This first selection is taken from chapter 6, "Numbers: Natural," and it introduces the famous Fibonacci numbers.

When you were a tadpole and I was a fish . . .

When the earth was young and the summers green and carefree, my cousin and I would roam the fields in search of whatever we might find. We had no prescribed duties, except perhaps bringing the cows to the barn at dusk; we were rather the keepers of berry patches and bluebird eggs. Since we weren’t in search of rare animals like moose, wolves, or bears, we were never disappointed. Hollow fence posts had to be inspected for birds’ nests, ponds for tadpoles, swamps for turtles, pools for gilled salamanders, and special hidden places we alone knew for snakes and the occasional blue skink.

We were unfettered in our natural interests. As far as we knew, the adults in our lives had little interest or knowledge of the world outside. Rarely did they speak about it — never a bird’s name or a flower’s location. Infrequently an aunt would express some fear or other, especially about poisonous snakes (we never found one), or skinks that might run up inside your pant leg to do great damage, or the ever-vicious wolves. On one occasion, a not too likeable aunt with a goiter asked my cousin and me to capture a snake large enough to wrap around her neck twice, and then to release it before sundown. Local wisdom affirmed this would cause the goiter to shrivel up and disappear. Since she feared snakes, and we didn’t particularly like her, we granted her wish. Trusting to adult wisdom, we fully expected the wretched disfigurement to vanish, but that never happened. So we learned grown-ups were not always wise.

The open fields were our special domain — the home of bumblebees, wasps, and innumerable wildflowers. We were young Pythagoreans for we loved to count. Oh, not as lovesick adolescents might with “she loves me, she loves me not” but just a straightforward enumeration of the flower’s petals. At that time what we found was consistency; years later this became a pattern, and presently we have an explanation.

Let’s begin at the beginning. The consistency lay in each healthy variety of flower always having a certain number of petals, even if the number was exceedingly large. As we shall see, this regularity also applies to leaves, branches, and seed spirals. Consider the examples below picked from the field of boyhood dreams.

Every summer evening when we had tired the sun with exploring and sent him down the sky, we would wander out in search of the cows. These warm, placid creatures had to be sheltered in the safety of our barn. Since the farm was heavily forested, they were usually hard to find, even with the help of our faithful dog Pal. So we would call to them in a smooth comforting voice: “Cooo‑bos, cooo‑bos, cooo‑bos.” Typically they would give themselves away with deep lowing sounds. Well it just so happens that bos or boV is the ancient Greek word for cow. This word had come down through the centuries — to two skinny boys chasing cattle in the darkening hinterland of Ontario — totally unchanged from the original Greek. That’s the wonderful consistency of a word through time and over space. But in the shadowy fields all around us was a design by nature that had endured millions upon millions of years and on every continent. And this is the pattern I wish to investigate.

Years passed and the dreams faded to the back of my mind for more pressing and immediate matters, but they never left me. I suspect we all have such special places, sacred in our memory. One afternoon all that changed! In my senior high school mathematics class the teacher wrote the following sequence of numbers on the blackboard and asked if anyone could discover the next term:

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

Without raising my hand or a moment’s hesitation I blurted out “It’s 89.” These were the numbers my cousin and I had found a decade earlier — see Figure 6.7.

The teacher was more amazed than annoyed by my outburst; he asked how I knew the answer so quickly. I told him a much-shortened version because one is protective of such precious memories. It was now his turn to amaze me, for the numbers we had stumbled upon as boys were none other than part of the famous Fibonacci sequence named for the medieval mathematician Leonardo of Pisa, also called Fibonacci (about 1170–1240). And the secret for finding the next term in his sequence — which as boys we had overlooked — is marvelously simple: add the preceding two terms to get the next. For example, 3 + 5 = 8, 21 + 34 = 55, and so on. As my teacher gave instance after instance where this sequence occurs in nature my surprise grew — a feeling that lives with me to the present moment. Come and share this wonder with me.

The second excerpt deals with the economy of nature both in the world of physics, which Newton saw, and the nature world, which Darwin knew.

Chapter — 7

Life on the Edge

Nature does nothing in vain, and more is in vain

when less will serve; for Nature is pleased with simplicity,

and affects not the pomp of superfluous causes.

Isaac Newton (1642-1727), Principia

intertime and the living is hard. Reynard is on the prowl. Because of the deep snow, he hasn’t hunted in two days. It’s – 40ºF/ºC; the red glow of dawn fills the east as he meanders through a field of wisteria bushes and poplar saplings. Reynard has no destination, he’s going nowhere, and everywhere — this is the hunt. It’s the prey that’s important, not his location; he follows his ears and nose. Since the snow is deep, he treads with great care conserving energy however possible. His track pattern is unusual for a four-legged animal: he leaves only half the expected paw prints. His smaller hind paw lands in the impression just made by the front paw on the same side. That is, he “double prints,” as we do through deep snow when following a friend because it’s less arduous. Reynard dreams of mice and voles or perhaps a plump partridge sleeping in its overnight snow cave. He hasn’t eaten in two days.

Fox Double Prints

To conserve energy in snow, many other animals such as wolves and coyotes also double print. Unnatural selection and a full food bowl have rendered dogs incapable of doing so — they commonly make a four-print pattern. Through a hundred thousand years, natural selection pressures have favored energy-conserving behaviors. In a million-year march, a minuscule pebble in a boot can be crippling.

These energy-saving behaviors, these maximizing and minimizing activities, are found across the entire spectrum of life. Often they’re subtle, usually they’re overlooked, but they’re always necessary for long-term survival. Consider the humble bee. Bernd Heinrich in his lucid and lively book Bumblebee Economics reveals their innumerable adaptive behaviors. On a summer’s day, in the same field of wisteria where Reynard conducted his winter hunt you can see bumblebees going about their tasks. Here’s one survival adaptation Heinrich noted:

The wider resources are scattered, the less efficient it is to recruit and defend specific items, and the more difficult it is to patrol and defend an area. Competitors then appear to work peacefully (without contact) side by side, but they may still compete relentlessly by trying to remove resources faster than the next individual. Aggressive encounters then become a liability, for even the winners lose — they have only expended time and energy that could have been used for foraging. The nonaggressors, which do not interrupt their foraging, reap more food energy and are competitively superior. Such competition, called scramble or exploitation competition, generally results in the depletion of resources to the very minimum of economic profitability. In turn, it selects for energy economy and foraging efficiency in the contestants.

This is natural selection at work, at all times, in all places. As the greatest evolutionary geneticist of our time, Theodosius Dobzhansky wrote, “Nothing in biology makes sense except in the light of evolution.”

This last excerpt is a humorous introduction to chapter 8 on the nature of patterns and how they are unexpectedly created by shattering symmetry.

Chapter — 8

Shatterer of Worlds

In Eden I

Madam, I’m Adam.

Eve. (She replies.)

Even in Eden, I win Eden in Eve.

Mad Adam! (Eve)

Tut tut. (Snake)

— a cautionary tale by Anonymous

n the beginning everything was beautiful, healthy, and symmetrical. Hence, Adam was uncertain whether he should speak left to right or right to left so he spoke in palindromic sentences; Eve and Snake responded in kind. And thus the first humans “deified” this wordplay that we have been burdened with ever since. Some zealots of the craft extend this to words with reversible meanings, declaring that when Genesis 1:1 says “heaven and earth” we are meant to interpret this as a three-word list with “and” as “DNA.” After all, they state, Hebrew does read right to left. (Perhaps I had best end this paragraph before my embarrassment makes my face even “redder”.)

On a serious note, mathematician Hermann Weyl in his classic little book Symmetry points out that this word is used in everyday language with two meanings. In one sense it denotes being well proportioned or balanced, beautiful. This first meaning is general and a little vague, but the second is particular and exact: it defines symmetry as “bilateral symmetry” — implying its essence is purely geometrical. In the natural world, this left-right equality is found in insects, birds, animals, and flowers. If a slide transparency or a negative of the natural world is flipped horizontally (bilaterally reflected), it’s usually impossible to know unless you’re familiar with the scene. This second sense has wide application in art and nature. It’s one of the birds in the aviary of Heraclitus that has been singing for millennia; let’s listen to its song. It sings as it flies.

In the first sense of this word, we unconsciously equate health with symmetry. A mother immediately checks her newborn for signs of asymmetry: hands and fingers, feet and toes — similarly for birthmarks. Once the infant has passed this visual examination, it’s judged healthy. Any animal in the wild that lacks left–right equality is sick, be it a limp, a closed eye, a tattered ear, or whatever. A wolf with a game leg will do its best to hide the limitation: it’s never to the animal’s advantage to reveal a weakness. Your dog will do the same. Every healthy animal grooms itself symmetrically.

As we age, symmetry trickles away, carrying our health along with it to be replaced by a cane, an arthritic hip, a mastectomy, a sling, a knee brace, and such. Although there is life after death in the form of mold, all our bodily symmetry will have vanished. First we decay, and when our bodies rise again they will be wildflowers, then rabbits, and then foxes and wolves. This is the cycle of life — as well as of symmetry, something we will investigate in depth. Cycles, like symmetry, pervade our world. The poet Shelley celebrated the mundane water cycle in his poem “The Cloud”:

I am the daughter of Earth and Water,

And the nursling of the Sky;

I pass through the pores of the oceans and shores:

I change, but I cannot die.