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."
