The Divine Proportion
A Study in Mathematical Beauty
by H.E.Huntley
(186 pages, pb, $5.95)
New York, Dover Publications, 1970
ISBN 0-486-22254-3
A small and elegant book with a persuasive argument that beauty is a function, not only of
consciousness, but of precise mathematical proportions that is hard-wired into the
universe. This is an excellent book for beginning architects, musicians, and anyone
interested in the mathematical harmony that presents a natural aesthetic.
Chapter Titles
Chapter I The Texture of Beauty
Chapter II The Divine Proportion
Chapter III Analysis of Beauty
Chapter IV Phi and Fi-Bonacci
Chapter V Art and the Golden Rectangle
Chapter VI Beauty in Mathematics
Chapter VII Simples Examples of Aesthetic Interest
Chapter VIII Further Examples
Chapter IX Patterns
Chapter X Pascal's Triangle and Fibonacci
Chapter XI The Fibonacci Numbers
Chapter XII Nature's Golden Numbers
Chapter XIII Spira Mirabilis
Selected quotes:
p. 1
The theme of this book is the aesthetic appreciation of mathematics. Poincare's remark
that " . . . but for harmony beautiful to contemplate, science would not be worth
following" is applicable also to his own discipline, mathematics. K. Weierstrass's
dictum that "No mathematician can be a complete mathematician unless he is also
something of a poet," recalls Poincard's: "The mathematician does not study pure
mathematics because it is useful; he studies it because he delights in it and he delights
in it because it is beautiful."
pp. 18-22
Music is the language of the unconscious mind par excellence. As we shall argue in chapter
VI, primordial racial memories are brought to the surface more readily by music than by
natural scenery or any other art; it seems to be possible to relate familiar features of
music to archaic experiences of humanity.
It is music that provides the strongest support for our thesis that aesthetic experience
consists in the interaction between the universal primordial images buried in the
unconscious and an external artifact or natural object which we call beautiful.
The incomparable power of music to move a listener to the depths of his being is
well-known; it will, on occasion, bring him to tears. What is the explanation of the power
of this stimulus which is unparalleled in the other arts? If our thesis is tenable it must
be that music is for some reason an unusually effective agent for bringing to the surface
archaic images and memories stored in the unconscious. As Jung remarks (see p. 77),
"The man who speaks with primordial images speaks with a thousand tongues.. . . That
is the secret of effective art." Now musical expression can stimulate archaic
emotional experiences very effectively - fear by agitato, mourning by molto legato,
excitement by prestissimo, sanctuary by rallentando succeeded by the tonic or home note,
and in similar ways. These universal emotionally charged experiences become effective when
they are raised from the deep unconscious to the surface mind, and it happens that music,
unlike any of the other arts, provides precise and powerful means of effecting this
transfer.
[...]
And now let us return to the consideration of the question of whether beauty serves a
purpose in the scheme of creation. We have already seen that it appears to serve no
utilitarian end. Many of our instincts and associated emotions have been evolved to ensure
our bodily survival, but the emotion aroused by a physical object such as a cloud or a
flower, or by a mental image such as an elegant mathematical theorem, has no such
objective. The answer to the question which we posed in crude terms: "What is beauty
for?" appears to be elusive. So much is this so that one is inclined to doubt whether
it has any purpose and to dismiss the matter by asking impatiently, "Must all things
have a raison d'etre? Is not a thing of beauty a joy for ever, and, so far from being a
means to an end, an end in itself?"
AESTHETIC PLEASURE UNIVERSAL
It seems to me to be important that we should have clear ideas in reference to this
question, and I hope the reader, before he proceeds to the chapters which follow, will
give careful consideration to the point of view now to be described. At first glance it
may appear that the contemplation and appreciation of the beauty of, say, a mathematical
theorem is an unimportant, even trivial, activity. On the contrary, it is, properly
regarded, one of great significance. It would seem to be unlikely, a priori, that the
whole human race should be endowed with the faculty to enjoy beauty unless it achieved
some noble consummation. "Earth's crammed with heaven and every common bush aflame
with God" to some purpose, surely? 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 that nature which distinguishes us from the animal creation. Thus, for an answer
to our question, we are driven back to the explanation o our nature given in Genesis 1, v.
26:
"And God said, Let us make man in our image, after our likeness."
Here, I suggest, is the clue. Man is by nature a creator. After the likeness of his Maker,
man is born to create: to fashion beauty, to originate new values. That is his supreme
vocation. 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 that of creative activity.
Ask the artist. Ask the poet. Ask the scientist. Ask the inventor or my neighbor who grows
prize roses. They all know the deep spiritual satisfaction associated with moment of
orgasm of creation.
CREATIVE ACTIVITY: EMPATHY
This deep joy has been thought by some to be the principal aim of education-more, the
chief end of human life. In The Education of the Whole Man, 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, but
suppressed, smothered, thwarted in most of us...."
This inborn love of beauty, our human heritage, must find expression if we are to be
happy. If the hunger for beauty remains unsatisfied, the effects are seen in loss of
physical and mental health, so deep is the need.
We now approach the final stage in the argument of this chapter. It underlines a truth
which it is important that all students of mathematics should understand, but (it is to be
feared) very few do. If it could be expressed in one word, that word would be empathy. The
German equivalent is Einfuhlung -" feeling into."
We have spoken of a common experience - the joy associated with any form of creative
activity, which a man has as a consequence of his having been made in the image of his
Creator. And we have interpreted the mystery of the nature and purpose of beauty by
recalling the familiar fact that the inborn faculty of aesthetic appreciation constitutes
the motive for the creation of objects of beauty. And now we have to meet the natural
objection that many would raise: they have had no experience of creative activity. They
have added nothing to the store of beauty, their own ideas have been neither new nor
original. They have never known the luminous moment of inspiration which widened the
bounds of knowledge. They can appreciate, but cannot create beauty.
The answer to this objection ran be stated briefly. The act of creation and the act of
appreciation of beauty are not, in essence, distinguishable. This is true whether the
lovely object is a work of art, a musical composition or a mathematical theorem. In the
actual moment of appreciation ("I see! Yes, indeed I see! How beautiful!"), the
beholder experiences those precise emotions which passed through the mind of the creator
in his moment of creation. With the help of the artist he himself shares the joy of
creation. This important fact has been expressed with characteristic clarity by J.
Bronowski:
"The discoveries of science, the works of art are explorations-more, are explosions,
of a hidden likeness. The discoverer or the artist presents in them two aspects of nature
and fuses them into one. This is the act of creation in which an original thought is bom,
and it is the same act in original science and original art.... rrhis view] alone gives a
meaning to the act of appreciation; for the appreciator must see the movement, wake to the
echo which was started in the creation of the work. In the moment of appreciation we live
again the moment when the creator saw and held the hidden likeness.... We re-enact the
creative act, and we ourselves make the discovery again....The great poem and the deep
theorem are new to every reader, and yet are his own experiences, because he himself
re-creates them. They are the marks of unity in variety, and in the instant when the mind
seizes this for itself, the heart misses a beat."
This passage, which illuminates the meaning of empathy, should be understood by all who
seek the aesthetic experience. In particular, the reader of the following pages, whether
his interest is focused on the golden cuboid, or the dodecahedron, or the logarithmic
spiral or the genealogy of the drone bee, should realize that, in the act of appreciation,
he is re-enacting the creative act and, attracted by beauty, is experiencing himself
thejoy of creative activity. He is in fact, in Kepler's phrase, "thinking God's
thoughts after Him."