by Paul Niquette
Copyright ©1997 Sophisticated:The Magazine. All rights reserved.

The Great Sonora Desert subsumes the Saguaro National Monument near Tuscon. Vast and shadeless, the SNM ranks high among Arizona's natural wonders, a cactus forest to prickle your mind. Spanish doesn't pronounce the g, by the way: SA-WA'-RO.

Elementary in its leafless form, almost Deco, the Saguaro grows straight up from Earth, a pure green cylinder with ribs and spines, hoarding moisture, standing stiff against the desert winds, the obligatory focus for many a desert painting. The saguaro grows arms, not branches. Each appendage juts out and immediately curves, forming an elbow, then reaches perfectly skyward, parallel to the trunk, as if guided by an internal static pendulum.

An opportunity to view thousands of saguaros may not enrapture everyone. For me, it has sophisticated -- mathematical -- meaning. Each of the saguaro's arms tends to emerge at an arbitrary level. But strangely never at . The saguaro refuses to abide by the Golden Section.

The symbol (phi, usually pronounce FAE) is a Greek letter, like (pi, usually pronounced PAE), commonly used to denote the ratio of a circle's circumference to its diameter, and 's value, 3.14159..., thanks to computers, has grown in precision to over 10 million decimal places. Likewise, also represents a ratio, called the "Golden Section."

    To find , start with the number 0 and the number 1. Add them together. You get 1. Now add that 1 to the previous 1. This time you get 2. Are we going too fast here? Now add the 2 to the immediately preceding 1, for a sum of 3. So far, we have produced the following series of 'natural' numbers: 0, 1, 1, 2, 3. Continuing in like manner, adding the 3 we just computed to the 2 from one step ago, and we get the next number in the series, 5. And so forth, 8, 13, 21, 34, 55,...
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...

These numbers form the 'Fibonacci Sequence.' Its discovery dates back to the early thirteenth century and to the most distinguished mathematician of the Middle Ages, Leonardo Fibonacci, also known as Leonardo of Pisa. He invented the first 'recursive number sequence.'

    Over the centuries, countless mathematicians have devoted their careers to the study of such integer sequences, discovering esoteric properties in them. Philosophers have joined in the fun, writing recondite monographs to one another about the mystical attributes of Fibonacci numbers. Today's number theorists and computer scientists continue the arcane tradition. As recently as 1962, a group of enthusiasts founded 'A Fibonacci Association' -- in California, of course.
Fibonacci numbers grow big fast. The 41st entry in the series amounts to more than 100 million. You can prove that for yourself in just a few minutes.

In the mid-eighteenth century, Robert Simson at the University of Glasgow discovered that the ratio of two successive Fibonacci numbers "tends" toward a constant. Figuring this out might have kept an old-time mathematician off the streets for weeks. With today's massive computing power under our fingertips, we can effortlessly reconfirm Simson's discovery in seconds. Actually, the proof is simple {HyperNote 1}

    Care to try? Take any Fibonacci number and divide it by the next number in the series. Use 21 and 34. You get 0.61764705. Or take the next pair, 34 and 55, which produces the ratio 0.618181818... By the 29th and 30th number, you will see that the ratio has settled down -- 'converged,' in math-talk -- so that you can perceive no change even with many digits of precision. The ratio approaches the constant value of...

Congratulations, you have found the value of. But why call it the "Golden Section"? And what does this have to do with the saguaro?

Draw a line one unit long (by "unit" can be meant an inch, a foot, or you can go metric if you like). Mark a point on it. You now have two joined line segments. Call their respective lengths and 1 - . Now, let the point move along the line so that the following statement applies:

    The length of compared to that of the whole line exactly matches the length 1 - (the part left over) compared to . {HyperNote 2}
The value of which satisfies this relationship has received the name "ancient mean" and the "extreme ratio." One finds the more colorful "Golden Section" used most today.
    For a line of, say, 100 centimeters, would measure a little more than 61.8 centimeters, about 62% of the length of your line.
Here ancient number theory and aesthetics strangely join. If the 100 centimeters just mentioned represents the vertical side of a piece of fine art, then the artist will position the focus of interest about 62 centimeters from the bottom.
    The viewer's eye will find that location most pleasing, according to adherents of -losophy (groan).
The same theory influences the location side-to-side of artistic emphasis. Thus do still-life artists place their baskets of fruit or flowers at the Golden Section. Hah, that desert scene: Did you happen to notice where the artist put the saguaro?

Look at the pictures around your room. Observe how often this 'rule' has held.

  • That horizon-line in a seascape, for example. You can expect to find it neither at the bottom of the canvas nor at the top -- not at the center, either. Coerced by the Golden Section, the artist divides the canvas with sky-above-ocean 62% of the distance from the bottom, and
  • foam in the foreground bursts upward, intersecting the horizon 62% of the way from left or right.
The frame-maker, too, appears to answer the call of , imposing the 62% ratio, horizontal to vertical or vice versa. Ancient Greek and Roman architects often designed their structures with the Golden Section firmly in mind.
    See for yourself: Take calipers to the Parthenon.
Curiously, all the artists I know retch at numerical deliberations such as those set forth above. Nevertheless, responding always to private mental maunderings, they paint and carve 62% proportions upon canvas and clay just the same.

In the dimensions of human face and figure, Golden Sections preponderate. Check with that other Leonardo (da Vinci) on that. Some might attribute such ratioing to Providential computations, others to Nature's aesthetic intuition.

    Which reminds me, mathematicians have classified along with as an "irrational number," an irrational choice of words for the case at hand, seems to me. Irrational numbers, cannot result from the ratio of whole 'integers' -- Fibonacci numbers.
The key to this paradox hides within that infinity business in the derivation provided below.
    Where parallel lines meet, so too do the rational and the irrational.
    -- Paul Niquette, 1997
But we must get back to the saguaro -- after just one more observation about the Fibonacci Sequence.  You will recall, we started our series with the number 0. An especially appropriate selection and a reminder of another debt of thanks we owe to Leonardo of Pisa. He first noticed zero's magnificent properties and gave us 'positional notation,' one of the greatest inventive leaps in history.
    Think about it. Mankind needed the 'nul cipher' for its ability to enable all manner of -- ta-dah! -- arithmetic. It was 0 that the Romans didn't have. Consequently, their numerals have long ago resigned themselves to non-arithmetic duties
    (I) marking clock faces,
    (II) performing unctuous service on cornerstones, and
    (III) enumerating typographic lists.
For the Fibonacci series: after 0, we chose 1 -- unity -- which is just as fundamental as 0. These two ciphers, 0 and 1, comprise the computer's 'binaries,' and with a string of fewer than a hundred of them, one can count the number of protons and neutrons in the universe! But I digress.
    Whatever their historical significance to mankind, 0 and 1 have little influence upon on the Fibonacci Sequence, as it turns out.
For no good reason, let's postulate a new family of Fibonacci numbers, using exactly the same recursive procedure, but started with different -- what shall we call them? -- "seeds." Try commencing your series with a couple of numbers that did not arise on the previous list, such as 4 and 7.
    Watch the entries grow: 11, 18, 29, 47,... Take out your calculator and figure those ratios of succeeding values. They tend to the same value, 0.618033988750! That Golden Section turns up again.
Experimentation with any number of seed numbers always shows that outcome. Consider, too, the series started by 100 and 2. Notice how fast the numbers grow. The 34th exceeds the population of the U.S.; the 40th, the population of the world. Yet, those ratios steadfastly approach the same value .
    You can plant a negative seed like -123 alongside, say, 77. After some vacillation, plus and minus, the numbers come out positive and sail off to infinity -- at Simson's fixed ratio still. Another amazement: By changing the 77 to 76, the consequent Fibonacci numbers stumble into a black hole of unlimited negativity -- but the ratio still holds!
Such a relentless 'emergent property' must have prevailed in many realms indeed. Most often observed is that the relative sizes of successive chambers in the nautilus obey ratio . Look for Fibonacci sequences and therefore Golden Sections
  • in the spirals of sunflower heads,
  • in pine cones,
  • in leaf buds, and
  • in animal horns.
You find all over the place.

Oh yes, and don't forget the elevation of the nominally normal navel in proportion to the height of healthy humans (the belly button of a 6-footer should be located almost exactly 44.5 inches above the floor).

SaguaroWe return at last to the saguaro. How astounding to observe that the saguaro, standing sublimely upright against the Arizona sky, uniquely resists the governance of the Golden Section! Unlike the sunflower and the pine, unlike the nautilis and the ram, unlike the human -- especially unlike the human! -- the saguara can defy .

Thus, we observe the eruption of the saguaro's arms at an elevation never equal to 61.8033988750 % of the prickly trunk's height (all data to the contrary should be disregarded).

As for the word prolix, you can plainly see that the person who wrote these paragraphs has no use for it.

Prolix adjective
  1. Wordy and tedious. 
  2. Tending to speak or write at great length.

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{HyperNote 1} For more about Simson, see the solution to Fibonacci Fantasy.

{HyperNote 2} You will find the derivation in the solution to Golden Section.