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The Divine Proportion
H. E. Huntley
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The Divine Proportion
H. E. Huntley
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Using simple mathematical formulas, most as basic as Pythagoras's theorem and requiring only a very limited knowledge of mathematics, Professor Huntley explores the fascinating relationship between geometry and aesthetics. Poetry, patterns like Pascal's triangle, philosophy, psychology, music, and dozens of simple mathematical figures are enlisted to show that the `divine proportion` or `golden ratio` is a feature of geometry and analysis which awakes answering echoes in the human psyche. When we judge a work of art aesthetically satisfying, according to his formulation, we are making it conform to a pattern whose outline is laid down in simple geometrical figures; and it is the analysis of these figures which forms the core of Professor Huntley's book.
For the philosopher, scientist, poet, art historian, music listener, artist, as well as the general reader who wants to understand more about the fascinating properties of numbers, this is a beautifully written, exciting account of the search for a naturally manifested aesthetic that has occupied man since he first asked the question `why?`
`This is a delightful book to read. . . . It wanders here and there through some of the most attractive byways of simple mathematics, returning always to the oddities and pleasures of the golden section. This is a browser's book — a happy, untidy traveling or bedside book for those who know how to enjoy the charm of numbers and shapes.` — Dr. J. Bronowski, The Salk Institute.
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CHAPTER I
The Texture of Beauty
Before launching out on our main topic, beauty in mathematics, it will be worth while to convince ourselves that the effort required to learn to appreciate aesthetic values is justified by the pleasure it offers. The conviction is born of experience; and we shall soon discover that it is shared by many of the worldâs wisest men. An example of ancient standing, written before the Christian era (Ecclesiasticus 43, vv. 11â12), magnifies one of the most familiar of beautiful objects:
Look upon the rainbow, and praise him that made it; very beautiful it is in the brightness thereof. It compasseth the heaven about with a glorious circle, and the hands of the most High have bended it.
A further quotation relating to the same example of beauty will serve to underline one of the important lessons of these chapters. For aesthetic appreciation there are two requirements: the first is given, the second acquired. The first is from natureâby inheritance; the second from nurtureâby education.
If the poet sees beauty in a rainbowâ
My heart leaps up when I behold
A rainbow in the sky...
âso does the physicist in the laws governing its manifestation:
His heart leaps up, too, as he discovers how the light of day is reflected, chromatically refracted, reflected again and dispersed by gently falling water spheres into a thousand hues, conforming the while to lovely theorems of mathematics so simple in some aspects that the schoolboy may understand, so complex in others as to defy analysis.1
The surface beauty of the rainbowââvery beautiful it is in the brightness thereofââis appreciated by all men: it is given. But the buried beauty, uncovered by the industrious researches of the physicist, is understood only by the scientifically literate. It is acquired: education is essential.
DEFINITIONS
It is difficult to define beauty, as we shall see; but there is much impressive testimony to the importance of the emotions that beauty calls forth. Mohammed said:
If I had only two loaves of bread, I would barter one to nourish my soul.
A more modern witness, Richard Jefferies, wrote:
The hours when we are absorbed by beauty are the only hours when we really live.... These are the only hours that absorb the soul and fill it with beauty. This is real life, and all else is illusion, or mere endurance.
Beauty is a word which has defied the efforts of philosophers to define in a way that commands general agreement. Yet it does not need a philosopherâs wisdom to utter a few meaningful words about it. One incontrovertible statement might be: beauty arouses emotion. This, being sufficiently indefinite, needs no qualification. The case would be different if we said, âThis lovely artifact always arouses pleasurable emotion in everyone who sees it,â for we know that some people, confronted by beauty which moves others, are entirely unresponsive to it. This appears to justify the familiar aphorism: âBeauty lies in the eye of the beholder.â Whether this be true or not, it is certain that no philosopher, however erudite, can contradict me when I say in sincerity, concerning some experience, âFor me, that is beautiful.â But whether beauty is subjective or objective or both is an unresolved metaphysical problem. According to the Shorter Oxford English Dictionary beauty is:
that quality or combination of qualities which affords keen pleasure to the senses, especially that of sight, or which charms the intellectual or moral faculties.
Only a part of this wide definition concerns us here. We are not interested at the moment in, for example, âthe beauty of holinessâ which âcharmsâ the moral faculties. Our interest lies in the combination of qualities which charms the intellect. âCombination of qualitiesâ reminds us that the experience of beauty is not a simple, but a complex experience. In mathematics it may be compounded of surprise, wonder, awe, or of realised expectation, resolved perplexity, a sense of unplumbed depths and mystery; or of economy of the means to an impressive result. When the mathematician refers to beauty in mathematics, we infer that he has had experience of some or all of these qualities.
Before we turn to the consideration of particular types of beauty, it is profitable to think of it in a wider context.
EVOLUTION OF AESTHETIC FACULTY
Taking a teleological view-point, we might begin by asking whether the universal human thirst for beauty serves a useful purpose. Physical hunger and thirst ensure our bodily survival. The sex drive takes care of the survival of the race. Fear has survival value. Butâto put the question crudelyâwhat is beauty for ? What personal or evolutionary end is met by the appreciation of a rainbow, a flower or a symphony ? At first sight, none. Why, if I have two loaves, should I âsell one and buy a lilyâ? Many of our appetites have been developed in the course of human evolution for a utilitarian purpose in the material environment of our mundane existence. Does this suggest a realm of another natural order? That it points to a definitive view of the nature of the human psyche is a conclusion which seems unavoidable. Before we develop this, let us remind ourselves of how some philosophers, both ancient and modern, have regarded beauty.
Plato, in the Symposium, has much to say about progress from aesthetic appreciation to the enjoyment of âabsolute beauty.â He recounts an inspired speech by Socrates in a dramatic dialogue at the âDinner Party.â Socrates modestly attributes his views to his âinstructressââa woman from Mantinea, called Diotima. The following excerpts are relevant to our subject:
The man who would apply himself to this goal must begin, when he is young, by applying himself to the contemplation of physical beauty.... The next stage is for him to reckon beauty of soul more valuable than beauty of body. . . . From morals he must be directed to the sciences and contemplate their beauty also. . . .[The man] who has directed his thoughts towards examples of beauty in due and orderly succession will suddenly have revealed to him as he approaches the end of his initiation a beauty whose nature is marvellous indeed, the final goal, Socrates, of all his previous efforts. This beauty is first of all external; it neither comes into being nor passes away; next, it is not beautiful in part and ugly in part, nor beautiful at one time and ugly at another.... He will see it as absolute, existing alone with itself, unique, external, and all other beautiful things as partaking of it....
This above all others, my dear Socrates, (the woman from Mantinea continued) is the region where a manâs life should be spent, in the contemplation of absolute beauty. Once you have seen that, you will not value it in terms of gold or rich clothing or the beauty of boys and young men.... What may we suppose to be the felicity of the man who sees absolute beauty in its essence, pure and unalloyed, who, instead of a beauty tainted by human flesh and colour and a mass of perishable rubbish, is able to apprehend divine beauty where it exists apart and alone ? Do you think that it will be a poor life that a man leads who has his gaze fixed in that direction, who contemplates absolute beauty with the appropriate faculty and is in constant union with it ?
Turning from an ancient to a modern philosopher, we may consider the views of the Italian philosopher, Benedetto Croce. His position is that beauty is an attribute of that which expresses feeling. Music, as Plato recognized, expresses human emotion very vividly; it is lento, vivace, con brio, etc. Beauty is seen in colors that are gay or somber. And there is the beauty of scenery:
Bright robes of gold the fields adorn,
The hills with joy are ringing,
The valleys stand so thick with corn
That even they are singing.
Clouds are lonely or angry; the morn is smiling; the oak is majestic; a mathematical theorem is elegant, its proof neat.
Wordsworth said of poetry that it was âemotion recollected in tranquillity.â
Following Croce, then, we may take it that the aesthetic experience supervenes when some material or mental entity, to which for that reason we attribute âbeauty,â stimulates pleasurable emotion.
Now emotions are regarded by psychologists as activities of the unconscious mind, so that the aesthetic experience is the resuscitation of subliminal emotions, and beauty is the power to evoke these emotions. This takes us into deep waters and we will postpone a discussion of the function of the unconscious in mathematical studies to a later chapter.
UNITY IN VARIETY
For a modern view of the nature of beauty we may turn again to J. Bronowski:
When Coleridge tried to define beauty, he returned always to one deep thought; beauty, he said, is unity in variety! Science is nothing else than the search to discover unity in the wild variety of nature,âor, more exactly, in the variety of our experience. Poetry, painting, the arts are the same search, in Coleridgeâs phrase, for unity in variety.2
A WORKING HYPOTHESIS
A student who aspires to gain an insight into a philosophy of the beauty that is latent in mathematics should fortify himself with some form of working hypothesis concerning beauty in a wider context, with the help of modern views on the nature of the human psyche as developed by Sigmund Freud, Carl Jung and other psychologists. Such an hypothesis will not have the status of a theory. While it will, of course, require modification and amplification as new knowledge is gained, it is not thereby invalidated as a viable frame of reference, of which the function is to maintain a logical sequence among numerous data.
Fundamental to such an hypothesis is the recognition that the aesthetic experience is an emotional, rather than a rational mental activity. Merely to state this basic fact is to realize that we shall not make much progress in understanding without admitting the relevance of what has been called âthe greatest discovery of the nineteenth centuryââthe subconscious mind. Though psychologists have found this topic a fertile source of differences of opinion, they are agreed concerning its importance in interpreting mental activity. It is invoked to explain such phenomena as hypnotic trance, dreams, narcosis, dual personality, mental disorders and much more. Its value for our present purpose is that it provides a clue to the understanding of aesthetic feeling.
STRUCTURE OF THE PSYCHE
Psychologists often use the wider terms psyche and psychic in place of mind and mental, which are normally applied to the conscious mind only. Psychic activity is no less real than physiological activity: the psyche has its own structure and is governed by its own laws. A montage of the psyche as seen by the pioneer psychoanalyst Carl Jung would cover four main concepts:
1. The conscious mind, or âsurfaceâ mind, the seat of conscious mental activity.
2. The preconscious, sometimes pictured as forming the periphery of the conscious mind, sometimes as the stratum below the surface mind. Our memories of recent events, now removed from the focus of attention, are stored herein. From here they may be voluntarily recalledârecollected.
3. The subconscious or personal unconscious. The conscious mind, according to Jung, is
based upon and results from an unconscious psyche which is prior to consciousness, and continues to function together with, or despite consciousness.3
The dipstick of introspection cannot plumb this layer of the psyche. Unconscious activity is only exceptionally recognized by the individual, despite the fact that unconscious motivation is one of the prime facts of life. In the unconscious are stored countless forgotten memories which, while they cannot be recalled at will, are nevertheless made manifest in dreams, in hypnotic trance and through other means.
4. The collective unconscious, according to Jung, forms a lower stratum of the psyche than the personal unconscious. It is the source of instinctive behaviour, an instinct being defined as âan impulse to action without conscious motivation.â Instinctive behavior is inherited: it is determined by the history of the race. So are what Jung calls âprimordial imagesâ or âarchetypes,â which were formed at low mental levels during the tens of thousands of years of the evolutionary history of primitive man, our remote human ancestors, by the constant recurrence of universal emotional experiences common to all, e.g., the alternation of day and night, seasonal changes, hunger and thirst, flight from danger, the mountains and the oceans, storm and tempest, the sanctuary of hearth and home.
As a mnemonic, the structure of the psyche may be compared to an ocean island. The land above the water surface represents the conscious mind, the area uncovered at low tide depicts the preconscious; the vast, hidden mass of rock below the ocean represents the unconscious which rises from an ocean bed standing for the collective unconscious. The scientists halt here, but the theologians (notably Tillich) speak of the deepest level of all, which undergirds the ocean bed, as âthe ground of our being,â and equate it with God.
EMOTIONAL ACTIVITY
If we may assume that the evolution of psych...