Arts: Beauty and Mathematics

The Classical Greek sculptor Polyclitus said, “Perfection comes about little by little, through many numbers.” And he lived by his credo. The two legendary Polyclitus bronzes that have come down to us as marble Roman copies – Doryphorous and Diadumenos – are mathematically scrupulous even second-hand.

Polyclitus literally measured with a ruler many strong, idealized Greek men and averaged the numbers. Then he created his sculptures based on this mathematical ideal.

I long have called Doryphorous the most significant and the most beautiful product of art history. I can barely imagine how magnificent the original was. And Polyclitus’ scholarly approach made me realize that our modern world, and all of the supreme order in the world, is unequivocally and irrevocably rooted in idealized forms.

This is not just an artistic vanity. It extends to the fields of architecture, design and engineering as well. Our modern economy would never be possible without refined forms. The Golden Gate Bridge would have collapsed long ago if its every subform – and its overall superform – were not idealized, perfectly conceived and then integrated with absolute clarity and certainty.

And the added bonus is that the Golden Gate is beautiful as well. Because form and function rise together in lockstep. The bridge is an aesthetic joy with its graceful swooping cables and its stately towers, all painted a daring red-orange.

So where does our essential concept of beauty come from?

It comes from mathematics. And one of the most significant mathematical entities on which beauty is based is called the Fibonacci Sequence which was codified in the 14th century by Italian Leonardo da Pisa although man has been aware of it in one way or another for millennia.

The sequence goes 0,1,1,2,3,5,8,13,21,34,55,89…. And on and on. It starts with zero and then goes on to the next number and then each ensuing number is based on the total of the previous two numbers (8 = 3 + 5) or (89 = 34 + 55).

Then when you calculate the ratio of any number in the sequence to the number just before it the ratio increasingly hovers around – alternately higher then lower than, and then closer and closer to – what is called The Golden Mean, which is 1.618…. The higher the numbers the closer the ratio gets to the ideal. But it can never arrive there because the Mean itself is unknowable in its mathematical irrationality.

The Egyptian pyramids are memorable because they are based on what is called The Golden Pyramid which also is based on 1.618… So even the Egyptians, after several centuries of experimentation with pyramid shapes that weren’t so hot, became aware of the power of what has come to be called The Golden Rectangle, which is not the color gold but is perfect mathematically. It is 1.618… times as long as it is wide.

The Golden Rectangle has appeared over and over throughout history. The Greeks embraced it repeatedly to proportion their sculptures and to scale their temple architecture. The mighty Greek amphitheater at Epidaurus – which just so happens to have been designed by Polylictus’ grandson around 380 BC – is a masterwork of refined proportion that depends centrally on the Golden Mean.

Today the Golden Rectangle is everywhere but we hardly know it. And if you showed 10 differently-proportioned rectangles to 1,000 different people and one of the rectangles was Golden most of the people would choose it out naturally as the most pleasing to their eye, i.e., the most “beautiful”.

Even the human head is a Golden Rectangle when viewed straight on. Do you think that we would see beauty in the human face if our heads were proportioned like a door?  The ideal human body too is based on Golden Rectangles.

Fact is that we have been subjected to mathematical beauty since the beginning of time – an alluring face, a handsomely proportioned body. These were the primal cues that still excite us today. I long have theorized that women love football because they are subliminally mesmerized by all those manly forms jammed into those tight spandex pants.

But wait. There’s more… The Fibonacci sequence – and thus obliquely the Golden Rectangle – can be applied directly even to the structure of plants. The spiraling seed arrangement in the sunflower head is calculated on a mathematical formula based on Fibonacci. It is called phyllotaxis.

Throughout my adult life as an artist, since my first drawing class with professor David Bumbeck at Middlebury College in 1973, I have pursued idealized descriptions of natural forms. I can never say for sure that my drawings have achieved the mathematical perfection of their subjects but I believe that I have come very close just by artistic intuition.

I have focused primarily on plants, flowers, portraits and the human form. And always I have sought to describe these natural forms so as to capture their dynamic essence and their telling proportions so that they never look stiff, artificial or illustrated. Because to express this ideal the artist must grasp through repeated experimentation the underlying essence of form as expressed through what Frank Lloyd Wright called the “hidden forms” that lurk “behind nature”.

In my drawing of a chrysanthemum from 2007 (colored pencil on paper) I have sought to “describe” the flower as accurately as possible. Because drawing in fact is “description”. It is never rendering, reproduction, representation, illustration, facsimile, copying, scribbling or cartooning as art is in the lower precincts of so-called ‘modernism’ (i.e., Picasso, Dali, Georgia O’Keefe, Warhol, de Kooning etc.).

In all of our high visual endeavors we are seeking to describe a natural order and force that will make the art, the architecture, the design or the feat of engineering “work”.

Would you fly on a jetliner that looked like it came out of a primitivist Picasso painting?

No. You would demand a plane with the most upright and straightforward integrity of form.

Through four decades I have built a body of art that delves into the essential nature of forms gleaned from knowledge gained in repetitive inquiry as a draughtsman. My oeuvre is never glamorous, extravagant, ostentatious or exhibitionist, all qualities that my Ancient Greek forefathers disdained and that nature abhors.

My Wooden Block Forms of 2012 are loose blocks that can be arranged at will by the viewer like children’s blocks. The arrangement shown here is only about 20 inches tall.

The blocks consist of three separate and ideal geometric forms – the square, the Golden Rectangle and what I have dubbed the Platinum Rectangle, which is based on the square root of 5, which is 2.236… In short it is 2.236… times as long as it is wide.

The Ancient Greeks loved the Platinum Rectangle in that the square root of 5 is the most irrational number of all. The stylobate (the footprint) of the Parthenon in Athens, considered the most geometrically perfect building ever constructed, is a Platinum Rectangle at 101 feet by 228 feet. The Parthenon also touted many Golden proportions.

My thesis behind my wooden constructions, folded paper cutouts and paintings is that specific, original, ideal and universal forms speak for themselves boldly and without reservation. They cannot be quelled or quieted. Ironically they speak softly because their power is understood, accepted and unchallenged. In short they know their own strength.

My thesis is that these forms are members in good standing of a cosmic vocabulary that has relentless and eternal energy that is discovered over and over while an infinite array of lesser forms is abandoned on the ash heap of history every minute of every day. And that any form created out of ideal subforms, like Wooden Block Forms is, will be ideal, universal and beautiful no matter how it is arranged because the subforms are ideal, universal and beautiful.

It all emerges from a passion for natural form which is a product of God-ly perfection through mathematics.

So the big question is: Did mathematics create ideal forms or did ideal forms create mathematics?

That would be a good issue for Polyclitus to address. Perhaps he did address it in writings long lost. Yet it is a question that we never will have a definitive answer for but that is a lot of fun to think about.

(Please bookmark this website. Thank you, Nikitas)

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