The emotion of geometry

The Emotion of Geometry

Introduction 04 Research Report 08 The Divine Proportion 20 The Analysis of nature 26 Harmony and Dynamic symmetry 30 Conclusion 34 Bibliography 38

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Introduction

From east to west, far in the antiquity, our ancestors already discovered the form and use of geometry from its existence in nature. Evidences such like the structures of ancient Greek buildings, the compositions in paintings to the revolutions of Renaissance. Like our ancestors, I’ve always been curious about the mysterious geometry that inevitably builds up our universe as well as its possibility to alter our perceptions through different structural arrangements. I believe my interest in geometry came in nature, which from my point of view was nothing more then ordinary; as a part of this geometrical constructed universe, it seems more than reasonable to be sensitive of the phenomena of geometry. We all have this experience of noticed some particular object around us is generated by geometric elements; it might just be the mug on your table, the tree outside the window, or the window itself. We stared, and for a very mere second, we emotionally reacted to the observed object, sometimes we reacted for a even longer period of time; we started to wonder why were we feeling the way we were feeling, what is that secure and steady sensation we perceived from geometrically constructed objects? Though I had so many questions, I had no intention to study into this subject matter before a research on the famous phenomenon of snowflakes.

We all know snowflakes are exquisite and beautiful miracles in nature, but actually studied into the details, deeply amazed me. They say there are no two same snowflakes in the world; every each of them are created in a geometric order of a six-fold symmetry hexagonal shape but formed in an irregularly way. I started to think about not only the fascinated existence of geometry in nature, even more, the mysterious constructional process. What is the underlying physical law behind it? How do mathematical rules applied? How does it evoke observer’s emotions like it evoked mine?

What I supposed was, since geometry is one of the basic elements that constructed the universe and human body, different geometrical shapes and their constructions must have the ability to deliver some sort of “feelings” that could stir up different emotions of human beings; and the emotional reactions must have their way to express themselves through human’s creativity. This creative power thus transformed into a variety of creations in our material culture through different pathways such as architecture, art and design. And if my suggestion here was correct, the feelings delivered from geometrical properties do have something to do with viewer’s perceptions, this furthermore rose up another controversial issue that I have always been curious about and has been discussed between art critics for a very long time: the indefinable reason of why some designs looks “right”, in other words, harmonious and balanced, but some looks “wrong”.

Editor Alva Wong (2008, p27) of IDN magazine, in The Geometric issue, stated her view of geometry, which encouraged me to start on the research: ‘it is a mistake to think of geometry as a cold, mathematically discipline, shape transmit feelings.’

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My aim is to investigate the relation of geometry and nature, along with it’s interplay with human psyche and creativity, therefore my research will include four main subject areas: mathematics, biology, psychology, and art.

It is important t that nature’s s conveys no mor loveliness hidd mathematics is it must be disco

to bear in mind surface beauty re than a hint of den within. The s not in its skin: overed.

(H. E. Huntley, 1970, p151)

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p re rch

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It is important to bear in mind that Nature’s surface beauty conveys no more than a hint of loveliness hidden within. The mathematics is not in its skin: it must be discovered. (H. E. Huntley, 1970, p151) My research started with exploring the relation between mathematics and nature. After snowflakes, another geometric phenomenon that caught my attention was the repetitive order of geometrical patterns in nature; the honeycombs for example, constructed from nearly identical hexagons, is an important sample of pattern formation in nature.

These patterns, obviously speak for themselves about the beauty of geometry, but there are many less obvious samples reside in nature; like the growing patterns of plants, the Saturn’s ring and the orbits of planets. However, I did not go further into exploring the formation of geometry in nature at this point; I realized my curiosity was not how geometry works in nature, but was why it works.

What is the relationship of geometry and this harmonious universe? How do we uncover

8 the mathematical rules and use it to decode the phenomena in nature?

Since the subject matter that I wanted to study here has very much to do with mathematics, my research turned exclusively onto mathematics, or more precisely, the definition of beauty in mathematics, which include a variety of different suggestions.

The most common instance of beauty in mathematics is, a brilliant step in an otherwise undistinguished proof. Every mathematician quickly becomes familiar with this kind of mathematical beauty. (Gian-Carlo Rota, Michele Emmer 1993, p4) Some mathematicians claim also that beauty acts as a guide in making mathematical discoveries and that beauty is an objective factor in establishing the validity and importance of a mathematical result. (James W. McAllister, Michele Emmer 1993, p15)

These are beauty in the process of solving a mathematical problem; the emotional reactions we experienced during the discovery of theorem or proof. However, ‘there are two classes of mathematical entities to which beauty may be attributed: process and product.’ (James W. McAllister, Michele Emmer 1993, p17) Instead of the mathematical process, I focused on the product, the beauty of the entities of theorem and proof.

‘A beautiful theorem may not be blessed with an equally beautiful proof; however, it is impossible to find a beautiful proof of theorems that are not beautiful.’ (Gian-Carlo Rota, Michele Emmer, 1971, p4) If we believed beauty does resides in mathematics, and so can produce a mental ferment to generate new ideas in mathematical process, therefore, it seems reasonable to believe mathematical products do have the ability to deliver the sense of beauty.

Geometry is another mathematical product. Geometrical constructions exhibit obvious aesthetic beauties, such as polygons, the Regular

9 soli d s , Golden Section and Symmetry, which is an important aesthetic property of geometrical constructions that can be manifested as regularity, pattern, proportion, or self-similarity and will be given detail later.

The word “Geometry” in the language of ancient Greek is a combination of two words: geo = earth and metria = measure; geometry from its very nature, means to measure the earth, concerned with questions of size, shape, relative position of figures and with properties of space. Euclidean geometry from the third century BC set a standard for many centuries to follow. One important geometrical construction that caught my attention during research was “The Regular solids”, which have been known since antiquity, some sources credit philosopher Pythagoras with their discovery. Euclid also gave a complete mathematical description of the Regular solids in The Elements.

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From plan to solid geometry, measurement and composition play capital roles in geometry; research into the famous theorem “The golden ratio” was inevitable. Ancient Greek mathematicians first studied golden ratio because of its frequent appearance in geometry, and again, they usually attributed discovery of this concept to Pythagoras, but Euclid’s Elements provides the first known written definition of what is now called the Golden ratio. Derived from the theory of ratio, proportion is the most important concept in composition. Artist Georges Vantongerloo assumed ‘One must have proportion to have art.’ The geometrical proportion is the mathematical aspect of the very general and important concept of analogy, which has influenced countless people and has been used commonly in the creativity world.

Further more, the most important plane figure in composition is the rectangle; the most important characteristic of a rectangle is indeed its proportion of ratio, the ratio between the longer and the shorter side.

This led my study towards the “Dynamic Rectangle” and Fibonacci numbers. At this point, I suddenly realized my research had gone from the phenomena in nature to mathematics, and now somehow, I am back to nature again; the relation between mathematics and nature is evidently unable to separate.

The Golden Section also give evidence related to nature; in the curves and diagrams connected with the growth and shapes of plants, flowers and some marine organisms like starfishes, and in the proportions of the human body. The Fibonacci number’s appearance in the spirals of the seeds in sunflowers is undoubtedly a striking appearance of the geometrical structure in nature.

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Just as many plants and animals share golden section proportions, human do as well. Some of the earliest surviving written investigations into human proportions are in the writings of ancient Greek scholar and architect Marcus Vitruvius Pollio. Vitruvius advised that the architectural temples should be based on the likeness of the perfectly proportioned human body where “harmony” exists among all parts. And “Harmony” here is an important concept in my research, which I will look at later. Renaissance artists Leonardo da Vinci’s and Albrecht Durer both studied in human proportion, we can capture the evidence from Durer’s Four Book of Human Proportion and Leonardo’s illustrations for mathematician Luca Pacioli, in his Divine Proportion. Individually, both Leonardo’s and Durer’s drawings clearly match to the proportioning system of Vitruvius. Later, Swiss architect Le Corbusier and his book Le Modular is another significant support in exploring the relation of Golden Section and human body.

Symmetry is one more essential concept when it comes to composition.

Mirror Symmetry is one of the simplest methods of creating a pleasure design, why it should instinctively please is not easy to say, but men’s penchant for symmetrical arrangements might be no more than an unconscious manifestation of his admiration for his own symmetrical self.

(Michael Holt, 1971, p72) This perception derived my study to psychologist Carl G. Jung’s “Collective unconscious”, which I will give details later. The earliest examples of consciously produced symmetry in art are the heraldic or bilateral symmetry of the Sumerians.

Even though I understood many theorist hold different opinions on whether these concepts I stated here are the real truth, as Mathematician Michael Holt (1971, pp16-17) suggests in his writing: Mathematical truth are, then, man-made, a one-sided reflection of culture. For instance, the Choctaw Indians never distinguished between the colours yellow and green, whereas a European might think such a distinction obvious and beyond man’s control.

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My research still carried on towards the same direction, neglected the varied opinions. And now it reached the point of the relation of mathematics and art. Michael Holt indicated the beginnings of mathematics and art in his writing: While mathematics is some five thousand years old, art, as we understand it is a surprising recent idea.

Before the advent of museums and galleries, art was expression of practical problems. Many of the Greek buildings that we admire as works of art were originally regarded simply as practical solutions to every day problems. (Michael Holt, 1971, p8) But I was not interested in the historical development of the concept of art, but was how mathematics has been applied to the development of art.

More precisely, how geometry has been applied to the harmonious composition among all visual matter. Possibly, the connection between them lies in the idea of a common culture. Painter Seurat: ‘Art

is harmony. Harmony is the analogy of opposites, the analogy of similar, of tone, of colour, of line…’

And mathematics, too, is concerned with relations of sameness, difference and order and with things like points, line, plane, and spaces. ‘Although mathematics is based on logic; the other is on an imaginative reconstruction of perceptual images; one is something we appreciate with our eyes and respond to with our emotions and the other with our minds and our intellect.’ (Michael Holt, 1971, p17) Here I came cross with an important concept in my study, “Emotion”, which will be explained later. Philosopher Alfred North Whitehead: ‘The laws of nature are mathematical laws in disguise.’ The mathematician uses symbols to see through the disguise, when the artist create new disguise for the nature.

Their common bond is to search for the structure of the universe, and this is the point where mathematics, art and nature all encountered and proved their inseparability. Painter Cezanne had heralded the revival of geometric art when he suggested that all nature could be represented by the ‘cylinder, the sphere, and the cone.’ And he came as near as one could to defining the artist’s and the mathematician’s job in one phrase: ‘I have not tried to reproduce nature; I have tried to represent it.’ Artist Mondrian saw nature as ‘a dictionary’ and thought he was representing the real world of science. These artists believed mathematics are the fundamental structure of universe, and are good examples of my study at this stage, but was not exactly what I was looking for.

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As I mentioned in my previous research, rectangle and symmetry are important elements in composition.

Dynamic rectangles that show irrational numbers in their proportions are most used. Could not think of design as purely instinctive, Artist Jay Hambidge turned to study dynamic symmetry, the law of natural design used by the Egyptians and the Greeks, which is based on the symmetry growth in human and plants. His hypothesis “Dynamic symmetry” is being widely used by artists now days; it is a very useful technique of creating a “harmonic composition.”

Computer scientist Jaroslav Nesetril: ‘What we would like to decide is how to formalize the fact that a picture or drawing is “harmonious” ’. Here I again, came crossed one of the key conception of my study, “Harmony”.

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We meant harmonious in the sense of being aesthetically pleasing. We prefer the word ”harmonious” to ”aesthetic”, which is probably more common, as the aesthetic feeling in certainly highly individual, and we cannot hope to define it accurately. The long tradition of art criticism is convincing on this point. (Jaroslav Nesetril, Michele Emmer, 1993, p35)

It is difficult to confine beauty to either objective or subjective categories. It seems to be more satisfactory to regard it as an interaction between the mind and an object or an idea, which arouses emotion. The discovery of beauty either in nature or in mathematics is indicative of some feature in the structure of the mind, and this concept directed my research into the psychological aspects. From the proverb ‘Beauty is in the eye of the beholder’, Mathematician James W. McAllister interpreted how we claim that a certain entity is beautiful is that the entity has a property named ”Beauty”, which the observer has perceived; that “Beauty” is a value that is projected into or attributed to objects by observers, not a property that intrinsically resides in objects.

Whether an observer projects beauty into an object is determined by two factors: the aesthetic criteria held by the observer, and the object’s intrinsic properties.

(James W. McAllister, Michele Emmer, 1993, p16) Emotion is regarded by psychologists as activities of the unconscious mind, so the aesthetic experience is the resuscitation of subliminal emotions, and beauty is the power to evoke these emotions; so aesthetic experience is an emotional, rather than a rational mental activity. (H. E. Huntley, 1970, p14)

When it comes to mental, I had to point out what has been called the greatest discovery of the 19th century, the Subconscious mind. The following excerpts from the writing of psychologist Carl G. Jung: There are definite reasons why I divide the unconscious into two parts. The personal unconscious contains everything forgotten, or repressed, or otherwise subliminal that has been acquired by the individual consciously or unconsciously. Such materials have an unmistakable personal stamp. But other contents are to be found, often enormously strange to the individual and bearing scarcely a trace of personal quality. You may discover such materials frequently in insanity, where they contribute not a little to the confusion and disorientation of the patient. In dreams of normal people such strange contents also occasionally appear…

It is not difficult to define what world these belong to: it is the world of the primitive mind which is deeply unconscious in cultured moderns so long as they are normal, but which rises to the surface when something fatal happened to the conscious.

“Collective” because it is not an individual acquisition, but rather the functioning of the inherited brain structure, which, in general, is the same in all human beings, and, in certain respects, is the same in all human beings, and, in certain respects, is the same to all mammals. The inherited brain is the inheritable of the ancient psychic life. It consists of the structural deposits of psyche activities repeated innumerable times in the lives of our ancestors. The collective unconscious is in no sense an obscure corner of the mind, but the all-controlling deposit of ancestral experience from untold millions of years, the echo of prehistoric world events to which each century adds an infinitesimally small amount of variation and differentiation.

“Collective unconscious”, according to Jung, forms a lower layer of psyche than the personal unconscious; it is the source of instinctive behavior, and its inherited, determined by the history of the race, which were formed at low mental levels during the tens of thousands of years of the evolutionary history of primitive man, by the constant reappearance of universal emotional experiences common to everyone, like the alternation of day and night. This instinctive behavior thus became an important concept in my study, as our reactions that motivated by our perceptions, to composition for instance, are instinctive behaviors.

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which he associates the human psyche with the physical nerves systems.

The power to appreciate beauty appears to be a human endowment and this suggests that we should seek its origin and its purpose in human nature, in the nature that distinguishes us from the animal creation. (H. E. Huntley, 1970, p20) He assume that the evolution of psychic potentialities through geological ages has run parallel to the development of the nervous system and the brain, it would appear that, historically, emotional life, which we share with the higher animals, must precede intellectual development and be associated with the primitive parts of the nervous system. He suggested that the personal unconscious, as well as the collective unconscious, is the arena of the emotions as well as the storehouse of emotive memory complex. Since the structure of the nervous system is inherited, it is reasonable to suppose that the physiological conditions favorable to the animation of primordial emotions of the collective unconscious are also handed down generation to generation. It is therefore natural to assume that the tendency of the psyche to make certain broad aesthetic judgments relating to the common human environment is inherited.

Psychologist H. J. Eysenck refers to the hypothesis, based on experiments, that ‘there exists some property of the nervous system which determines aesthetic judgment, a property which is biologically derived…one deduction, for instance, might be that this ability [aesthetic judgment] should be very strongly determined by heredity’.

Additionally, an explanation of our human nature is given in Genesis 1, v.26: ‘And God said, “Let us make man in our image, after our likeness”.’ According to this, Huntley suggested, ”Man is by nature a creator”.

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17 After the likeness of his Maker, man is born to create, to fashion beauty and to originate new values. This truth awakens a resonant response deep within us, for we know that one of the most intense joys that the soul of man can experience is the spiritual satisfaction associated with the moment of orgasm of creation. This inborn love of beauty, our human heritage, must find expression if we are to be happy. (H. E. Huntley, 1970,pp20-21) In The Education of the Whole Man, Philosopher L. P, Jacks writes: ‘What then is the vocation of the whole man? So far as I can make out, his vocation is to be a creator: and if you ask me, Creator of what? I answer, creator of real values. And if you ask me what motive can be appealed to, what driving power can be relied on, to bring out the creative element in men and women, there is only one answer I can give; but I give it without hesitation, the love of beauty, innate in everybody.’ In addition, Huntley furthermore explained for those that claimed have never experienced this creative motivation, that by the act of appreciation of beauty, a man is re-enacting the creative act. Besides, being attracted by beauty is experiencing the joy of creative activity itself.

To here, I had in conclusion finalized my positions in the research; I have explored the mystery of nature along with mathematics, the purpose of beauty by recalling the familiar fact that inborn ability of aesthetic appreciation constitutes the motive for creation of object of beauty. However, in order to present the entire conception, more details has to be given in the following writings.

The Divine Proportion The Golden Section

The ancient Greeks studied the phenomenon of golden ratio because of its frequent appearance in geometry, which resides all over in nature. An important phenomenon found by psychologist Adolf Zeising, present evidences of the existence of golden ratio in nature. He found the golden ratio expressed in the arrangement of branches along the stems of plants and of veins in leaves. He extended his research to the skeletons of animals and the branching of their veins and nerves, to the proportions of chemical compounds and the geometry of crystals. In these phenomena he saw the golden ratio operating as a universal law. Further evidences of golden ratio in nature will be explained soon after.

The modern history of golden ratio started with Renaissance mathematician Luca Pacioli, who defined golden ratio as the “Divine proportion” in his Divina Proportione [On the Divine Proportion] of 1509, which was illustrated by Leonardo da Vinci. This publication captured the imagination of artists, architects and mathematicians of the notion of golden ratio in Renaissance; a body of literature on the aesthetics of this topic had developed. As a result, architects, artists, book designers, and others have been encouraged to use the golden ratio in the dimensional relationships of their works, believing this proportion to be aesthetically pleasing. Mathematician Johannes Kepler describes the golden ratio as a “precious jewel”:

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‘Geometry has two great treasures: one is the theorem of Pythagoras, and the other is the division of a line into extreme and mean ratio; the first we may compare to a measure of gold, the second we may name a precious jewel.’

The Golden Ratio also seems to been used by ancient Greek architects and sculptors. The Parthenon in Athens is an example of the Greek system of proportioning. The Parthenon’s facade as well as elements of its facade and elsewhere are say to be circumscribed by golden rectangles. Classical buildings or their elements are proportioned according to the golden ratio, this might indicate that their architects were aware of the

golden ratio and consciously employed it in their designs.

Alternatively, it is possible that the architects used their own sense of good proportion, and that this led to some proportions that closely approximate the golden ratio.

Some psychologists, starting with Gustav Fechner, had been devised to test the idea that the golden ratio plays an important role in human perception of beauty. A rectangle, the sides of which are in the golden ratio, is called the golden section rectangle, which shape appear to have aesthetics attractions superior to that of other. A thorough yet casual experiment of Gustav Fechner supports this observation.

Fechner made literally thousands of ratio measurements of commonly seen rectangles, including playing cards, windows, writing paper pads, book covers, and found that the average was close to Phi, the golden ratio. He also extensively tested personal preferences, and finally established that most people prefer a certain rectangle that the proportions of which lie between those of a square and those of a double square. If the result means what it means, it point out unambiguously to a popular preference for a rectangle shape closely approximate to the golden rectangle. The observation offered here may be related to the well-known fact that certain musical intervals are more acceptable by the mind than the others because they are more harmonious. Fechner’s experiments were repeated later in a more scientific manner by others, and the results were remarkably similar.

It is not surprising that many writers have dismissed the whole subject as nonsense. Nevertheless, it is hard to believe the golden section rectangle, supported as it is by many experiments, are entirely nonsense. It is wiser to regard this difference of opinion as just another example of the notorious difficulty of finding a rational explanation for aesthetic preferences. But the difficulty of accounting for a phenomenon does not invalidate its reality.

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Le Corbusier and The Modular Swiss architect Le Corbusier, famous for his contributions to the modern international style, focused his design philosophy on systems of harmony and proportion. Le Corbusier’s faith in the mathematical order of the universe was closely bound to the golden ratio and the Fibonacci series, which he described in his Towards a New Architecture as:

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Rhythms apparent to the eye and clear in their relations with one another. And these rhythms are at the very root of human activities. They resound in man by an organic inevitability, the same fine inevitability that causes the tracing out of the Golden Section by children, old men, savages and the learned.

Le Corbusier explicitly used the golden ratio in his “modular” system for the scale of architectural proportion. He took Leonardo da Vinci’s suggestion of the golden ratio in human proportions to an extreme: he sectioned his model human body’s height at the navel with the two sections in golden ratio, then subdivided those sections in golden ratio at the knees and throat. He saw this system as a continuation of the long tradition of Marcus Vitruvius Pollio, Leonardo da Vinci’s “Vitruvian Man”,

Leon Battista Alberti, and others who used the proportions of the human body to improve the appearance and function of architecture. In his book The modular: A harmonious Measure to the Human Scale Universally Applicable to Architecture and Machines, chronicled his proportioning system on the mathematics of the golden section and the proportion of the human body.

An important feature of Le Corbusier’s modular is the use of “Regulating line”, a mean to create order and beauty, and for achieving visual balance.

Despite the critics argued that ‘with the regulating lines he is killing imagination, and make a god of a recipe’, Le Corbusier believed the understanding of the underlying organizational principle of geometry brings to a creative work a sense of compositional cohesiveness. It gives insight into the process of realization and a rational explanation for many decisions, whether the use of organizational geometry is intuitive or deliberate, rigidly applied or casually considered. His modular is one that is valued for all artists, designers and architects.

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Golden section spiral The golden section rectangle can be subdivided, which concept will be explained later on. An important idea is when subdivided a golden section, its reciprocal is a smaller proportional rectangle and the area remaining after subdivision is a square. The proportionally decreasing squares can produce a spiral by using a radius the length of the sides of the square, which we called the Golden section spiral.

The power of the golden section to create harmony arises from its unique capacity to unite different parts of a whole so that each preserves its own identity, and yet blends into the greater pattern of a single whole. (Gyorgy Doczi, 1994, Kimberly Elam, 2001, p8) The golden spiral can be found in the pattern of growth in living things such as plants and animals. A famous instance is the logarithmic spiral found in the contour spiral shapes of shells; one of the most beautiful of mathematical curves. An interesting property of this spiral is worth noting: However different two segments of the curves may be in size they are not different in shape; the spiral is without a terminal point: it may grow outwards or inwards indefinitely, but the shape remains unchanged.

The shells attract observers both by the lustrous exterior and by the perfection of the spiral curves, it has been of common occurrence in the natural world for millions years. I was reminded once again that aesthetic appreciation of any sort has two aspects; the immediate emotion evoked by beauty is a common human experience,inborn and lying in our unconscious; but this satisfaction can be developed further by education.

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The Expressive line One of the gentle satisfactions enjoyed by all our ancestors, which must have left its mark on the unconscious mind, is the smooth sweep of eye along the curves found in nature. The smoothness of their contours is associated with the ease and comfort of the eye’s muscular effort. Jagged and jerky lines have been shown by psychologists to produce an opposite mental effect. The curves that the human gaze has followed for a million years include the sea horizon, the rainbow, the parabola of the waterfall, the arcs traced in the sky by the sun and the crescent moon, the flight of the bird and many others. Such purely sensuous please is an ingredient of the aesthetic joy found in the geometry of the circle, the ellipse and other conic sections.

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The following is Psychologist Johan Lundstrom’s experiments in the area of the “Expressiveness of line”.

When ask to draw a beautiful line, Lundholm’s subjects tried to make one that was smooth, curved, and symmetrical continuous; for an ugly line they drew an unorganized mass without continuity, and when they wished to express merriment, playfulness or fury, they drew sharp waves or zigzags.

Lines, one of the basic elements in constructing not only the curves, but all geometrical shapes and compositions; are to be believed having the possibility to alter our perceptions as other geometrical elements could have done.

e tur Na of is lys na eA Th 26 Pythagoras and The Five regular solids Some of the earliest references to the pleasure of mathematics are linked with the name of the Greek philosopher Pythagoras, who observed certain patterns and numbers relationship occurring in nature.

The explanation of order and the harmony of nature were to Pythagoras, to be found in the science of numbers. One of the famous facts of Pythagoras is to the “Five regular solids”: the tetrahedron, cube, octahedron, icosahedron and dodecahedron. Developed from regular polygons, the solids are obviously associated with geometry. The Dodecahedron for instance, which is the development of the pentagon in three dimensions, is dominated by the golden section proportion.

Despite the obvious mathematical facts, I was more interested in how the solids associate with nature.

Also called the “Platonic solids”, the solids have long been regarded as having a special status in revealing the divine perfection underlying the universe.

Plato, who was deeply influenced by Pythagoras, had related them to the important entities of which he supposed the world to be made. He associated each of the four with the classical elements with a

regular solid; earth was associated with the cube, air with the octahedron, water with the icosahedron, and fire with the tetrahedron. It was an intuitive justification for these associations: the heat of fire feels sharp and stabbing; like little tetrahedron. Air is made of the octahedron; its minuscule components are so smooth that one can barely feel it. Water, the icosahedron, flows out of one’s hand when picked up, as if it is made of tiny little balls. The highly un-spherical solid, the hexahedron, represents earth. The dodecahedron has been seen as the quintessence of the universe as a whole. The dodecahedron ‘s twelve regular facet corresponded to the twelve sign of zodiac; it was a symbol of the universe.

It is not surprising that the Greeks took a mystical view of the solids, as the forms are beautiful themselves. No mathematical sophistication is needed for the appreciation of the charm of their outward appearances, which are given element of their beauty.

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Renaissance In Renaissance natural philosophy, for example, to describe the world mathematically meant to represent it by geometrical figures or individual numbers. (Michele Emmer, 1993, p18) The Renaissance was a movement that deeply affected European intellectual culture. Beginning in Italy, and spreading to the rest of Europe by the 16th century, it influenced all aspects of the human culture, including art, science and mathematics; those of which I would like to investigate here. An important fact of Renaissance is the employment of humanist, searched for realism and human emotion in art.

The ubiquity of the golden ratio also aroused many interests during Renaissance. For Leonardo da Vinci and to all draughtsmen of Renaissance, “drawing� was the fundamental discipline that signaled a mastery of design in its principle and practice. It was based upon the measurement of things according to their form, number, proportion, motion and harmonious composition.

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Drawing was the supreme tool that serves the eye, and when Leonardo praised the eye, he was essentially making claims about the ultimate reach of drawing as a tool of investigation and exposition (Martin Kemp, 2006, p96):

Do you not see that eye embraces the beauty of the world? The eye is the commander of astronomy; it makes cosmography; it guides and rectifies all the human arts; it conducts man to the various regions of the world; it is the prince of the mathematics; its sciences are most certain; it has measured the height and the size of the starts; it has generated architecture, perspective and divine painting‌And its nature are finite, but the works that eye commands of hands are infinite.

For Leonardo da Vinci, in terms of drawing, it mean: diagrams of the celestial system; the theory and practice of the system of proportions that governs all artistic beauty; the drafting of scaled maps; the visual demonstration of the truths of geometry, both in themselves and as manifested in the sciences of nature; the analysis of dynamics and statics in the behavior of earth, water, air and fire; the design of build-

ings using plans, elevations, sections and perspectives. For hum, the science of painting is with its roots in nature.

The development of highly realistic linear perspective and the understanding of lights are important art revolutions in Renaissance. Linear perspective was formalized as an artistic technique in the writings of Renaissance scholar Leon Battista Alberti, who regarded mathematics as the common ground of art and the sciences.

Alberti stressed that ‘all steps of learning should be sought from nature; the ultimate aim of an artist is to imitate nature.’ By imitate nature, he meant the artist should be especially attentive to beauty, ‘for in painting beauty is as pleasing as it is necessary.’ Beauty was for Alberti ‘the harmony of all parts in relation to one another,’ and subsequently ‘this concord is realized in a particular number, proportion, and arrangement demanded by harmony.’ Alberti’s concepts on harmony were not new, they could be traced back to Pythagoras, but he set them in a fresh perspective, which fit in well with the contemporary aesthetic judgments.

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In addition, painters also developed understanding in light, shadow, and in the case of Leonardo da Vinci, human anatomy, his “Vitruvian man” is a significant example analyzing human body proportions and his “head of an old man” reveals his intense interest in the bone beneath the skin. His works are significant instances of the intermingled of science and art in Renaissance, in which we can see the changes in arts, were mirrored by the changes in sciences. Painter Albrecht Durer also has important contributions in analyzing human body; his illustrations of the human body proportions in Four Books on Human Proportion, is entirely composed of proportions derived from the three unique division of unity into the Arithmetic, Harmonic, and Geometric Proportion.

There is a general agreement that the desire to explain natural phenomena is a key motivation of Renaissance revolutions, as they saw significant changes in the way the universe was viewed and the thoughts needed their way out.

Harmony and Dynamic Symmetry Mathematician Paul Montrel was to write an introduction for an art exhibition held in Paris in 1963, in the introductory text, reads: It might be surprising that there is a connection between art and mathematics, between the world of qualities and the world of quantities. However, there are close links between these two modes of representation. In fact, both mathematical research and artist creation can been seen as two side of the same coin.

Symmetry is an important idea in the relation of mathematics and art. For ancient Greek and Roman, also Gothic architects, symmetry means the linking of all the elements of the planned whole through a certain proportion or a set of related proportions. As Vitruvious stated in what is the key sentence of his treatise on architecture: ‘When this symmetry or co-modulation between the elements and between the elements and the whole is achieved in the right way, we obtain eurhythmy.’

(Matila Ghyka, 1956, p8) Symmetry clearly has a very well long tradition associated with the impression of harmonious composition.

Following the hints of Vitruvius that the geometrical proportion is the tool necessary for the realization of symmetry or harmonious co-modulation, and that the questions of symmetry have to be solved by the use of irrational proportions, that is by dynamic rectangles.

Dynamic rectangles, are rectangles with the ratios between their longer and their shorter sides equivalent to numbers such as √2,√3,√5 or √φ (Golden rectangle).

Although the linear elements used in the rectangle are irrational, the surfaces built up on them may be commensurable, linked through a rational proportion. Rectangles 1 and 2 can be considered either static or dynamic, however, static rectangles do not produce a series of visually pleasing ratios of surfaces when subdivided. The reason is, dynamic rectangle allow much more flexibility and a much greater variety of choice than the static rectangles, especially when used in order to establish the co-modulation by proportion of the elements and the whole of an architectural, pictorial or decorative composition.

The notion of relationship between rectangles, that is, between their proportions, derives it’s importance from a rules of composition already mentioned by Renaissance scholar L. B. Alberti and rediscovered by Hambidge, the rule of the “non-

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mixing of proportions or themes” in a plane composition; which means in such a composition, only related must be used.

(The rectangle √5, √φ and φ are related, but they are not related to √2 and √3.)

Ancient Greek architects probably used the method “Dynamic Symmetry”, which was re-introduced by artist Jay Hambidge; this method takes into consideration the fact that the overall frame of a two dimensional plan or composition, whether architectural, pictorial or decorative, is generally a rectangle or a complex of rectangles. In his method of harmonic analysis, the rectangles are ”treated” by the diagonal. Which means, when we draw a diagonal; and from on of the opposite summits a perpendicular to this diagonal; this can be repeated with the other diagonal, and from the points of intersection between diagonals, perpendiculars and sides, are drawn parallels to the sides. This process can be continued with many variations, and every diagram thus obtained is what Hanbidge calls “Harmonic Subdivision”.

That is, produces a perfect co-modulation of surface, and obeys the “law of non-mixing themes”. An important property of this method is that the perpendicular to the diagonal produces inside the original rectangle a smaller rectangle similar to it.

According to Hambidge, root 2, 3, 4 and 5 rectangles are often found in Gothic and Classical Greek and Roman art, objects and architecture, while rectangles with aspect ratios greater than root 5 are seldom found in human designs. According to Mathematician Matila Ghyka, Hambidge’s dynamic rectangles can produce the most varied and satisfactory harmonic subdivisions and combinations. The principle is commonly applicable today both in analyzing classic design and in original creation.

All dynamic rectangles can be subdivided; they produce a variety of harmonic subdivisions and combinations that are always related to the proportions of the original rectangle, the same property that could be found in dynamic symmetry. Root 2 rectangle for this reason is the basic for the European DIN, a system of paper sizes, which has been commonly used nowadays. The system is not only efficient but also optimizes the use of pape rthrough an arrangement that has no waste of paper.

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My research and studies have clarified my initial concepts of this subject area; I have understood the ultimate source of aesthetic sensibility is to be sought for in the unconscious mind or even frequently in the collective unconscious by ability that is inborn. The mental processes evoked in all human beings for a million years past by their physical environment have deposited a soil in which the roots of the psyche are deeply and securely implanted. It is to the emotionally charged experiences of a thousand generations of our ancestors that we must look in order to discover the sources of aesthetic pleasure in art, in poetry, in music, in mathematics, and in all other artistic forms. It is possible to guess what some of these experiences must be which, either because their repetition is so frequent or because they evoke strong mental excitement, have left their traces on our mental structure; these traces are the fixed part of our human inheritance and the ground of our aesthetic appreciation.

I have learnt that beauty is not a property that resides in the observed object, but they stir buried memories, which rise to awaken feeling in the conscious levels of the

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Conclusion

mind. When the sensibility of beauty needs to find a way out, it could be expressed through a variety of man-made or organic methods; and when the viewer sees in these creations, it once again provoke the emotional reaction on the person to project “beauty” into the object. For instance, our perception of harmony are inherited from our ancestors and lying in our unconscious, when an object has a certain property that fit out requirements to what is harmonious, our emotions being evoked and we project the feeling of “harmony” onto the object. This conclusion is slightly different form my initial opinion, in which I thought beauty is something that resides in the property of the observed object and this property have the ability to transmit “feelings” towards the observer. But here I understand the “feeling” is actually something that we project onto the observable object.

I have seen that beauty in nature and in mathematics are closely associated. But there is a difference. Nature’s beauty dies. The day dawns when the nautilus is no more. The rainbow passes, the flower fades and the star grows cold. But the beauty in mathematics; the divine proportion, the golden rectangle, the logarithmic spiral, endures for evermore, and the creative ability provoked by them will keep influence the manmade culture and structure our universe.

The eye has to be educated to see. How much of the beauty the eye misses for ack of training! “Having eyes, they see not.� Even the most highly trained mathematician must remain unmoved by much of the splendor because it is hidden from his keenest sight.

Clearly it eye but the sees.

is not the e soul that (Francis Youngusband, Matila Ghyka, 1956, p89)

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Bibliography Brian Thomas (1971) Geometry in Pictorial Composition Oriel Press: UK Bruno Munari (2006) The Triangle: Discovery of The Triangle 2nd ed. Grafiche SiZ: Italy Bruno Munari (2006) The Square: Discovery of The Square 2nd ed. Intergrafica Verona: Italy David Prakel (2006) Composition AVA Publishing SA: Switzerland Gyorgy Kepes. Ed (1965) Structure in Art and in Science Studio Vista Ltd: London Gyorgy Kepes. Ed (1996) Module, Symmetry, Proportion Studio Vista: London H. E. Huntley (1971) The Divine Proportion Dover Publication Inc: New York.

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Jay Hambidge (1948) Dynamic Symmetry in composition: As used by the artists Yale University Press: USA Kimberly Elam (2001) Geometry of Design Princetion Architectural Press: New York.

Martin Kemp (2006) Leonardo da Vinci: Experience, Experient and Design V&A Matila Ghyka (1956) A Practical Hand Book of Geometrical Composition and Design Alec Tiranti Ltd: London.

Michele Emmer. Ed (1993) The Visual Mind II MIT Press:Cambridge: .

Michael Holt (1971) Mathematics in Art Studio Vista: London.

P. H. Scholfield (1958) The Theory of Proportion in Architecture The Syndics of the Cambrige University Press: London Richard Padovan (2003) Proportion: Science, Philosophy, Architecture Spon Press: London