## Fibonacci numbers & the Golden Proportiion

FIBONACCI NUMBERS & THE GOLDEN PROPORTION

There are, of course, aspects of measured, proportional relations that numbers describe quite wonderfully, but I abandoned numbers long before I tried giving up the verb to be: “1 + 1 = 2” doesn’t offer much room for improvisation, much less beauty. But geometry I can draw out on paper, and by the same physical, relational method, geometry has given me a new appreciation for number. So bear with me, and return to your drawing of little boxes. Take the first box as a unit measure of one; use that unit of length to measure the long side of each successive rectangle. You should get a repeating series of lengths whereby each new number is the sum of the preceding two:

0+1 = 1
1+1 = 2
2+1 = 3
3+2 = 5
5+3 = 8
8+5 = 13
13+8 = 21
21+13 = 34
34+21 = 55

and so on…

This particular arrangement of numbers is famous as a “Fibonacci series,” named after Leonardo of Pisa, who helped introduce the system of Arabic numerals to the West. (His nickname is abbreviated from the Latin phrase “filius Bonacci,” or “son of Bonacci.” “Bonacci” means “good nature,” in Italian, and shares a Latin root with “beauty.”) In the 1170s, when Bonacci’s mother brought him into the world, cut the cord, and eventually loosed him to his businessman father, accounting was dominated by the Romans. As you may recall from grade school, Roman numerals consist of not only ones and tens, but also 5s and 50s. They further confuse things by representing the numbers 4 and 9 as “5 minus 1” and “10 minus 1”. It makes adding, subtracting, or multiplying notoriously difficult, but that was the system in which Fibonacci was raised. However, because his father traded overseas and served as a custom official in Pisa and Bugia (Algeria), Leonardo’s teachers included an Arab who taught him to use the “nine Indian figures,” or Arabic numbers, which were much easier.

Traveling around the Mediterranean some years later, Fibonacci observed how the Arabic system facilitated trade, and thought that Europeans might benefit by learning it, so in 1202, he published Liber Abaci, or Book of the Abacus. (Livio, 92-3.), including an example of the numeric series we now know by his name. And while he neither invented, discovered, nor named it (a 19th century French mathematician did that), Fibonacci series show up in everything from the arrangement of seeds and leaves in plants, to the lengths of your limbs, to the sizes of planetary orbits.

As a simple list of numbers, they may seem to hold little interest, simply indicating the lengths of a series of rectangles:

1, 2, 3, 5, 8, 13, 21, 34, 55, 89, etc.

However, each number, when combined with its antecedent, produces the following number in the series. OK, big deal. It could go on forever (and it does). Dividing each number by both the following and preceding one produces two more series, as follows:

1. 1/2 = 0.5; 2/1 = 2.0
2. 2/3 = 0.666…; 3/2 = 1.50
3. 3/5 = 0.6; 5/3 = 1.666…
4. 5/8 = 0.625; 8/5 = 1.60
5. 8/13 = 0.6154; 13/8 = 1.6250
6. 13/21 = 0.6191; 21/13 = 1.6154
7. 21/34 = 0.6177; 34/21 = 1.6191
8. 34/55 = 0.6182; 55/34 = 1.6177
9. 55/89 = 0.6180; 89/55 = 1.6182
10. 89/144 = 0.6180; 144/89 = 1.6180

Notice that the product of each series of divisions approaches a common constant that is numerically close to 0.6180, or 1.6180. Amazingly, however, this number never resolves itself into a final and absolute quantity. As with the difficult concept of infinity, the further down the Fibonacci series you go, the closer you get to the Golden Mean, but no matter how far you get, the possible new number pairs stretch out in front of you farther than they do behind you. You can never actually arrive, either at infinity, or at the Golden mean. (Interestingly, you get the same result no matter what two numbers you start with. Try it!)

Long before Fibonacci, Euclid noted and named the same constant but he did it geometrically rather than numerically. He called it the “extreme and mean ratio,” and defined it not as a number, but as a point on a line where the short part has the same relation to the long part as the long part has to the whole. There are numerous other geometric examples and proofs for the Golden Mean, but whether you define it by Euclid’s geometry or Fibonacci’s “nine Indian figures,” the Golden Mean challenges the solid, physical terms by which we typically try to grasp the concept of measure. The stability of the Golden Mean rests on relationships between birth, growth, and physical form. So when poets talk about learning the secrets of the sea from meditating on a dewdrop, or seeing the universe in a grain of sand, they are in fact being just as truthful and, I would argue, as correct as any mathematician talking about Pi or the Golden mean. Indeed, a Brittanica article on “number” admits that our practical usage of numbers creates “embarrassing difficulties” that can’t be resolved without setting up an “abstract” system made of “pure logic.” Or, as Einstein put it: “as far as the laws of mathematics refer to reality, they are not certain; and as far as the are certain, they do not refer to reality.” (in Livio, p12)

We call number and geometric figure both “ratios” from, again, the Indo-European root, “ar,” “to fit together.” Life, like art, proceeds in finite increments – days, minutes, years, months; idea, material, work – but each incremental step must fit with all that came before it. Art and beauty come of how we fit ourselves into the time we have. In this sense, I think, the Golden mean reminds us to tell the story, rather than trying to simply “explain” it with a not-so-simple number.

Life and materials involve so many different factors that are beyond our capacity for analytical understanding that logic requires contradictory concepts like irrational numbers. As a number, the Golden Mean cannot correct ugliness or imbalance. We can use it, analytically, to understand how some things fit together (or not), but in itself, it merely shows that the material limits and forms of life balance the growth of life. Beyond those limits life cannot go. If Einstein was right about the laws of mathematics not definitely referring to reality, how could we possibly ask for definite certainty from such indefinite things as art, beauty, or the Golden Mean?

However, if we accept the mystery of irrational numbers then so-called “vague and unreliable” things like “gut feelings” and intuition begin to be worthy of greater exploration. Intuition, infact, is not nearly so vague and unreliable as numerical logic would make it seem. Like the tuition fees you pay for private schooling, the root of the word is tueri, which means “to look at, watch, protect.” So tuition is, essentially, what you pay the baby-sitter (which raises serious questions about the baby-sitting institutions we call schools, but that’s another story).

Intuition, then, is that same attention focused on yourself, your surroundings, and your work. Intuition enables an athlete to anticipate his opponent’s next move, or a martial artist to break solid rock with soft, fragile flesh. Intuition enables an artist to go beyond what she can see; to choose Truth without assembling all the facts required for proof. But if you want to achieve what intuition enables, you must be willing to hear and obey what no one else may notice.

So anything I have said here can’t and won’t be true unless you make it so by giving it new life. Even if I had the whole U.S. military at my command, I couldn’t enforce the Golden Mean any more than King Canute could reverse the tides. So rather than trying to obey something as mysterious as the Golden Mean, it seems to me that art must simply obey the rules and limits of materials, the edges of the frame, the limits of time and money and patience and discipline and knowledge. And then we proceed by putting paint to canvas, chisel to stone, pen to paper. As Kathleen Norris suggests, “obedience is an active form of listening.” Buckminister Fuller said “When I am working on a problem, I never think about beauty. I think only of how to solve the problem. But when I have finished, if the solution is not beautiful, I know it is wrong.” (Livio, 10) And of course, once you achieve beauty, why stop?