The Mathematics of Beauty

Introduction

... in the kindgdom of transformation. When something's let go of, it circles; and though we are rarely the center of the circle, it draws around us its unbroken, marvelous curve.     - Rainer Maria Rilke

Auguries of Innocence

To see a world in a grain of sand,
And a heaven in a wild flower,
Hold infinity in the palm of your hand,
And eternity in an hour.
           - William Blake     more...

A Skeptic

Rudolf Wittkower, in his Architecture in the Age of Humanism says:

". . . in trying to prove that a system of proportions has been deliberately applied . . . one is easily misled into finding . . . those ratios which one sets out to find. Compasses in the scholar's hand do not revolt."

In other words, we tend to find what we're looking for, whether its there or not. We will hope to avoid that pitfall by questioning everything. source

The Geometry of Art and Life

And it was then that all these kinds of things thus established received their shapes from the Ordering One, through the action of Ideas and Numbers. (Plato, Timaeus)

This pronouncement of Plato, and his conception of Aesthetics, has set the precedence for Western Thought and Art and the development of our culture in general. in the same way that Plato conceived the "Great Ordering One" as arranging the Cosmos harmoniously according to preexisting, eternal, paradigms, archetypes or ideas, so the Platonic view of Art conceived the Artist as planning a work of Art according to a preexisting system of proportions, as a "symphonic" composition, ruled by a "dynamic symmetry" corresponding in space to musical eurhythmy in time. This technique of correlated proportions was in fact transposed from the Pythagorean conception of musical harmony: the intervals between notes being measured by the lengths of the strings of the lyra, not by the frequencies of the tones [but the result is the same, as length and numbers of vibrations are inversely proportional], so that the chords produce comparisons or combinations of ratios, that is, systems of proportions. In the same way Plato's Aesthetics, his conception of Beauty, evolved out of Harmony and Rhythm, the role of Numbers therein, and the final correlation between Beauty and Love, were also bodily taken from the Pythagorean doctrine, and then developed by Plato and his School. A great factor in Plato's Mathematical Philosophy--and, in a subsidiary manner, in his [p. ix] system of Aesthetics--was the importance given to the five regular bodies and the interplay of proportions which they reveal; we shall see this point of view transmitted all through the Middle Ages to the Renaissance and beyond, with the study, and the application to artistic composition, of the same proportions.