Steps on a spiral staircase

kw: nature, mathematics

By an interesting coincidence, both the book I most recently reviewed and the one I am now reading place a certain emphasis on epigenetics. The word means "beyond genetics" and refers to influences on the development of a creature that depend on something more than pure genetics. One example given is of elk that inhabit the Yellowstone ecosystem, which is very optimal for elk, with lots of open space (they can see predators from far away) and forage. They grow big there. By contrast, elk that live in more forested areas such as the Black Hills are smaller, though genome sequencing shows they are the same species, even the same subspecies. Offspring of elk from a forested area taken to Yellowstone do not grow as big as Yellowstone "natives", which is taken as evidence for epigenetic inheritance.

I'd want to see how the grandchildren do, because maternal nutrition affects the development of her offspring. We see this in humans. Children of a poorly nourished mother are smaller and develop later than those of a better-fed mother. Conversely, many immigrants to the US who bear children at least a few years after immigration find that those children are larger and develop more rapidly than siblings born in the "home country". They frequently grow to tower over their parents. This is not evolution in action, it is the expression of potential that is either released or repressed by environmental influences.

In the current book, much is made in one chapter of the spiral patterns seen in many plants and animals, and their relationship—not always so faithful—to the Fibonacci Series. This famous mathematical series is produced by successive addition of terms: 1, 1, 2, 3, 5, 8, 13, and so forth, each term being the sum of the prior two. The ratio of successive terms converges on the value Φ (Phi), the Golden Mean or Golden Ratio, which is, to eight decimals, 1.61803399. Phi is also equal to one plus the square root of five, all divided by two, (1+√5)/2.

Do this: take off your shoes, and mark with a pencil on the wall the level of the base of your rib cage. This is your waist height. If you don't know your height accurately, mark that also. Measure. Divide your height by the height of your waist. For me, the two values are 72" and 44". 72/44 = 1.64. For most people, this ratio will be between 1.55 and 1.7. For a lot of people, the average is about 1.62, so that is often stated as an example of the Golden Ratio in human growth.

I guess Michael Phelps, the swimmer, is a big exception. His waist is 4" (10 cm) closer to the ground than it should be. In one article it was stated this way: He has the torso of a 6'-8" man, and the legs of a six-footer. He is 6'-4" (193 cm). From this I infer that his waist height is, rather than the 47" we'd expect, about 43", and his height/waist-height ratio must be about 1.77. This departure from the norm, however, makes him "built for swimming", with longer arms and stronger legs than anyone else his size.

Plants exhibit more exact Fibonacci Series values than animals. This sunflower bloom has spirals formed by the ovules (soon to be seeds). If you print this image and count the spirals, you'll find there are 55 that sweep to the left as they spiral outward, and 34 that sweep to the right. These are the 10^th and 9^th Fibonacci numbers, a Fibonacci pair. Different flower species have different numbers of spirals, but always, barring damage during growth, they form a Fibonacci pair.

We could say the flower head has a problem to solve, to develop seeds that pack together so as to fill the space without wasting space. The solution developed over time by natural selection is to have the seeds gradually increase in size and plumpness from center to edge, and to segment the seed head according to chemical gradients based on Fibonacci pairs. Underlying this segmentation is a mechanism that begins with 2 and 3, the 3^d and 4^th Fibonacci numbers, and advances stepwise to the values used to place the ovules, each of which develops into one seed.

The series doesn't have to proceed nearly as far to generate the spirals seen in the base of a pine cone. This one has 8 and 13 spirals. Even though the sunflower plant is an angiosperm and the pine is a gymnosperm, similar genetic mechanisms underlie the development of these structures. The use of Fibonacci pairs to pack things into spherical or circular spaces was developed very early.

It reminds me of the principle of a knurler. This is a machine tool used to put the crosshatch grooves on a metal knob. I learned to use one many years ago. It has two smooth rollers and one with the crosshatch knife pattern, and works like a plier. The piece to be knurled, typically an aluminum knob, is centered in a lathe and is rotated at about 1-2 turns per second. Beginning with a light touch, the knob is grasped with the knurler and the machinist looks at the pattern to see how well it fits around the knob. It helps to start with a knob with a dimension that will be a good fit, but the fit will change as the grooves are deepened, so you have to start with a slightly oversize knob.

The knurler's handles can be twisted to make the tool bite at a little different angle, to improve the fit. When there is a near match, the machinist squeezes hard, and the pattern entrains and deepens. Four or five seconds is enough to get a good knurl. When it is done right, you get a nicely knurled knob without chatter marks caused by misfit.

The pattern of a sunflower blossom or pine cone is determined very early. The growth of the entity then fixes the pattern and retains the fit. This is how natural systems work. The sunflower ovules or cone scales, as they grow, are controlled by their surrounding fellows, so all grow evenly and the final product is very uniform.

But the most important learning from these patterns is this: They result naturally from simple rules followed when you have a finite number of items to pack into a space. They are digital items, and only pack well when they are arranged according to a Fibonacci pair. You are welcome to try to make a sunflower head that contains 9 and 17 spirals, or 40 and 60, for example. Of course such things can be drawn, but they'll look rather odd. More to the point, they can't be generated by simple addition from starting numbers like 2 and 3.

Plants like this succulent look a lot like an unopened pine cone when they are very small. They express Fibonacci numbers in another way. The problem to be solved here is to give each leaf maximum sun exposure. This is done by using the square of Φ, which is about 2.618 (in exact terms, Φ² = Φ+1). 360°/Φ² ≈ 137.5°. If you start with the top leaf, the next one down is about 137.5° around the plant, and so forth. For this particular plant, I counted round and round until I found a leaf that is just below the small one that points down and to the right. It is the 13^th leaf after that first one, and the number of turns around the plant stem is five. Five and 13 are not an immediate Fibonacci pair, but a pair with a skip.

Many plants have similar arrangements. Sunflower leaves tend to repeat after eight leaves and three turns. This leads to an angle of 135° from leaf to leaf. If any plant were to have exactly 360°/Φ² from leaf to leaf, then no leaf would ever be directly below any other. However, nature really isn't that exact. A 3-for-8 or 5-for-13 ratio is good enough.

If you make squares of dimension 1, 1, 2, 3, 5, 8 and so forth, they will pack together as shown in this diagram. The smooth curve is the Fibonacci spiral, and it is very close to a logarithmic spiral with a parameter of Φ⁴ (the parameter is the ratio between one turn's width and that of the prior turn). The turn-to-turn distance increases by Φ each quarter turn.

Curves that closely approximate logarithmic spirals abound in nature, because many creatures grow according to their current size, adding a certain percent each month, for example. This is different from the pre-patterning that leads to sunflower head spirals, but has a similar result: spirals that open up as they go out. If you compare the Fibonacci spiral here with the spiral patterns on the sunflower picture above, it is clear that the sunflower spirals of both directions are more open; they have larger parameters.

The chambered nautilus shell is a great example of a logarithmic spiral, as exact as nature is able to produce. Though it has often been stated that nautilus shells follow a Fibonacci spiral, this is not so. The blue line in the image is a Fibonacci spiral. Its parameter is Φ⁴, or about 6.85. The nautilus shell's spiral has a parameter nearer to 3. Snails also have spirals with parameters in the range around 3, which seems to be optimum for an animal to increase in size and have a living chamber that is deep enough to retreat into, but the shell doesn't get too heavy to lug around. No small-number addition here, just growing by proportion. Fibonacci ratios don't seem to have much to do with animal growth.

These examples are considered to express "laws of form" that nature imposes. I guess you can call them that. They emerge naturally from the growth and development of a creature subject to the constraints of its environment: gravity, nutrient availability and space. I suspect a chambered nautilus that experienced a period of near-starvation would have some kind of variation in the curve of its shell. Nothing stays the same, and just as one can grow a cubical watermelon by growing it into a Plexiglas box, we find that genetics provides the potential, and environment either nurtures or hinders it. But it would take one heck of an environmental upset to produce a sunflower that had straight radial lines and concentric circles instead of spirals, if it were possible at all!