Saturday, October 16, 2010

Mathematics and Beauty: A Golden Connection

Beauty is mysterious.

“Beauty is in the eye of the beholder.” “Beauty is an illusion.” These ideas make beauty seem so subjective, so transient. Surely many prefer it that way. But what of us who crave the eternal? What of the scientists and philosophers who seek truth, the artists and engineers who sense all of time in one raised eyebrow or perfect pediment? Is that for naught?

What if there was some magical property that justified the beautiful, that widened, not narrowed our definitions thereof? What if the whole world was beautiful?

There may be no magic spell, but there is however, the next best thing: an equation. The Greek letter phi is defined as the following: (a+b)/a=a/b. If we simplify these terms, we are soon left with Phi^2-Phi-1=0. As we can see, this is a quadratic equation, and the only positive solution is (1+√5)/2=1.61803399.

What does this have to do with beauty? This is where things get interesting. As we will soon see, this ratio seems almost omnipresent. We can see it in faces, in flowers, in fruit, and even in fountains. All these beautiful things often have this ratio as a common thread between them. One may argue that this is just as subjective as anything else. As we examine the history of the ratio, we shall see why this particular philosophy of beauty deserves merit.

We start with Phidias, who was born in about 430 BCE in Greece. Phidias was a sculptor, painter, and architect, generally regarded as the greatest of his time. Favored by the leader Pericles, it is generally believed that Phidias designed the statues that reside in the Parthenon. We do not know much about Phidias, and cannot say for certain he knew of the ratio, but he seems to have employed it in his work. It is for this reason we call the ratio phi, the transliterated first letter in Phidias’ name.

Our next character is the philosopher Plato or Platon, born in Athens, Greece in about 427 BCE. In one of Plato’s dialogues, Honor, he describes 5 solids, all convex regular polyhedrons. Some of these solids are related to the golden ratio. In fact, it is possible to combine the solids and find even further examples of the golden ratio. Once again however, we do not know if Plato was aware of these relationships or not.

The first record of the golden ratio we are aware of occurred in the mathematician Euclid’s (or Euklidis’) book Elements. This book is so influential, we are still required to study it in secondary schools. Euclid, who was also Greek and born in about 300 BCE, actually defines the golden ratio. Euclid described it as the extreme and mean ratio.

It wasn’t until the middle-ages and the work of Italian mathematician Fibonacci that we were able to learn more about the ratio. In his The Book of the Abacus Fibonacci describes a sequence of numbers where each number is the sum of the two numbers before it, e.g. 3,5,8,13,21 etc. It can be observed that the farther we take this sequence, the closer the ratio of any two consecutive numbers (3/5, 5/8) gets to phi. While he did not discover this sequence, he popularized it, and it is now known as the Fibbonacci sequence.

Luca Bartolomeo de Pacioli, Italian mathematician and friar, wrote The Divine Proportion, published in 1509. This book was illustrated by Leonardo Da Vinci and explored how the ratio related to art and architecture. It also showed how the proportion applied to people-with proportions on the face and body being close to phi. The following quote shows Pacioli’s thoughts on the ratio:

The Ancients, having taken into consideration the rigorous construction of the human body, elaborated all their works, as especially their holy temples, according to these proportions; for they found here the two principal figures without which no project is possible: the perfection of the circle, the principle of all regular bodies, and the equilateral square.

(Emphasis added)

Johannes Kepler was a German mathematician, astronomer, and astrologer born in 1571. Kepler was mainly concerned with uniting astronomy and physics, and in doing so was able to show that planets did not move in orbs (circles) as previously thought, but in orbits (ellipses). We were able to find that golden ratios often show up in ellipses, and Kepler later said

Geometry has two great treasures: one is the Theorem of Pythagoras, and the other the division of a line into extreme and mean ratio; the first we may compare to a measure of gold, the second we may name a precious jewel.

So far we have seen the ratio in shapes, architecture, art, people, and in the shape and movements of an ellipse. Those who think Phi is just a coincidence are losing their arguments. In the eighteenth century, Charles Bonnet, a Swiss naturalist and philosopher is finding it in plants. The spiral phyllotaxis (or leaf arrangement along the shoot of plants) going clockwise and counter-clockwise is often two successive Fibonacci sequences. We also find this in the spirals of the nautilus shell and the petals and seeds of the sunflower. German mathematician Martin Ohm begins to describe this ratio as “golden”.

French mathematician François Édouard Anatole Lucas studied the Fibonacci sequence and is the namesake of the related Lucas sequence, whose ratios also get closer and closer to phi. Lucas came up with a formula for the nth Fibonacci term, which is as follows. Fsub (n+1) = Fsub (n) + Fsub (n-1). He also gave the sequence the name Fibonacci Sequence.

It wasn’t until about 1909 that phi became the symbol for the ratio, as given by Mark Barr, an American mathematician. Today even more exciting connections are being made to the golden ratio by individuals such as Roger Penrose among others.

Gustav Theodor Fechner did studies that show that 75% of people prefer golden ratios, and subtle (highly unscientific) evidence in our culture seems to support this. Others, like George Markowsky, say it is all coincidence. But we may find that is not even the point. The point is that we find things such as order and symmetry to be worthy of our time. The point is that when we see beauty, we joy in it. H.E. Huntley says this is a response driven by an inner desire to create, and to see that it is good. He says:

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 the moment of orgasm of creation.

I would add that beauty shows us our values. Do we value order & intelligence or do we value the dark primordial chaos from whence we came? Do we value this beauty we find even on ourselves? Do we value that we are perfectly made, by God or nature, to be beautiful?

I will end with the ratio as defined by Adolf Zeising.

[A universal law] in which is contained the ground-principle of all formative striving for beauty and completeness in the realms of both nature and art, and which permeates, as a paramount spiritual ideal, all structures, forms and proportions, whether cosmic or individual, organic or inorganic, acoustic or optical; which finds its fullest realization, however, in the human form.

Is that not something we can value?