a series of similar right triangles all values ​​of the red and blue rows. As a result, the geometric construction and the numerical regularity of the "face and back of the carpet", in the words of Crussard, merged into one.

Corbusier was by no means a mathematician and scrupulously mentions the creative contribution of everyone who participated in the mathematical side of the work on the Modulor or helped with their advice. Speaking about the fact that at school he solved problems badly, with disgust, Corbusier writes: “... every day, for all my naivety, I became more and more convinced that my art is subject to certain laws. I gladly acknowledged the existence of these rules, began to treat them more respectfully ... "

The head of the first part of Modulor-1 " Mathematical Foundations” sounds like a hymn: “Mathematics is the main tool created by man for knowing the universe ... the divine world, where the keys to knowing the greatness of the universe are stored. These doors lead to a world of wonders... He ended up in a world of numbers... the brightness of the light is almost unbearable...”

Sometimes Corbusier gets carried away, suggesting, for example, just like J. Deyer, to use the Modulor to express cosmic distances and magnitudes of the microworld, but usually his sense of proportion does not fail him and he is aware of the danger of abusing mathematical calculations.

Without deifying mathematics, but emphasizing the importance of mathematical patterns in solving specific practical problems, Corbusier says that we are talking only about “a tool called Modulor, lying on the drawing table next to a pencil, T-square and square” and “Modulor is a working tool, a whole range of numerical dimensions that can be used to design ... mass-produced products industrial production, as well as to ensure the unity of large architectural structures". The first major experiment on the use of Modulor was carried out by Corbusier in 1946-1950. in the process of designing and building a Marseille house-complex (“Marseille residential unit”). Column grid, room width, built-in equipment, even complex composition volumes on the flat roof of a Marseille house are calculated according to the Modulor. But most of all, the proportions of the Modulor are felt in the composition of the facade, directly perceived by the eye.

The column pitch of 419 cm is made up of two dimensions along the blue row - a clear distance of 366 cm and a structure thickness of 53 cm. The height of the rooms is 226 cm and the ceiling thickness of 33 cm (floor height 259 cm) also corresponds to the blue row. From these dimensions given by the author, it follows that the main grid of the facade, both in cleanliness between the structures 366 x 226 cm, and in the axes 419 x 259, corresponds to the proportion of the golden section. The main grid gets additional divisions vertically, it would seem complex rhythm which is achieved only by three sizes along the red row; one of these dimensions, equal to half the floor height of 113 cm, is divided in the golden section into 70 and 43 cm.

As a result, a characteristic system arises - a kind of "order" of Corbusier, which then varies in a residential building built in Nantes and in some other projects.

The rhythm of division of the plane covered by the gaze is, perhaps, the most striking manifestation of the possibilities of the Modulor as a tool for harmonizing proportions. This applies to the facade of the Marseille house and to the composition of the wall-fences in the lobbies and halls, composed of various combinations of five types of modular elements, as well as to the planes of glazing and decorative carpet patterns on the walls of the Palace of Justice in Chandigarh. This is where the “play of planar panels” (Le jeux de panneaux), explained at the end of the first book “Modulor-1”, takes place, folding a mosaic of modular elements. Tables of drawings arranged by Corbusier and employees of his workshop on the street. Sevres, 35, show an almost infinite variety of possibilities for filling planes with the selection of various combinations of elements according to the Modulor, and then with various permutations of the selected elements. The game unfolds in breadth by varying the shape and size of the planes to be filled, each of which can give rise to new series of combinations and permutations. This is followed by equally wide possibilities for varying textures and colors. The play of planar panels is noticeable at first glance at the facades, stained-glass windows, decorative panels of Corbusier, in which, for all their orderliness, one feels dynamism, a departure from the simple multiplicity of divisions characteristic of Modulor.

TO best examples include wall carpets of the Palace of Justice in Chandigarh, composed of panels of three main sizes 140 cm wide and 226 cm high (including the bottom row, equal to the height of the door); 333 and 419 cm - upstairs for the Supreme Courtroom *. From one edge, when laying out the typical dimensions of the panel, a discrepancy is formed, which Corbusier, freely introducing the necessary adjustments, compensates with additional elements with a width not according to the Modulor.

* All dimensions are calculated according to the blue Modulor series, but two of them are derivatives and consist of two values: 419 cm = 366 cm + 53 cm (i.e. the height of the rooms surrounding the hall, in total with the thickness of the ceilings) and 333 cm = 366 cm - 33 cm.

The composition of the pattern of carpets is to a certain extent subordinated to their structure, but has considerable freedom. The boundaries of the color planes sometimes coincide with the division of the plane into elements - panels, and sometimes they cut them, divide them again based on the Modulor numbers. The details of the drawing are complemented by square and rectangular spots - “dots” and symbolic drawings.

The carpets are only a detail, but they reflect some of the general features of the proportional constructions incorporated in the design of Chandigarh. Describing the proportional system adopted by him, Le Corbusier formulates three concepts corresponding to the methods he uses - "arithmetic", "structural" and "geometric". The first of them means the repetition of identical quantities, i.e., the presence of simple multiple modular relations; the second is Modulor relations related to structure and size human body; the third is geometric constructions.

With these definitions, followed by a consideration of proportional constructions in the planning and development project of Chandigarh, Corbusier shows that his creative method is by no means limited to the use of Modulor, but also implies the presence of other numerical and geometric patterns inherent in the peculiarities of solving each compositional problem.

The construction of the general plan of Chandigarh is based on a simple numerical relationship with a division into "sectors" measuring 800-1200 m with the administrative center - the Capitol, arranged in two squares 400 x 400 m, one of which is located in a larger square 800 x 800 m. relations are also, according to Corbusier, the basis for the designation of the dimensions of the halls of the Supreme Court 12 x 18 m, 12 m high and the judicial chambers 8 x 12 m, 8 m high in the Palace of Justice. However, the above diagram (Fig. 28) also shows the width of these halls in Modulor numbers and the length obtained by geometric construction by subdividing the square, which, apparently, was the final solution *.

* This, in particular, can be seen from the data given by Corbusier on the layout of carpet elements on the end walls of the large hall 8 x 140 + 133 = 12.53 m, the small hall 5x140 + 0.72 = 7.72 m. The numbers are almost exactly (width of the small hall diverges by 2 cm) correspond to the dimensions of the width of the rooms according to the Modulor, shown in fig. 82.

The main construction of the facade (Fig. 31 and 32) also corresponds to simple arithmetic laws, but the articulation of stained-glass windows and sunscreens is determined by the structural relations of the Modulor.

The same methods were applied in the projects of the building of ministries, temporary administrative buildings and a shopping arcade, conceived by Corbusier in Chandigarh, a museum in Ahmedabad, a factory in Saint-Dieu. The height of the premises is taken, as a rule, equal to 2.26 m or 1.83 x 2 \u003d 3.66 m, or 2.26 x 2 \u003d \u003d 4.52 m or more. The distance between the axes of columns or load-bearing walls in the plan (clean) of various buildings is selected from a number of 2.26; 2.96; 3.66; 4.79; 5.92; 7.75 m

When dividing the Modulor dimensions into equal parts, for example, when dividing the total height of the trading arcade in Chandigarh 7.75 m into two and three parts, small discrepancies are formed with the Modulor dimensions, which in most cases do not lie among themselves in simple multiples relations (with the exception of pairs from the blue and red rows 43 and 86; 70 and 140; 113 and 226; 183 and 366 cm, etc.). These discrepancies fall on the thickness of floors, walls, columns or are compensated by additional elements. Additional elements also appear when using geometric constructions. The development of the main idea of ​​constructing a system of proportional sizes corresponding to the scale of a person is accompanied by many observations and descriptions of creative searches given in the book.

Corbusier repeatedly emphasizes that the initial Modulor value of 2.26 m is associated with the required height of the premises of the minimum volume for a person, the small dimensions of which would be compensated by highly efficient engineering equipment, as in the cabin of an ocean packet boat. In Modulor 1 and 2, he returns to this idea again and again, talking about the height of the premises of residential buildings in Marseille and Nantes, about his office in the workshop on the Rue Sevres, about the “hut” he built and the holiday houses on the Côte d'Azur.

Le Corbusier experiments, uses the Modulor for large structures and small forms, determines the dimensions of the "open hand" - a monument at the entrance to Chandigarh, works on the proportions of exhibition stands and decorative panels, formats of publications and illustrations, on the designs of collapsible scaffolding and the dimensions of containers for transportation of goods.

The Modulor emblem itself becomes a decorative motif. The emblem is repeated on the memorial wall and on the walls of houses in Marseille, Nantes, and later, after the death of Corbusier, it will be placed in an ensemble with a pavilion built according to his sketch in Zurich. Corbusier speaks in the book about his sketches of the building of the UN in New York, the plan of the right bank of Antwerp, the business part of Algiers, the small industrial city of Saint-Dieu, work on which he was not instructed to continue. In these sketches, however, no system of application of the Modulor can be discerned.

In the very general view references are also given to the use of the Modulor in the chapel at Ronchamp, Corbusier's most poetic and unscheme work. In connection with the work on the project of this chapel, he says: “In principle, I am against any modules if they fetter the creative imagination ... I reject the canons ... plastic images do not obey schoolboy or academic proportions,” but then in conclusion: “A huge pleasure was the opportunity to use in the work all the richness of the combinations provided by the Modulor.

But where and how are the Modulor relations applied in Ronchamp? Do they determine the construction or, naturally, arise in the same way as any other ratios in one or another place of the bend of the curvilinear outlines of the plan and volume? The author does not answer these questions, and they require further interpretation.

The data given by Le Corbusier about his measurements of architectural monuments are also not very convincing, which, according to the author, confirm the objective regularity of the Modulor values. The use of golden section ratios or the Fibonacci series in Egyptian bas-reliefs is well known and corresponds to some Egyptian canons of division of the human figure, but in most of the other examples considered, the coincidences with the Modulor are very approximate, with a discrepancy of up to 5-10%. In Modulor-2, Corbusier gives examples of the use of the system of proportional construction proposed by him in the works of other authors. But there are few such examples, and only a residential building built by A. Vozhensky using dimensions according to the blue row of Modulor is considered in detail.

How should one evaluate the prospect of using Modulor, which has so far remained aloof from the mainstream of modular coordination of dimensions used in construction? Does it have a rational principle that determines the expediency of the direct application of Modulor, and especially the further development of the principles laid down in it?

An affirmative answer to this question should be sought, first of all, in the expressiveness of the rhythm of the partitioning of the facades of the "Housing Units" in Marseille and Nantes, the clarity of the proportional system of the composition of buildings in Chandigarh, the stained-glass windows and carpets of the Palace of Justice.

This is also evidenced by some letters of scientists, architects, engineers, and artists cited by Corbusier.

Architects H. L. Sert and B. Wiener write: “We have used Modulor with excellent results”; A. Vozhensky: “The use of Modulor has never hampered me or limited me in my work”; father and son Auger: "Thanks to the Modulor, which was the basis of the project, a complete agreement was established between us, since we both used the same well-tuned instrument." Albert Einstein, who was told about his work by Modulor's author, writes to him: "It is a gamut of proportions that makes bad things difficult and good things easy to achieve." Objecting to those who pointed out that Einstein's statement does not have the character of a scientifically based conclusion, Corbusier regards it as a foresight and a friendly gesture of a great scientist.

The well-known mathematician Le Lionnet spoke more cautiously: “... As you know, I reproached many authors for attributing too much of great importance bordering on mysticism - the use of the golden section. I hasten to assure you that this does not apply to you (probably because Corbusier considered his system only a working tool, and not a guarantee of the perfection of a work - D. X.) ... Obviously, even if the Modulor does not become the only mandatory, directive in the field of plastic arts, it has a number of other qualities that, along with other numerical values, can attract the attention of both artists and architects.

Very important for understanding the essence and meaning of the Corbusier system is the statement of Siegfried Giedion, who showed that the Modulor is not invented, but “... is based on great systems of proportions; he managed to tie them together. Giedion refers to these sources of Modulor systems based on the golden ratio, on some geometric constructions and on the canons of the human figure. Corbusier himself repeatedly speaks about the golden ratio and the Fibonacci series. Geometric constructions affected the graphic models of Modulor. The question remains about the connection of Modulor with the historical canons of human proportions.

Corbusier took as the basis of the Modulor the height of a man - 6 feet, referring only to a tall man in English detective novels, but this value exactly matches the standard of Vitruvius, which indicates that the foot, i.e. foot, is 1/6 of human height *.

* The absolute value of a foot in antiquity and in modern England differs, but the expression of a person's height and other sizes in feet and inches remains the same.

With the arm raised, the man becomes taller by a cubit, i.e., according to Vitruvius, by ¼ of his height, and reaches a height of 7½ feet, as in the scheme of Corbusier, reduced by Serralta and Meissonier to feet and inches. The analysis shows that this scheme coincides with the ancient canon in other main articulations. The height from the solar plexus to the foot here, just as according to Vitruvius, is ½ the height of a person with a raised arm, i.e. 33/4 feet \u003d 90 "; the height from the crown to the solar plexus is 2 ¼ feet \u003d 27", etc. Compared with the original Corbusier scheme (without the roundings adopted by Serralta - Meissonier under his leadership), the indicated values ​​\u200b\u200bdifference by ½ "or 1", i.e. by 1.2-2.5 cm.

Working on the system of proportions, Corbusier did not proceed from ancient canons, just as at the very beginning of his journey he was not yet completely based on the golden ratio. Corbusier invented the Modulor, guided by intuition and experience, analyzing historical monuments, examining the dimensions that are functionally necessary for a person, and testing them in his creative laboratory. As a result, however, he again approached the knowledge of some previously found objective patterns of proportional constructions in architecture, but in the aspect of their application to solve modern architectural problems.

This gives an answer to the question about the place and significance of Modulor in solving the problem of proportions, which Le Corbusier himself constantly posed: “... if Modulor paves the way to the wonderful properties of numbers, is it directed along only one randomly turned up path out of many others that exist? or could be, or by a lucky chance, exactly the path that is needed was found?

So, Modulor is not accidental; it is a link in the development of the theory of architectural proportions, based on earlier known systems, which were developed in antiquity, in the Middle Ages, in the Renaissance and today.

What is new in the Modulor is not only a clearer and clearer combination of the golden section size scale and the canon of the human figure compared to previous constructions, and not only the modern dynamic scheme of a “moving in space” man with a raised hand, which Gidion speaks of. What is also new is the treatment of the Modulor as a working tool and the transformation of an abstract scheme into a working method, the application of which is shown in a number of practical examples. Search creative method the appointments of proportional dimensions of buildings and their parts are also characteristic of some other architects, but they did not receive such a crystal clear completion as in Modulor. In particular, the method of I. V. Zholtovsky, based on the use of patterns of growth, decrease and alternation of the ratios of the golden section, unfortunately, is known only from individual statements master and retelling from his words.

Modulor, its emblem, expressive and clear schemes of proportional sizes, functionally necessary for a person, Corbusier's practical examples attract more and more attention of architects and designers. There are a growing number of examples of Modulor being used.

But what nevertheless explains that for all positive qualities Modulor, he did not receive wide practical application?

Here, apparently, a combination of several reasons, and above all the contradiction between the Modulor and the metric system of measures. All Modulor values ​​are approximately given in millimeters and rounded to centimeters, but they received a kind of random expression, not related to the main division of the meter and with the installed building modules, based on the initial value M \u003d 10 cm or 4 "≈ 10.16 cm. Last , apparently had special meaning, since even in countries with a foot-inch system of measures, the Modulor has not become a generally accepted working tool, although its initial value is expressed in English feet, and the values ​​\u200b\u200bare significantly: they are more easily expressed in inches than in centimeters. Corbusier disparagingly speaks of the 10 cm module, talking about the "wretched system of standardization associated with it, which excludes the manifestation of creative imagination." However, the decimeter is just a measure of length, and the corresponding modular grid serves only as a canvas for assigning dimensions, which, moreover, can receive further enlarged or, if necessary, fractional division. As for proportional ratios, the series of sizes that are multiples of the currently accepted modules M \u003d 10 cm, 3M \u003d 30 cm, as well as larger or fractional modules, make it possible to choose values ​​​​in relation to Fibonacci numbers, for example 50, 80, 130, 210 cm or 150, 240, 390, 630 cm, providing the same approximation to the golden ratio as the ratios of the Modulor numbers.

Another reason is the contradiction of Modulor with the principle of composing the whole from equal or commensurate parts, which is characteristic of any construction carried out using finished products(large panels and blocks, as well as bricks, stones of the same size, logs of a certain length), and leads to simple numerical modular relations. Modulor's capabilities here are very limited. Each value of the red row corresponds to a doubled value of the blue row, but further multiplication, dividing in half, into three or more parts, as well as a simple numerical relationship between individual pairs of quantities, are not provided by the Modulor system.

The contradiction between the Modulor values ​​and the need to use simple multiple ratios was also felt by Corbusier himself, who, considering the experience of designing Chandigarh, speaks of a combination of arithmetic, structural (according to Modulor) and geometric ratios. The idea here is only outlined and carried out with the use of individual additional elements that compensate for the discrepancies, but it is of undoubted interest and deserves further development.

The path to solving the problem of combining arithmetic, structural (according to Modulor) relations is also suggested by the Serralt-Meissonier scheme, on which the red and blue Modulor scales are supplemented with size scales in cubits, half-cubits, feet and spans, recommended for use if additional divisions are required. Such a combination, however, was possible with some compromise * and only for the foot-inch system of measures, since the original Modulor value is associated precisely with this system.

* The height of the figure of a man with a raised arm is 5 cubits, or 7½ feet, i.e. 228.6 cm versus 226 cm Modulor (for a full match, you need to take 7 feet 5 inches)

Of significant interest is also the still insufficiently appreciated proposal of the architect M. Rogier, who, as follows from the analysis of the text and the diagram given by Corbusier, proposed to introduce half and double values ​​of all Modulor numbers (and not just doubling the values ​​of the red or dividing the values ​​of the blue series in half). Thus, inside the system of numbers connected by the proportions of the golden section, additional ones appear that are necessary in practical project work rows of successive doubling and dividing in half *. Not fully resolved contradictions with the metric system of measures and with the method of composing the whole from equal or commensurate parts, necessary in prefabricated construction, apparently served as the main reasons limiting the use of Modulor. To this should be added the insufficient development of the system, which has retained its experimental character and is, in essence, a scale of sizes without clear rules for their application. The initial values ​​also remain debatable - the conditional height of a person is 183 cm, with a raised hand determining the height of the room 226 cm **.

* This main content of Rogier's proposal usually escaped the attention of researchers, since it is veiled by the author's indication that in addition to a person's height of 183 cm according to the Modulor, heights of 173 and 193 cm are introduced. These values ​​do appear in the Rogier series accompanying the main Modulor series, but they are not the cause, but the effect of his idea.

** The room height of 226 cm and, accordingly, the floor height of 2.5-2.6 m most closely correspond to the English standard, according to which the floor height of residential buildings is 8 feet 6 "≈ 2.59 m and is rounded due to the transition to the metric system of measures up to 2.6 m. In the USSR, the norms set the height of the floor of residential buildings at 2.8 m (in some cases 3 m), the minimum height of residential premises is 2.5 m; the same or slightly lower (2.35-2.4 m) the height is accepted in most other European countries.

Many people think and write about overcoming the contradiction with the metric modular system and further development of the Modulor idea with the aim of its wider practical use.

Literature about Modulor in the USSR and abroad is extensive. Many articles contain an analysis of its construction and examples of its application *.

* Kazarinova V., Romm N. Modulor Le Corbusier in theory and practice. - "Architecture of the USSR", 1968, No. 3; Put E. Modulor Le Corbusier. - "Housing construction", 1971, No. 5.

Books and articles on the modular system in architecture contain an assessment of the Modulor, criticism of its individual provisions, and corrections to its construction are proposed *. The proposal for the development and adjustment of the Modulor is also the content of a number of special articles.

* Khazanov D. V. Module in architecture. - In: Questions of the theory of architectural composition, No. 2. M., 1958; Khazanov D. B. Modular coordination in design. M., 1959; Khazanov D. B. Modular coordination of dimensions in architectural composition. - In the book: Architecture of a residential complex. M., 1969.

S. Vasilevich proposes to use, if necessary, a series of values ​​that are average between the values ​​of the red and blue series, increasing exponentially with a denominator √1.618≈1.272. Le Corbusier cites the proposal of Martineau-Lagarde, who proposes to change the pattern of constructing the Modulor and accept the increase in series with the denominators of the Renard geometric progression, now accepted in international and national standards (including the USSR*) of preferred sizes for industrial products. The fifth row of preferred numbers, increasing with the denominator of the progression 5√10≈1.585, indeed gives values ​​very close to the Modulor values, but here their main advantage is completely lost - the additivity inherent in the Fibonacci numbers and the proportions of the golden section, in which each subsequent value is equal to the sum the two previous ones. The absence of such dependence makes it difficult to combine elements in the construction of buildings or the aggregation of equipment.

* GOST 8032-56 "Preferred numbers for industrial products".

The same shortcoming is characteristic of the proposals of the Indian architect D. D. S. Ahluwalia *, who chose slightly different characteristic points of the human figure than Corbusier, and on the basis of his measurements of people of different sex, age and height, he established the denominator of the geometric progression of increasing values ​​1, 66, or 5/3. At the same time, if we take the initial height of a person with a raised arm as 226 cm, then the subsequent values ​​​​of the Ahluwalia series correspond less to the functionally necessary heights various kinds fences, window sill, table, chair than the more advanced range of Modulor values. Only when the initial height is reduced to 200 cm (with a person's height of 162 cm) do the subsequent values ​​turn out to be relatively more favorable, but some of the necessary dimensions are also missing among them **.

* Ahluwalia L. J. J. S. Het preportionels stelsel Bouw. 1963, III, 11

** The author also proposes, regardless of the proportions of a person, to take a number of vertical dimensions (window sill, window, room height), increasing exponentially with a denominator of 1.6, and a number of horizontal dimensions corresponding to Fibonacci numbers.

Often, the validity of the choice of a unified conditional human height of 6 feet ≈183 cm, adopted for Modulor, raises doubts.

Design artist V. Pakhomov proposes, when assigning sizes for furniture and equipment, to proceed from the options for a person’s height of 160, 170, 180 cm *, taking values ​​derived from them according to the Modulor principle, but bringing them to modular sizes, a multiple of 5, 10 or 20 see This is achieved by using, in addition to the Fibonacci series, also other recurrent series close to it in properties**.

* Pakhomov V. Modular coordination in instrumentation. - "Technical aesthetics", 1968, No. 8; Pakhomov V. Technical and economic essence of the modular system. - Sat. Creativity in artistic design, LDNTP, L., 1969; Pakhomov V. On the issue of modular coordination of the geometric parameters of products, LDNTP, L., 1969.

** In particular, the series 1, 3, 4, 7, 11, 18, each member of which, as in the Fibonacci series, is the sum of the two previous ones, moreover, the ratios of magnitudes approach the ratio of the golden section as they increase.

According to the proposals of the designer A. Melnikov *, a person’s height can also be taken as 160, 170, 180 and generally a multiple of M = 10 cm, but he comes to this in the opposite way, taking the initial modular values ​​½ M = 5 cm, M = 10 cm, 2M, 3M, 4M and further up to 12M, increasing each of them in relation to the numbers of the Fibonacci series and summing up all the obtained values ​​**.

* Melnikov A. Composition and standard. VNIITE, M, 1971.

** Such a proposal, but only for the first three modules - ½M, M, 2M - was made by the Greek delegation to the UN European Productivity Agency.

Differentiation of the initial Modulor values ​​is also proposed by other authors in the USSR and abroad. Such proposals are obviously justified when assigning such sizes of furniture and equipment that are intended for a strictly defined category of people, for example, only for men, only for women, or for children of various ages. The same applies to the solution of certain specific problems, which was shown by Corbusier himself, who used a special version of Modulor with initial dimensions of 165-204 cm to divide windows 204 cm high in the old building he reconstructed.

For solutions common tasks construction and equipment of buildings, Corbusier rightly considers a single conditional height necessary, since "an architect must make such doors so that a person of great stature can pass through them." The same applies to built-in equipment, wardrobes, and even to such pieces of furniture as chairs, tables, armchairs, which cannot, as a rule (with the exception of children's furniture), be designed to an individual measure. Moreover, if the height of objects should be assigned “without a margin”, since it is inconvenient to sit on a chair 50 cm high, then some freedom is always useful in the width of a chair, armchair, bed, desktop, achieved by designing according to a fairly “tall” Modulor.

Le Corbusier emphasizes that the Modulor is not only an architect's tool, but also a standardizer, and that it is necessary to take into account the conditions of the increasingly developing international exchange of industrial products, establishing a system of sizes that is uniform for all countries.

The original height of a person, according to Modulor, 183 cm, found by Corbusier as a result long searches, meets the specified conditions. Based on this initial size, the height of the chair is 43 cm, the table is 70 cm, the balcony railing or the location of the window sill is 86 cm, the barrier is 113 cm, etc. regardless of Modulor, including the norms of planning elements and dimensions of equipment published in the USSR.

* Neufert E. Structural design. Per. with him. M., Stroyizdat, 1965.

In the conditions of mass construction of buildings, carried out with the widespread use of prefabricated elements, the values ​​calculated by the Modulor must be brought into line with the modular system established by Soviet and international standards.

In Soviet regulatory documents, starting from 1960 *, and then in the recommendations of the CMEA for socialist countries and in 1968-1969. in the recommendations of the International Modular Group (Commission No. 24 of the International Building Council - CIV) and the International Organization for Standardization -180, a single series of enlarged modules derived from the main module M = 10 cm is established, and the nominal dimensions of building elements should be taken as multiples of these modules.

* A single series of derivative modules - enlarged and fractional, approved by the USSR State Construction Committee in 1960 and included in the chapter SNiP P-A.4-62 “Unified modular system; the main provisions of the design. M., 1962.

Of the established values, the module 3M = 30 cm is of particular importance, which is the basis for assigning multiple sizes to it in the layout of the premises of residential buildings and some types public buildings, most of all related to the size of a person, as well as for the nominal sizes of windows, doors and for dividing structural elements in all types of construction.

The same 3M module interconnects all larger, multiples of the installed modules 6M = 60 cm, 12M = 120 cm, used for the main planning parameters (longitudinal and transverse steps), mainly in the construction of buildings from large panels, blocks and bricks, and also 15M = 150 cm; 30M = 300 cm and 60M = 600 cm, adopted for large public halls and frame-panel public buildings, the designs of which allow significant freedom in the placement of non-bearing partitions, regulated by smaller modules. For longitudinal and transverse steps of columns of industrial and warehouse buildings, the multiplicity of 60M and 30M is used.

An important feature of the 3M = 30 cm module, and to a certain extent also of the 6M and 12M modules, is the connection with a person, with the dimensions necessary for his movement, work and rest. This is like a modern foot in the metric system of measures, equal to 30 cm, i.e. average value foot, adopted at different times in different countries (in ancient times, for example, the size of a foot ranged from 295 to 308 mm). From here, it seems quite natural to round off the conditional height of a person according to the Modulor, equal to 6 English feet = 183 cm, to a height of 6 modules of 3M = 180 cm. and then 183cm).

The value of the module 3M = 30 cm is easily divided into 2, 3, 4, 5, 6, 8, 10, 12 or more parts, multiples of the main module M = 10 cm, or fractional modules ½M; 1/5M; 1/10M, established by Soviet norms and recommendations of the CMEA; in necessary cases, it is also allowed to use a fractional module ¼ M, which is, as it were, a “metric inch”. Thus, the modules used combine the possibilities of all simple divisions inherent in the metric and foot-inch systems. All Modulor values ​​can be saved with their conversion to the accepted metric modular system, but with rounding of feet to 30 cm and inches to 2.5 cm, which practically does not change the values ​​of absolute dimensions *.

* In England, a decision was made to switch to the metric system of measures, i.e. practically a foot in construction is replaced by a module of 30 cm.

When designing special types equipment and furniture, other Modulor series can also be used, reduced to one or another height of a man, woman or child, rounded to multiples of M or ½M (10 or 5 cm). For construction purposes, on the contrary, along with a scale corresponding to a person's height of 180 cm, it is possible to use conditional size scales of a larger scale. The original Modulor size of 226 cm was adopted in accordance with Corbusier's recommended height of living quarters. In practice, however, in most countries this height is taken equal to 240 or 250 cm, and the floor height is 260, 270, 280, 300 cm. which can be used to divide the dimensions of buildings, especially vertically *. The direct connection with the growth of a person, of course, is weakened, and all sizes acquire, as it were, a certain reserve.

* One of these scales, but with the initial value of the height of the premises of 3.28 m, adopted in India, and with a progression denominator of 1.6, was proposed by D. D. S. Ahluwalia.

The division of such scales, especially at the initial values ​​​​of 240 and 300 cm, with a slight rounding, corresponds to sizes that are multiples of building modules 12M, 6M, 3M, 1½M *, M, ½M.

* The ½M module is not provided for in SNiP, but in practice it appears in those cases in which it is necessary to divide in half the sizes containing odd number 3M modules, for example, when dividing a floor height of 2.7 m.

Bringing the Modulor to the metric modular system is essentially similar to the scheme of bringing it to feet and inches according to the Corbusier-Serralt-Mesonnier scheme and represents the same possibility of combining Modulor dimensions with the necessary additional dimensions corresponding to one or another system of measures. In this case, such measures are not meters or feet, but the module M and its derivatives, enlarged and fractional modules. Plan dimensions, especially of rooms in general, have less relationship to human height than vertical dimensions. Therefore, there is no need to take such steps of load-bearing walls as 226, 296, 366, 479, 592 cm according to the Modulor, which can be rounded up to 240, 300, 360, 480, 600 multiples of 6M and 12M. It remains, of course, the possibility of using other additional sizes that are multiples of 3M.

Larger values ​​according to Modulor 957, 1253, 1549, 2028, 2507, 3281 cm are even less related to the size of a person and can be replaced by a number of established modular sizes of large spans 9, 12, 15, 18 (21), 24, 30 m etc.

So, the Corbusier Modulor series of values, with a slight adjustment of their sizes, can be used in the modern metric modular system and applied in combination with the usual full series of modular values, built on the principle of arithmetic progression or with the so-called stepwise increase (i.e., with the enlargement of modules according to as the series increases). This is fully consistent with the recommendation of Le Corbusier himself to combine arithmetic, structural (according to Modulor), and sometimes (for example, with curvilinear configurations) and geometric systems. The expedient field of application of rows built according to the Modulor principle within the framework of the modern modular system is determined by the very idea of ​​​​Modulor, which is associated with the physical dimensions of a person, and with the aesthetics of the proportions of the golden section.

The first, i.e. connection with a person, determines the feasibility of applying the laws of Modulor in the design of furniture, various types of equipment, the interior of living rooms, workplaces in rooms intended for mental and physical labor, recreation areas. When designing furniture, Modulor can be used quite freely, but when aggregating (blocking, combining) equipment elements, it is necessary to choose dimensions, taking into account the additive properties of Modulor numbers. Here, however, in many cases it may be more appropriate to use the usual modular proportionality, which allows for a freer division into equal and multiple parts than Modulor, or, finally, a combination of one or another system.

The second, i.e., the aesthetic side, provides a basis for using the Modulor when choosing proportions and dismembering the planes and volumes covered by the viewer’s gaze or revealed in front of him in the process of gradually walking around the building. This primarily applies to the segmentation of the planes of facades, interiors, decorative panels, and in the latter case, bringing the modulor to the metric modular system is not necessary. It is characteristic that this kind of task was most successfully solved by the author of Modulor himself when creating the facade of the Marseille house, stained-glass windows and wall carpets in Chandigarh.

It is also necessary to remember the words of Corbusier that the Modulor is not a canon. “Sometimes they show me unsuccessful, poorly arranged projects, justifying them by the fact that it was done according to the Modulor. So much the worse for Modulor, I answer... If Modulor leads you to this disgrace, throw it out. Your eyes should serve as your only criterion...” This is how Modulor is assessed by its author himself. We should also evaluate it. Modulor is not the ultimate truth, it is not the end of the road, but the creative proposal of one of the greatest masters of architecture, based on proportional systems created in the process of historical development.

Modulor deserves the closest attention, conscious and creative application as one of the possible methods of assigning proportions and sizes in the design of buildings, but taking into account the metric modular system, which is necessary condition associated with modern industrial construction methods.

The word, as Corbusier says, is for those who use the Modulor, as well as for all architects, engineers, designers working on the basis of the accepted modular system, and for those who must further improve this system, taking into account not only functional, technical and economic, but also aesthetic requirements.

Le Corbusier's modulor principle- a universal auxiliary measuring scale based on the average size of the human body and the "golden row" (in which each subsequent number is equal to the sum of the previous two). “Modulor is a gamma,” he wrote. "Musicians have a scale and create music according to their ability - banal or beautiful." In accordance with the modulor, the proportions of a large multi-storey residential building in Marseille with two-level apartments (1947-1952) were calculated. For the first time, the architect decorated the wall of this building with the Modulor relief. (By the way, since 1945, Le Corbusier has become addicted to another type of creativity - sculpture.) And after that he left the image of his ideal person, as if greeting everyone approaching him with a raised hand, on the walls and other "residential units" in Nantes-Reze ( 1955), West Berlin (1957), Brie-en-Foret (1961), Firmini (1968). However, the main thing is that he constantly put into practice the proportions developed on the basis of the modulor.

It was a kind of experiment in the field of public construction for post-war France, and for Corbu himself - the embodiment of it. utopian ideas about the "car for housing", an attempt to put into practice the "modulor" - a system of harmonic quantities developed by him in the 1940s, used as a tool for the proportional construction of architectural forms. In the "unit" up to 1600 people live in a single block, a kind of "vertical village", with an inner street, shops, recreation areas, a children's studio theater and a rooftop pool.

In the middle of the 20th century, a residential building in Marseille became for Corbusier a real opportunity to embody their theoretical calculations related to the search for harmonic proportioning. We are talking about a house for a person, proportionate to a person. That is, according to the architect, about the experience of “a universal harmonious system of measures proportionate to the scale of a person, applicable both in architecture and in mechanics”, about a modulor (the name was coined in 1945). The system was based on the height of a person of average height with a raised hand. Initially, he suggested focusing on 2.20 meters. In the final version, a height of 2.26 meters was taken (the 1940s, alas, did not imply future accelerators). The mathematical model included the construction of two squares with sides of 1.13 meters, which constituted a rectangle, inside of which a right angle was inscribed. This last divides the rectangle exactly in the middle.

How important the modulor was to the architect is indirectly confirmed by the emblem immortalized in the counter-relief on the concrete wall of the house - a schematic figure of a man with a raised hand. The human principle is the basis of his new geometry.

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. More and more often in scientific literature there is a fruitful influence on a person of structures proportional to the golden section. 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: ʼʼ... 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. From these initial data, the rest ʼʼ were calculated (Fig. 8).

I will return to this modulor base, 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 geometrically (by dividing the segment in extreme and average ratios), or by successive approximations along the Fibonacci number series. (I note that there are many such series, Fibonacci was the author of the first recorded series, and all of them, before A.A. Pilecki, seemed to be single. The first double row and 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 in the form of 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

If we now shift the numbers of the blue line to the red line, then 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 into the number of the blue line standing diagonally below and to the left of it, then with each division we will get the same coefficient 1.306, and when dividing the numbers of the red line into those standing on the left and below from them the numbers of the blue line - a coefficient of 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 a variant linked to a person’s height. The fact that the modulor was limited to only two rows of the 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

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 are filled (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 are not recommended for use in proportioning objects.

Modulor Le Corbusier allows, of course, to obtain some common types of proportions of the golden number: Ф = 1.618; 2/F = 1.236; F 2 /2 = 1.309; 2/F 2 = 0.472 ...

Not dwelling on them 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 limitedness 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 it in connection with this that 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, however, to carry out a sequential division of each number into each other 2.260 / 1.829 \u003d 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 let's define for each of these numbers that ĸᴏᴛᴏᴩᴏᴇ 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 included in 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 lesson of the golden ratio" - One of the most beautiful works ancient greek architecture is the Parthenon (5th century BC). A wonderful example of the "golden section" is a regular pentagon - convex and star-shaped. "Golden section" in sculpture. "Golden Ratio" in photography. "Golden section" in architecture.

"Golden Ratio in Mathematics" - Line segment. Golden ratio. Leaf arrangement. The principles of the "golden section". The history of the golden section. Nature. Basic geometric shapes. Dividing a segment into two parts. Squares. Human interest. Sequence member. Spreading. Introduction to principles. Numbers forming a sequence.

"Proportions of the golden section in life" - Regular geometric shapes. An example of the use of the "golden section" in painting. Pentagram. The structure of the hand. Aesthetic enjoyment. Senate in the Kremlin. Five pointed star. Establish relationships between geometric concepts and the surrounding world. Image of the Parthenon temple. The shape of a regular pentagon can be found in wildlife.

"Harmony in the golden section" - Division of the segment. Separate part of the body. Sculpture of Apollo Belvedere. Attitude. The golden ratio in the pyramids. The standard of beauty. Pattern. The golden ratio in art. A work of ancient Greek architecture. Golden ratio. The body of a man. Golden ratio in nature. The word "proportion". The golden ratio in clothes.

"Golden Ratio in Human Proportions" - Eyes. The skirt is long. Dentistry. Proportions of the "golden section". The height of the supralabial fold. Knowledge of the work of all organs. Navel. Class. proportions of the human body. The principle of the "golden section". Separate parts of the body. The proportions of the human head. Method " golden ratio". Man aspired to beauty.

"Proportions of the golden section" - Zabolotsky. Leonardo da Vinci (1452-1519). Pakistan. Plato. There is something that disturbs me a little: A magic sign at your doorstep. Mauritania. Sandro Botticelli "The Birth of Venus" (circa 1485). God - the father "protects" the universe, which has the shape of a dodecahedron. Guinea - Bissau. Five regular polyhedra - five elements.

In total there are 9 presentations in the topic

The Golden Ratio The laws by which works of art are created are usually called the laws of harmony. These include balance, the law of unity and subordination. law Means of harmonization also include rhythm, contrast, nuance, identity, as well as proportion and scale. Let's pay attention to one of the most important means of harmonization - proportions (connections of parts and the whole). Continuing the theme of unity a holistic work, we argue that proportions are exactly the means, which is based on the idea of ​​the relationship of the whole and the parts that make up this whole. Proportion is understood as the ratio of the parts of the whole between themselves and this whole.

The golden section The golden section is such a proportional division of a segment into unequal parts, in which the smaller segment is related to the larger one, as the larger one is to everything. a: b = b: c or c: b = b: a.

The Golden Ratio For example, in a regular five-pointed star, each segment is divided by an intersecting segment in the golden ratio. These ratios are equal to 1.618. The number 1.618 is called the golden ratio. Let's build the segments in the proportions of the golden section. In a rectangle with an aspect ratio of 1:2, a diagonal is drawn, on which the smaller side is superimposed by rotation. The remainder of the diagonal rotates around the vertex of the rectangle until it coincides with the position of the upper base. Thus, the upper base was divided into two unequal segments in proportion to the golden section.

The golden section It is generally accepted that the concept of the golden section was introduced into scientific use by Pythagoras. There is an assumption that Pythagoras borrowed his knowledge from the Egyptians and Babylonians. Indeed, the proportions of the Cheops pyramid, temples, bas-reliefs, household items and decorations from the tomb of Tutankhamun indicate that the Egyptian craftsmen used the ratios of the golden division when creating them.

Fibonacci sequence. The name of the Italian mathematician monk Leonardo from Pisa, better known as Fibonacci (son of Bonacci), is indirectly connected with the history of the golden ratio. In 1202, Fibonacci's mathematical work "The Book of the Abacus" (counting board) was published, in which all the problems known at that time were collected. One of the tasks read "How many pairs of rabbits in one year from one pair will be born" . Reflecting on this topic, Fibonacci built the following series of numbers: Months 0 1 2 3 4 5 6 7 8 9 10 Pairs of rabbits 0 1 1 2 3 5 8 13 21 34 55 11 12 etc. 89 144 etc.

Fibonacci sequence. With the discovery of the Fibonacci series, the main property of the golden section was discovered - the unity of additivity and multicativity. This is the essence of the golden ratio. It contains the key to the phenomenon of shaping, openly lying on the surface of mathematical knowledge. In mathematics, the concept of "additivity" means that in the number series Ф 1, Ф 2, Ф 3, Ф 4. . . Фn-1, Фn each subsequent term is equal to the sum of the two previous ones. Moreover, any two numbers can be taken as the beginning of such a series, for example, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610. . . Multiplicativity means that in the number series Ф 1, Ф 2, Ф 3, Ф 4. . . Fn 1, Fn all members of the series are connected in geometric progression: F 1: F 2 = F 2: F 3 = F 3: F 4 =. . . = Фn-1: Фn = const.

Modulor Le Corbusier Modulor is a measuring scale (a system of harmonic quantities) created by Le Corbusier in 1942-1948 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.

Modulor Le Corbusier Description of modulor: 1 Scale of three intervals: 113, 70, 43 (cm), which are consistent with φ (golden section) and the Fibonacci series: 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 Le Corbusier Le Corbusier published the first volume of his work in 1948: Modulor / Modulor The second volume of Modulor was published in 1954. In the book, he outlined the results of his research, ongoing since 1942. Modulor - according to its developer - helps the architect to choose the optimal dimensions of the designed house and its elements, corresponding to the height and proportions of a person. The first house calculated using the modulor was built in Marseille by 1952. The house stood on pillars, it had 337 two-story apartments, a roof-deck with a garden, kindergarten, swimming pool, gym, etc.

Modulor Le Corbusier Harmonized size series according to the Modulor Le Corbusier: 4, 6, 16, 27, 43, 70, 113, 183. 13, 20, 33, 53, 86, 140, 226.

Printed graphics An example of a newspaper modular grid. Newspaper "Evening COURIER" Modular grid of the first page. It is slightly different from the general modular grid with the top block. For a clearer perception of the logo, it had to be distinguished by very significant pauses. But there are no problems with reading and highlighting it.

Printed graphics An example of a newspaper modular grid. Newspaper "Evening COURIER" In addition to a well-defined upper block, the pages have a well-defined "basement" - this is how newspapermen call the lower part of the page. It helps to divide the information on the page into thematic zones.

Printed graphics Modular grids are often based on a square - a very convenient module. The double square has long been known as a module of the traditional Japanese house, where the dimensions of the rooms were in accordance with how many times a tatami mat having the proportions of a double square will fit on the floor. In applied graphics, the square is used for formats of prospectuses for albums, children's books, but it also defines the interior space of these publications. An example of using a square module in a square format: with a three-column typing, the entire area allocated for text and illustrations is divided into 9 squares. If the column width is set to 1, then the square will be 1 x 1. At the same time, illustrations can occupy areas: 1 x1, 1 x2, 1 x. Z, 2 x2, 2 x. Z, Zx. Z, 2 x1, etc., that is, we will have ample opportunities for combining illustrations and text in the layout.