Encyclopedia of Perception (from the publisher's website)
Photography belongs to the fine arts. A lot of copies were broken, however, now this can be definitely stated. The following is a translation of an article from - generally non-fiction - Bruce Goldstein's Encyclopedia of Perception. I stumbled upon this book by accident: I was “played” by Richard Zakia - a book simply required reading for people related to photography - Richard Zakia “Perception & Imaging / Photography: A way of seeing” - and I rushed to look for it or a replacement for it . That's how I came across Goldstein.
I’ll make a reservation right away: the translation is practically without editing, make a discount on this.
The article has been translated and posted with the permission of the copyright holder. Copyright by SAGE Publications Inc.
Original article: E. Bruce Goldstein's Encyclopedia of Perception, Aesthetic Appreciation of Pictures. pp. 11-13 Copyright 2010, SAGE Publications Inc.
Viewing works of fine art, while having a strong emotional impact, remains a completely personal process. In the discussion of the aesthetic perception of a painting within the study of perceptual processes, an attempt is made to bridge the gap between a clear understanding of the processes of the lower level - visual and cortical * perception of the objective characteristics of the image, such as color and shape, and a less clear understanding of the higher level of visual phenomenology, or subjective experience.
Over the centuries, the definition and content of the concept of "aesthetic experience" have been presented by people in completely different ways. Usually, within the study of perceptual processes (perception studies), aesthetic evaluation is determined through a preference based on the perceived beauty of the image in question. The study of perception thus draws from the approaches to aesthetics of both David Hume and Immanuel Kant, in terms of taste and beauty that they discuss. Factors that influence the aesthetic response to a painting are considered to be both the physical characteristics of the work itself, which are "inside the frame", and contextual influences, such as the title of the work and the way it is shown (presentation), which exist "outside the frame".
Research into the problems of aesthetic perception is still based on group methods (nomothetic approach), however, there is an opinion that only the study of the individual / individual (or ideographic approach) can serve as a starting point if the goal is to fully understand the process. This article focuses on how the aesthetic is measured, defines the objectivist and subjectivist approaches to aesthetics, and talks about the use of these approaches by researchers.
Measuring aesthetic
The origins of empirical aesthetics are usually attributed to Gustav Fechner and his book Elementary Aesthetics, and Daniel Berlyne is credited with reviving interest in the application of scientific methods to the study of aesthetics in the 1970s. These early experiments were aimed at identifying individual preferences of the subjects through the evaluation of large sets of artificially created stimuli, called "polygons" (polygon).Polygons differed from each other by a given set of quantitative (calculable) variables divisible by collative (for example, complexity), psychophysical (e.g. color) and environmental (e.g. meaning/meaning) According to Berline's psychobiological approach, aesthetic experience/perception should be higher for the average level of arousal, with arousal calculated as the sum of the properties involved: thus, for example, multi-sided polygons should contain less color than polygons with fewer sides.
These early studies identified approaches to measuring aesthetic experience using a simple numerical scale (also known as the Likert scale) whereby images are asked to be sorted or ranked from least preferred/beautiful to most preferred/beautiful. Despite the fact that this method lends itself to easy criticism due to the inaccessibility of the entire range of assessments to the subjects, such subjective measurements underlie the perceptual study of the aesthetic. Over time, subjective assessments of aesthetic experience have been supplemented with objective metrics, such as time spent looking at a single image and blood oxygenation in the brain, to provide convergent data for understanding aesthetic experience.
Aesthetics "Inside the frame"
The first experiments aimed at understanding aesthetics through the study of perception showed a significant simplification of the approach. It was assumed that one could come to understand the origins of the beauty of the work of art in question by studying individual reactions to the basic elements of visual perception. At the same time, the overall assessment of the painting was divided into the study of the preference of its individual components: color combinations, orientation of lines, sizes and shapes. A common limiting factor for many psychological studies is the discrepancy between the ability to control the proposed materials within the walls of the laboratory, and therefore the ability to generalize the data obtained, and the much more diverse and rich examples of visual art that exist in the real world. Research based on abstract visual stimuli means that the subjects had no prior exposure to the images, and this limits the aesthetic experience to the primitive side, where the influence of schema or memory is excluded, and the image is evaluated only through stimuli. And these types of stimuli are far from real: will the study of polygons tell us anything about the work of Picasso?
William Turner, The Shipwreck
An opportunity to explore at the intersection of lower and higher levels of visual experience is given by the work of Piet Mondrian, in which pictorial elements are superimposed in a special way on basic visual forms, such as line orientation and color. They gave the researchers the ability to consistently change the distance between the lines, their orientation and thickness, the placement and combinations of colors within the picture in question, to assess the level of change to which the subjects found the original Mondrian composition aesthetically more attractive than the modified one. The results showed that even subjects with no training in the visual arts gave higher marks to original paintings, suggesting that aesthetic perception is partly due to the placement of visual elements in the picture. Other studies have shown that the aesthetic preference for original over unaltered paintings also applies to works of a representative nature, although the preference for original works was revealed only after significant changes were made. These observations suggested that aesthetically preferable would be a picture in which the artist has achieved the best arrangement (or balance) of elements, and this compositional balance is easily perceived by non-artists. The findings are in excellent agreement with Prägnanz's Gestalt psychology principle (also known as "visual correctness") and provide evidence for universalism in aesthetic experience.
Marcel Duchamp
Aesthetics "outside the frame"
In contrast to the objectivist approach to the empirical study of aesthetics, in which the beauty of a painting is considered to be hidden in the organization of the visual elements themselves, the subjectivist approach emphasizes the role of external factors in determining what is beautiful and what is not. The need for a subjectivist component in aesthetics will be clear to anyone who has had the misfortune of accompanying a lover of Renaissance painting through the gallery of modern art. The fact that individuals can have vastly different responses to the same visual stimuli suggests that attitudes towards art and preparedness have a significant impact on aesthetic perception. Comparisons between the perceptions of untrained viewers and those of art critics are common in the scientific literature, although understanding of what it means to be an "art critic" or "what constitutes an art critic" has never been achieved. Based on the differences between figurative art and abstract art, original color or altered black and white, the aesthetic preferences of "newbies" tend to be towards figurative art in color, while art critics tend to have a much broader range of preferences.
Edward Munch
It is believed that the name of the picture has an impact on the aesthetic response of the viewer. However, this influence depends both on the content of the title and on the type of image to which it refers. Adding a descriptive title for representative paintings can be redundant (e.g. William Turner's The Shipwreck, The Shipwreck), but more abstract works (e.g. Marcel Duchamp's Nude Descending a Staircase) title can help the viewer unblock individual ambiguous elements on the canvas. Moreover, additional information about the origin, style, or interpretation of a work can significantly influence an individual's response. Thus, the information that in the work of Edvard Munch's "The Scream" (1893) (Edvard Munch's The Scream) the character in the foreground of the picture does not actually scream, but rather tries to protect himself from the cry of nature, can radically change the aesthetic perception of the canvas . There have been studies comparing responses to untitled and descriptive or explanatory titles. Descriptive titles are often helpful in getting the picture right, while explanatory titles are more likely to lead to a deeper aesthetic response. Another external factor influencing the aesthetic perception is the place where the picture is viewed. In the interests of the purity of the experiment, individuals taking part in experiential aesthetics research are most often asked to view images on a computer monitor for a limited amount of time. This is fundamentally different from viewing paintings in a gallery, where they are presented in their original size; viewing distance is often carefully calculated, and viewing time is not limited. There are few data comparing the perception of originals and reduced copies, and they indicate no significant differences in perception; however, it can be assumed that some of the optical or scale effect intended by the artist may be lost when the size is reduced. For example, Mark Rothko's large color canvases may be judged differently if their dimensions are not preserved. It has been experimentally deduced that a person usually spends half a minute looking at a picture. Time constraints can also limit the depth of analysis of a painting, leading to an aesthetic assessment of only the general properties of the image.
Is the sense of taste measurable?
Comparison of objectivist and subjectivist approaches to the aesthetic perception of works of art led to the beginning of the unification process; the new approach is called interactive. In defense of the objectivist approach, both representational and abstract painting evoke an aesthetic response, and as such, the relationship between approaches must be seen through the lens of the painting itself, not its content. In defense of the subjectivist approach, identical visual stimuli can lead to different aesthetic preferences. It becomes clear that alternatives to the nomothetic approach to empirical aesthetics need to be considered. By breaking down complex visual stimuli into their main components, researchers have found it difficult to create a group model of aesthetic satisfaction that adequately reflects personality. Moreover, the clinical application of aesthetics tends to lean toward an ideographic approach. For example, the palliative benefit of viewing healthcare art is based more on the personal as opposed to the institutional. Although patients in the early stages of Alzheimer's disease differ among themselves in image classification, their aesthetic preferences may remain stable over a two-week period, while explicit memory does not remain stable during this period. Finally, the existing examples of images of male and female bodies, reflecting ideas about the ideal, showed that aesthetic assessments to a large extent depend on a number of socio-psychological factors inherent in the time of creation of these images. Understanding aesthetics on both an individual and a group level promises to lead to a more intense and beautiful understanding of the environment. Research in empirical aesthetics has proven that it is indeed possible to find a dimension to taste, although some of the most important aspects of the aesthetic experience remain elusive.
Ben Dyson
I will add from myself.
Perception- (from Latin perceptio - representation, perception) the process of direct reflection of objective reality by the senses.
figurative art(from lat. figura - appearance, image) - works of painting, sculpture and graphics, in which, unlike abstract ornamentation and abstract art, there is a fine beginning
Cortical - pertaining to the cerebral cortex, cortical
Likert scale- named after Rensis Likert - a preference scale used to identify preferences in surveys.
Pregnanz(clear, clear) - refers to the Law of pregnancy, formulated by Ivo Köhler, one of the founders of Gestalt psychology. The law of pregnancy or "closure" is that "the elements of the field are isolated into forms that are the most stable and cause the least stress" (Forgus). So, if the image of a broken circle flashes on the screen with a high frequency, we will see this circle as a whole.
Understanding the object of measurement
the plate is borrowed from psylib.org.ua. Author - O.V. Belova
I recently received a message from a person who said that he liked my photos, but, unfortunately, he does not have a “photographic eye”. This prompted me to write the following article on the basics of aesthetics in photography.
Express your opinion
When we talk about aesthetics, we mean that some images are more attractive to our eyes, whether they are photographs, paintings or sculptures.
The difference between a photographer and any other person is not the ability to notice beauty, but that the photographer must be able to explain why certain elements are pleasing and others are not. Everyone has an understanding of aesthetics. Anyone can see it, but only a few can analyze the picture and explain the compositional techniques that create a beautiful image.
These techniques were not "invented" by expert artists. They have been found in a wide variety of disciplines. For example, the golden ratio has meaning not only in photography or painting, but also in architecture, mathematics, and even flower arranging. This means that we can apply some of these universal rules to create images that most people perceive as visually harmonious.
Composite elements
Leading lines
The viewer's eye is automatically guided by leading lines and other geometric shapes. Leading lines help put emphasis on the object that becomes the center of attention. If the eyes naturally follow the lines and finally stop at the object, a very harmonious impression is created.
Rule of thirds
The rule of thirds is based on a simplified principle of the golden ratio and divides the image into three equal areas. It helps to place the subject off-center and create a nice effect.
The ideal areas for placing objects are four points formed as a result of the intersection of lines parallel to the sides of the frame. In street photography, it is desirable to use high points. They will allow us to show more of the subject on which we want to focus.
triangles
Geometric shapes help create dynamic movement in a shot. They form an auxiliary basis that enhances the perception and unites the individual elements of the frame into a single whole. For example, geometric objects such as triangles and circles are popular.
odd rule
The previous photo already shows an example where three objects form a triangle. But the viewer is pleased to perceive not only three objects. 5 or even 7 points of interest can greatly increase the aesthetic value of an image.
This strange rule is explained by the fact that if objects are easy to arrange, put in pairs (2, 4, 6, etc.), then our brain becomes uninteresting.
break the symmetry
A symmetrical picture is a great achievement, but a 100% symmetrical frame is too clear. To make it more interesting, you can just place the object to the left or right of the section axis.
Summing up
These compositional techniques will help you create aesthetically pleasing photographs. You don't have to be born with some "exceptional" eyes to see interesting images. Every person has an aesthetic sense. The difference lies in being able to explain and recreate eye pleasing photographs or paintings.
The ground rules are an easy way to create a certain glow in the image, avoiding total chaos. In other words: an aesthetically successful image does not automatically become great. It's just a great base to lay out the plot.
movement: aestheticism
fine art type: painting
main idea: art for art
country and period: England, 1860-1880
In the 1850s in England and France there was a crisis of academic painting, the fine arts needed to be updated and found it in the development of new trends, styles and trends. A number of movements emerged in England in the 1860s and 1870s, including aestheticism, or aesthetic movement. Artists - aesthetes considered it impossible to continue to work in accordance with classical traditions and patterns; the only possible way out, in their opinion, was a creative search beyond the boundaries of tradition.
The quintessence of the aesthetes' ideas is that art exists for art's sake and should not be aimed at moralizing, glorifying or anything else. Painting should be aesthetically beautiful, but plotless, not reflecting social, ethical and other problems.
Sleepers, Albert Moore, 1882
The origins of aestheticism were artists who were originally supporters of John Ruskin, who were part of the Pre-Raphaelite Brotherhood, who by the early 1860s had abandoned Ruskin's moralizing ideas. Among them are Dante Gabriel Rossetti and Albert Moore.
"Lady Lilith", Dante Gabriel Rossetti, 1868
In the early 1860s, James Whistler moved to England and became friends with Rossetti, who led a group of aesthetes.
Symphony in White #3, James Whistler, 1865-1867
Whistler is deeply imbued with the ideas of aesthetes and their theory of art for art's sake. Whistler appended a manifesto of aesthete artists to the lawsuit filed against John Ruskin in 1877.
Whistler did not sign most of his paintings, but drew a butterfly instead of a signature, organically weaving it into the composition - Whistler did this not only during the period of passion for aestheticism, but throughout his entire work. Also, one of the first artists, he began to paint frames, making them part of the paintings. In Nocturne in Blue and Gold: The Old Bridge at Battersea, he placed the “signature” butterfly in a pattern on the frame of the picture.
Other artists who adopted and embodied the ideas of aesthetes are John Stanhope, Edward Burne-Jones, some authors also classify Frederick Leighton as aesthetes.
“ Pavonia” , Frederic Leighton, 1859
The difference between aestheticism and impressionism
Both aestheticism and impressionism appear at approximately the same time - in the 1860s and 1870s; aestheticism originates in England, impressionism - in France. Both of them are an attempt to move away from academicism and classical patterns in painting, and in both, the impression is important. Their difference is that aestheticism transformed the impression into a subjective experience, reflecting the artist's subjective vision of the aesthetic image, while impressionism transformed the impression into a reflection of the momentary beauty of the objective world.
The successes of modern natural science are inevitably associated with the development of physical and systemic pictures of the world, which are usually presented in the form of a natural hierarchy. At the same time, human consciousness, moving towards the study of the macro- and microworld, discovers more and more laws of motion, variability, relativity, on the one hand, and constancy, stability and proportionality, on the other.
In the eighteenth century the world of randomly and spontaneously arising whirlwinds of already known and yet unknown laws of nature was replaced by the world and the principle of an unchanging mathematical law. The world he ruled was no longer just an atomistic world where one arises, lives and dies by the will of an aimless chance. The picture of the metaworld, the megaworld appeared some sort of orderly formation, in which everything that happens can be predicted. Today we know the Universe a little more, we know that stars live and explode, and galaxies arise and die. The modern picture of the world has destroyed the barriers that separated the sky from the Earth, united and unified the Universe. Accordingly, attempts to understand the complex processes of interfacing with global patterns inevitably lead to the need to change the research paths along which science moves, because the new scientific picture of the world inevitably changes the system of concepts, shifts problems, and questions arise that sometimes contradict the very definitions of scientific disciplines. One way or another, the world of Aristotle, destroyed by modern physics, was equally unacceptable to all scientists.
The theory of relativity changed the classical ideas about the objectivity and proportionality of the Universe. It has become highly probable that we live in an asymmetric universe in which matter predominates over antimatter. The acceleration of ideas that modern classical physics has reached its limits is dictated by the discovery of the limitations of classical physical concepts, from which the possibility of understanding the world as such followed. When randomness, complexity and irreversibility enter physics as a concept of positive knowledge, we inevitably depart from the former very naive assumption about the existence of a direct connection between our description of the world and the world itself.
This development of events was caused by unexpected additional discoveries that proved the existence of universal and exceptional significance of some absolute, primarily physical, constants (the speed of light, Planck's constant, etc.), which limit the possibility of our influence on nature. Recall that the ideal of classical science was a "transparent" picture of the physical Universe, where in each case it was assumed that it was possible to indicate both the cause and its effect. But if there is a need for a stochastic description, the causal relationship becomes more complicated. The development of physical theory and experiment, accompanied by the emergence of more and more new physical constants, inevitably predetermined the increase in the ability of science to search for the One Principle in the diversity of natural phenomena. Repeating in some way the speculations of the ancients, modern physical theory, using subtle mathematical methods, as well as on the basis of astrophysical observations, strives for such a qualitative description of the Universe, in which an increasing role is no longer played by physical constants and constant quantities or the discovery of new elementary particles, but numerical relationships between physical quantities.
The deeper science penetrates at the level of the microcosm into the mysteries of the Universe, the more it reveals the most important the unchanging ratios and quantities that determine its essence. Not only man himself, but also the Universe began to be presented in exceptionally and surprisingly harmonious, proportional both in physical and, oddly enough, in aesthetic manifestations: in the forms of stable geometric symmetries, mathematically constant and precise processes that characterize the unity of variability and constancy . Such, for example, are crystals with their symmetry of atoms, or the orbits of the planets so close to the shape of a circle, proportions in plant forms, snowflakes, or the coincidence of the ratios of the boundaries of the colors of the solar spectrum or the musical scale.
This kind of invariably repeating mathematical, geometric, physical and other regularities cannot but encourage attempts to establish a certain commonality, a correspondence between the harmonic regularities of material and energy nature and the regularities of phenomena and categories of a harmonious, beautiful, perfect in the artistic manifestations of the human spirit. It is no coincidence, apparently, that one of the outstanding physicists of our time, one of the founders of quantum mechanics, the Nobel Prize winner in physics, V. Heisenberg, was simply forced, in his words, to “renounce” the concept of an elementary particle altogether, as physicists were forced in their time " discard" the concept of an objective state or the concept of universal time. As a consequence of this, in one of his works, W. Heisenberg wrote that the modern development of physics turned from the philosophy of Democritus to the philosophy of Plato; “... if we continue,” he noted, “to divide matter further and further, we will ultimately come not to the smallest particles, but to mathematical objects defined using their symmetry, Platonic solids and underlying them triangles. Particles in modern physics are mathematical abstractions of fundamental symmetries"(emphasis mine. - A. L.).
When stating this amazing by nature conjugation between heterogeneous, it would seem at first glance, phenomena and laws of the material world, natural phenomena, there is enough reason to believe that that both material-physical and aesthetic regularities can be expressed to a sufficient extent by force relations, mathematical series and geometric proportions similar to each other. In the scientific literature, in this regard, attempts have been repeatedly made to find and establish some universal objectively given harmonic ratios found in the proportions of the so-called approximate(complicated) symmetry, similar to the proportions of a number of natural phenomena, or direction, tendencies in this higher and universal harmony. Currently, several basic numerical quantities are distinguished, which are indicators of universal symmetry. These are, for example, the numbers: 2, 10, 1.37 and 137.
And magnitude 137 known in physics as a universal constant, which is one of the most interesting and not fully understood problems of this science. Many scientists of various scientific specialties wrote about the special significance of this number, including the outstanding physicist Paul Dirac, who argued that there are several fundamental constants in nature - the electron charge (e), Planck's constant divided by 2 π (h), and the speed of light (c). But at the same time, from a series of these fundamental constants, one can derive a number, which has no dimension. On the basis of experimental data, it has been established that this number has a value of 137 or very close to 137. Further, we do not know why it has this value, and not some other. Various ideas have been put forward to explain this fact, but no acceptable theory exists to this day.
However, it was found that next to the number 1.37, the main indicators of universal symmetry, which is most closely associated with such a fundamental concept of aesthetics as beauty, are the numbers: = 1.618 and 0.417 - the “golden section”, where the relationship between the numbers 1.37, 1.618 and 0.417 is a specific part of the general principle of symmetry. Finally, the numerical principle itself establishes the numerical series and the fact that the universal symmetry is nothing but a complicated approximate symmetry, where the principal numbers are also their reciprocals.
At one time, another Nobel Prize winner, R. Feynman, wrote that “we are always drawn to consider symmetry as a kind of perfection. This is reminiscent of the old idea of the Greeks about the perfection of circles, it was even strange for them to imagine that the planetary orbits are not circles, but only almost circles, but there is a considerable difference between a circle and an almost circle, and if we talk about the way of thinking, then this change is simply huge. A conscious theoretical search for the basic elements of a symmetrical harmonic series was already in the center of attention of ancient philosophers. It was here that aesthetic categories and terms received their first deep theoretical development, which were later laid down as the basis of the doctrine of shaping. In the period of early antiquity, a thing had a harmonic form only if it had expediency, quality factor, utility. In ancient Greek philosophy, symmetry acted in structural and value aspects - as a principle of the structure of the cosmos and as a kind of positive normative characteristic, an image of what should be.
The cosmos as a certain world order realized itself through beauty, symmetry, goodness, truth. The beautiful in Greek philosophy was considered as a kind of objective principle inherent in the Cosmos, and the Cosmos itself was the embodiment of harmony, beauty and harmony of parts. Given the rather debatable fact that the ancient Greeks “did not know” the very mathematical formula for constructing the “golden section” proportion, well known in aesthetics, its simplest geometric construction is already given in Euclid’s “Elements” in Book II. In Books IV and V, it is used in the construction of flat figures - regular pentagons and decagons. Starting from Book XI, in the sections devoted to solid geometry, the "golden section" is used by Euclid in the construction of spatial bodies of regular dodecagons and dodecagons. The essence of this proportion was also considered in detail in the Timaeus by Plato. By themselves, the two terms, argued the expert on astronomy Timaeus, cannot be well paired without a third, because it is necessary that between one and the other a certain connection uniting them should be born.
It is in Plato that we find the most consistent presentation of the main aesthetic formative principles with his five ideal (beautiful) geometric solids (cube, tetrahedron, octahedron, icosahedron, dodecahedron), which played an important role in the architectural and compositional representations of subsequent eras. Heraclitus argued that hidden harmony is stronger than explicit. Plato also emphasized that "the relations of parts to the whole and of the whole to the part can only arise when things are not identical and not completely different from each other." Behind these two generalizations, one can see a completely real and time-tested phenomenon and the experience of art - harmony rests on an order deeply hidden from external expression.
The identity of relations and the identity of proportions connect forms that are different from each other. At the same time, belonging of different relations to one system is spontaneous. The main idea, which was carried out by the ancient Greeks, who laid down the methods for calculating harmonically uniform structures, was that the quantities united by correspondence would not be either too large or too small in relation to each other. Thus, a way was discovered to create calm, balanced and solemn compositions, or area of average relations. At the same time, the greatest degree of unity can be achieved, Plato argued, if the middles are in the same relation to the extreme values, to what is more and to what is less, and there is a proportional relationship between them.
The Pythagoreans considered the world as a manifestation of some identical general principle, which embraces the phenomena of nature, society, man and his thinking and manifests itself in them. In accordance with this, both nature in its diversity and development, and man were considered symmetrical, reflecting in the connections “numbers” and numerical relations as an invariant manifestation of a certain “divine mind”. Apparently, it is no coincidence that it was in the school of Pythagoras that not only repeated symmetry was discovered in numerical and geometric ratios and expressions of numerical series, but also biological symmetry in the morphology and arrangement of leaves and branches of plants, in a single morphological structure of many fruits, as well as invertebrate animals.
Numbers and numerical relations were understood as the beginnings of the emergence and formation of everything that has a structure, as the basis of a correlatively connected diversity of the world, subordinate to its unity. The Pythagoreans argued that the manifestation of numbers and numerical relations in the Universe, in man and human relations (art, culture, ethics and aesthetics) contains a certain single invariant - musical and harmonic relations. The Pythagoreans gave both numbers and their relations not only a quantitative, but also a qualitative interpretation, giving them reason to assume existence at the foundation of the world. some faceless life force and the notion of an internal connection between nature and man, constituting a single whole.
According to historians, already in the school of Pythagoras, the idea was born that mathematics, mathematical order, is a fundamental principle by which the entire multiplicity of phenomena can be justified. It was Pythagoras who made his famous discovery: vibrating strings, stretched equally strongly, sound in tune with each other if their lengths are in simple numerical ratios. This mathematical structure, according to W. Heisenberg, namely: numerical ratios as the root cause of harmony - was one of the most amazing discoveries in the history of mankind.
since the varieties of musical tones are expressible in numbers and all other things seemed to the Pythagoreans to be modeled figures, and the numbers themselves - primary for all nature, the heavens - a set of musical tones, as well as numbers, understanding of the entire richly colored variety of phenomena was achieved in their understanding through awareness of the inherent in all phenomena unifying principle of form expressed in the language of mathematics. In this regard, the so-called Pythagorean sign, or pentagram, is of undoubted interest. The Pythagorean sign was a geometric symbol of relations, characterizing these relations not only in mathematical, but also in spatially extended and structural-spatial forms. At the same time, the sign could manifest itself in zero-dimensional, one-dimensional, three-dimensional (tetrahedron) and four-dimensional (hyperoctahedron) space. As a consequence of these features, the Pythagorean sign was considered as a constructive principle of the world and, above all, of geometric symmetry. The sign of the pentagram was taken as an invariant of the transformation of geometric symmetry not only in inanimate, but also in living nature.
According to Pythagoras, things are an imitation of numbers, and consequently, the whole Universe is a harmony of numbers, and only rational numbers. Thus, according to Pythagoras, the number either restored (harmony) or destroyed (disharmony). Therefore, it is not surprising that when the irrational "destructive" number of Pythagoras was discovered, he, according to legend, sacrificed 100 fat bulls to the gods and took an oath of deep silence from his students. Thus, for the ancient Greeks, the condition for some kind of sustainable perfection and harmony was the need for the obligatory presence of a proportional connection or, in the understanding of Plato, a consonant system.
It was these beliefs and geometric knowledge that formed the basis of ancient architecture and art. For example, when choosing the main dimensions of a Greek temple, the criterion for height and depth was its width, which was the average proportional value between these dimensions. In the same way, the relationship between the diameter of the columns and the height was realized. In this case, the criterion that determines the ratio of the height of the column to the length of the colonnade was the distance between the two columns, which are average proportional values.
Much later, I. Kepler succeeded in discovering new mathematical forms for generalizing the data of his own observations of the orbits of the planets and for formulating the three physical laws that bear his name. How close Kepler's reasoning was to the argument of the Pythagoreans can be seen from the fact that Kepler compared the revolution of the planets around the sun with the vibrations of strings, spoke of the harmonic coherence of various planetary orbits and the "harmony of the spheres." At the same time, I. Kepler speaks of certain prototypes of harmony, immanently inherent in all living organisms, and of the ability to inherit prototypes of harmony, which lead to shape recognition.
Like the Pythagoreans, I. Kepler was fascinated by attempts to find the basic harmony of the world, or, in modern terms, the search for some of the most general mathematical models. He saw mathematical laws in the structure of pomegranate fruits and in the movement of the planets. Pomegranate seeds represented for him important properties of the three-dimensional geometry of densely packed units, because in pomegranate evolution gave place to the most rational way of placing as many grains as possible in a limited space. Almost 400 years ago, when physics as a science was just emerging in the works of Galileo, I. Kepler, we recall, referring to himself as a mystic in philosophy, quite elegantly formulated, or, more precisely, discovered, the riddle of building a snowflake: “Since every time, as soon as it starts to snow, the first snowflakes are in the shape of a six-pointed star, then there must be a very definite reason for this, for if this is an accident, then why are there no pentagonal or heptagonal snowflakes?
As a kind of associative digression associated with this regularity, we recall that back in the 1st century. BC e. Marius Terentius Varon argued that honeycombs of bees appeared as the most economical model of wax consumption, and only in 1910 did the mathematician A. Tus offer convincing evidence that there is no better way to implement such a stacking than in the form of a honeycomb hexagon. At the same time, in the spirit of the Pythagorean harmony (music) of the spheres and Platonic ideas, I. Kepler made efforts to build a cosmographic picture of the solar system, trying to connect the number of planets with the sphere and Plato's five polyhedra in such a way that the spheres described near the polyhedra and inscribed in them coincided with planetary orbits. Thus, he obtained the following order of alternation of orbits and polyhedra: Mercury is an octahedron; Venus - icosahedron; Earth - dodecahedron; Mars is a tetrahedron; Jupiter - cube.
At the same time, I. Kepler was extremely dissatisfied with the existence of huge tables of figures calculated in his time in cosmology and was looking for general natural patterns in the circulation of the planets that remained unnoticed. In two of his works - "New Astronomy" (1609) and "Harmony of the World" (circa 1610) - he formulates one of the systemic laws of planetary revolution - the squares of the time of a planet's revolution around the Sun are proportional to the cube of the planet's average distance from the Sun. As a consequence of this law, it turned out that the wandering of the planets against the background of "fixed", as it was then believed, stars - a feature previously not noticed by astronomers, bizarre and inexplicable, follows hidden rational mathematical patterns.
At the same time, a number of irrational numbers are known in the history of the material and spiritual culture of man, which occupy a very special place in the history of culture, as they express certain relations that are of a universal nature and manifest themselves in various phenomena and processes of the physical and biological worlds. Such well-known numerical relations include the number π, or "non-Peer number".
One of the first who mathematically described the natural cyclic process obtained in the development of the theory of biological populations (for example, the reproduction of rabbits), corresponding to the approximation to the "golden ratio", was the mathematician L. Fibonacci, who back in the 13th century. deduced the first 14 numbers of the series, which made up the system of numbers (F), later named after him. It was at the beginning of the Renaissance that the numbers of the "golden section" began to be called "Fibonacci numbers", and this designation has its own background, repeatedly described in the literature, so we only briefly give it in a note. .
The Fibonacci series has been found both in the distribution of growing sunflower seeds on its disk, and in the distribution of leaves on the trunk and in the arrangement of the stems. Other small leaves framing the disk of the sunflower formed curves in two directions during growth, usually the numbers 5 and 8. Further, if we count the number of leaves located on the stem, then here the leaves were arranged in a spiral, and there is always a leaf exactly located above the lower one. sheet. In this case, the number of leaves in the coils and the number of coils are also related to each other, as is the adjacent number Ф. This phenomenon in wildlife has received the name phylotaxis. The leaves of plants are arranged along the stem or trunk in ascending spirals so as to provide the greatest amount of light falling on them. The mathematical expression of this arrangement is the division of the "leaf circle" in relation to the "golden section".
Subsequently, A. Durer found the pattern of the "golden section" in the proportions of the human body. The perception of art forms created on the basis of this ratio evoked the impression of beauty, pleasantness, proportion and harmony. Psychologically, the perception of this proportion created a feeling of completeness, completeness, balance, calmness, etc. And only after the publication in 1896 of the well-known work by A. Zeising a thorough attempt to revisit the "golden section" as a structural one, first of all - aesthetic invariant of the measurer of natural harmony, in fact, synonymous with universal beauty, the principle of the “golden section” was proclaimed the “universal proportion”, which manifests itself both in art and in animate and inanimate nature.
Further in the history of science, it was discovered that not only the ratios of Fibonacci numbers and their neighboring ratios lead to the "golden ratio", but also their various modifications, linear transformations and functional dependencies, which made it possible to expand the patterns of this proportion. Moreover, it turned out that the process of arithmetic and geometric "approximation" to the "golden ratio" can be counted. Accordingly, we can talk about the first, second, third, etc. approximations, and all of them turn out to be associated with the mathematical or geometric laws of any processes or systems, and it is these approximations to the "golden division" that correspond to the processes of sustainable development of almost all without exception natural systems.
And although the very problem of the “golden section”, the remarkable properties of which, as proportions of average and extreme ratios, were tried to be theoretically substantiated by Euclid and Plato, of an older origin, the curtain over nature itself and the phenomenon of this wonderful proportion has not been completely lifted to date. Nevertheless, it became obvious that nature itself, in many of its manifestations, acts according to a clearly defined scheme, implements the search for optimization of the structural state of various systems not only genetically or by trial and error, but also according to a more complex scheme - according to the strategy of the living series of Fibonacci numbers. The "golden section" in the proportions of living organisms was found at that time mainly in the proportions of the external forms of the human body.
Thus, the history of scientific knowledge associated with the "golden proportion", as already mentioned, has more than one millennium. This irrational number attracts attention because there are practically no areas of knowledge where we would not find manifestations of the laws of this mathematical relationship. The fate of this remarkable proportion is truly amazing. It not only delighted ancient scientists and ancient thinkers, it was deliberately used by sculptors and architects. The ancient thesis about the existence of single universal mechanisms in man and nature reached its highest general humanitarian and theoretical flowering during the period of Russian cosmism in the works of V. V. Vernadsky, N. F. Fedorov, K. E. Tsiolkovsky, P. A. Florensky, A. L. Chizhevsky, who considered man and the Universe as a single system, evolving in the Cosmos and subject to universal principles, which make it possible to accurately state the identity of both structural principles and metric relations.
In this regard, it is quite significant that for the first time such an attempt to highlight the role of the "golden ratio" as a structural invariant of nature also did the Russian engineer and religious philosopher P. A. Florensky (1882-1943), who in the 20s. 20th century The book “At the Watersheds of Thought” was written, where one of the chapters contains reflections on the “golden section” and its role at the deepest levels of nature, exceptional in its “innovation” and “hypotheticality”. This kind of variety of AP appearances in nature testifies to its complete exclusivity, not only as an irrational mathematical and geometric proportion.
The role played by the "golden section", or, in other words, the division of lengths and spaces in the middle and extreme ratio, in matters of aesthetics of spatial arts (painting, music, architecture) and even in non-aesthetic phenomena - the construction of organisms in nature, has long been is noted, although it cannot be said that it has been revealed and its final mathematical meaning and significance are unconditionally determined. At the same time, most modern researchers believe that the "golden section" reflects the irrationality of the processes and phenomena of nature.
As a consequence of its irrational property, the inequality of the conjugated elements of the whole, connected by the law of similarity, expresses the “golden section” measure of symmetry and asymmetry. Such a completely unusual feature of the "golden section" allows you to build this mathematical and geometric treasure in a row invariant essences of harmony and beauty in works created not only by mother nature, but also by human hands - in numerous works of art in the history of human culture. Additional evidence of this is the fact that this proportion is referred to in the creations of man. in completely different civilizations, separated from each other not only geographically, but also temporally - millennia of human history (the Cheops pyramid and others in Egypt, the Parthenon temple and others in Greece, the Baptistery in Pisa - the Renaissance, etc.).
- derivatives of the number 1 and its doubling by additive addition, give rise to two famous in botany additive rows. If the numbers 1 and 2 appear at the source of a series of numbers, the Fibonacci series appears; if at the source of a series of numbers the numbers 2 and 1, there is a Lucas series. The numerical position of this pattern is as follows: 4, 3, 7, 11, 18, 29, 47, 76 - Luke's row; 1, 2, 3, 5, 8, 13, 21, 34, 55 - Fibonacci series.The mathematical property of the Fibonacci series and the Lucas series, among many other amazing properties, is that the ratios of two adjacent numbers in this series tend to the number of the "golden section" - as you move away from the beginning of the series, this ratio corresponds to the number Ф with increasing accuracy. Moreover, the number Ф is the limit to which the ratios of neighboring numbers of any additive series tend.
