The Work of Art

ARTS & MEASURE:
NUMBER, PROPORTION, BEAUTY

The fairest thing we can experience is the mysterious. It is the fundamental emotion which stands at the cradle of true art and science.

– Albert Einstein (quoted in The Golden Ratio, by Mario Livio)

In order to talk about art, we need to examine our terms and assumptions, to establish a common language by which to build understanding. That requires common experience. Otherwise, confusion reigns.

For example, after I moved out to the country, I went hunting with my neighbor Tim, who grew up tracking deer and elk through the hills. When we had walked into the woods a ways, he suggested we split up so as to squeeze out any hiding deer, and said he’d meet me at a bench on the opposite side of the valley. I knew that he didn’t mean a park bench, but I also didn’t know exactly what I was looking for, and after I’d been wandering around for awhile, I started to worry that he was playing a practical joke. When he finally found me, I asked if I was anywhere near the right spot, and he explained exactly which flattened part of the hillside was “the bench.” And while he didn’t intend it as a joke, we laugh about it now because we still run into differences in how we understand the same word, especially when he’s talking about the land where he grew up and has lived most of his adult life.

So rather than starting with the terms themselves, let’s talk about the processes by which we might know them, processes in which we all participate every day. The processes themselves will provide understanding far deeper than either the dry, lifeless approach of the academic, or the childish rule of personal preference (“I don’t know anything about art but I know what I like”). “Number” and “proportion” come first, because we need them as tools to talk about everything else.

NUMBER

I used to wonder if perhaps counting began when women noticed how their menstrual cycles related to the cyclically changing shape of the moon in the sky, perhaps inspiring them to count the days of their months. It made a nice fuzzy kind of theory, but number may come “hard-wired,” as they say, in the synapses of our brains. Certainly every human being, male and female, carries number and measure in their anatomy: 

• 2 comes to us through paired eyes, ears, hands, feet, arms, legs;

• fingers and toes give us multiples of 5 and 10;

• and 3 knuckles on each of 4 fingers gives us multiples of 12.

So humans have variously counted by twelves, twenties, tens, and twos. There’s the 12 inch foot and 12 month year (related to hexagonal and circular geometry that uses 60° angles and a 360° circle). There’s the French “quatre-vingt” (“four-twenty”), and base-20 phrases meaning “a whole man” (“a 20”), in Inuit as well as Mayan cultures. Ten fingers and toes makes a decimal system. And base two, easily expressed in electronics as either “on” or “off,” “yes” or “no,” underlies both our hi-tech, computer society, as well as ancient dualities like dark and light, good and evil, yin and yang.

From such basic experiences of number, we developed “the art of counting,” or “arithmetic” (literally, from the greek “arithmos,” number, and “tekhne,” skill. The initial “ar” of “arithmos,” of course, shares the Indo-European root “ar” with art, order, ratio, and related ways of “fitting together” experience and reality.)

PROPORTION

Though we often express it numerically, proportion, like measure, refers to relationship. When we say something “is in proportion,” we mean that that all the parts work together well, that they balance harmoniously. A nicely proportioned figure has arms and legs that are right for the trunk; fingers right for hands, etc. Literally, proportion means “for her or his share.” It suggests a fair division of parts, but fair doesn’t have to be one for one. Proportion can be 2 for 1, or 1 for 3, or 7 for 12. It can be a percentage. It can be a fraction. It can be skinny, fat, tall, or round (bodies are most often justly proportioned, as anyone who has studied the figure can tell you; often clothes just confuse and hide their logic). On the surface, bodies, numbers, and the material world seem solid, firm and definite. In fact, however, the deeper we look, the more mystery we find. In the early days of number, when seekers looked into mathematical problems, they didn’t find answers but profound philosophical questions.

For example, take the number Pi, or 3.14159…. Pi is the number of diameters that fit into the circumference of a circle. Every circle contains Pi diameters. The problem is that Pi simply cannot be expressed, either as a whole number, or as a ratio between two whole numbers – which is why you get the series of dots at the end – properly expressed, the digits after the decimal go on forever.

Say you have a box that’s 12 inches on each side. If I ask you to cut it in half, you can cut it precisely through the center and get two rectangles exactly 12 x 12 x 6 inches. We describe the proportion of each half as exactly 1 to 2, precisely one half, or 0.50. Similarly, you can cut the box into three or four equal parts and end up with proportions of 1 to 3 or 1 to 4, precisely one third or one quarter, 0.666… or 0.750.[1] These relations fit together in a precise and finite ratio – they’re rational.

Pi, however is an irrational number – because the fit between a circle and its circumference is, simply, not something we can count – it’s not finite. Relationships that we can’t adequately describe with the fingers on our hands and toes we label “irrational” – which makes for an interesting contradiction, since a number is supposed to be a rational and definite quantity. In fact, number fails to define many natural relationships.  We can, however, easily see that the relationship between diameter and circumference is uniquely stable! But that stability requires a whole story of unity and harmony beyond what number can describe.

In 1502, a mathematician named Luca Pacioli wrote that “Just as God cannot be properly defined, nor can be understood through words, likewise our proportion cannot be ever designated by intelligible numbers, nor can it be expressed by any rational quantity, but always remains concealed and secret, and is called irrational by the mathematicians.” [quoted in Livio, p. 132] The Divine Proportion, a book Leonardo DaVinci illustrated, explored another irrational number known as Phi, or the Golden Mean. This particular and “unintelligible” proportion that Pacioli called “our proportion” tries to express the unique relationship between growth and form, between spirit and matter. It’s a measure that we find throughout nature, and it’s uniquely related to beauty.

[1] While the ratio 2:3 works out to an endless 0.666666…, in base 3 it works out as exactly 0.1, a trick that won’t work with Pi or the Golden Mean, neither of which find expression as whole number fractions.