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THE GEOMETRY OF FLOWERS Life By The Numbers, p.70-76, Keith Devlin John Wiley & Sons (New York: 1998) |
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THE GEOMETRY OF FLOWERS Life By The Numbers, p.70-76, Keith Devlin John Wiley & Sons (New York: 1998) |
The different ways dinosaurs moved, the patterns of spots and stripes on the skins of animals, the way viruses cause DNA molecules to tie themselves into knots-from the large to the small, mathematics can help us to understand the world of living creatures. We can likewise use mathematics to help us understand the other living world, the world of plants.
For instance, how do you describe the shape of a flower? For a daisy, you could say it was circular. Thus, any mathematical description of a circle will give you a description of the shape of a daisy.
One way to describe a circle is as "the set of all points in a plane that are the same distance from a fixed point." (The "fixed point" is the center of the circle. The "same distance" is the radius of the circle.) This description corresponds to the way we draw a circle using a compass. The sharp point of the compass is placed on what will be the center of the circle. The compass point is sharp precisely so that the center is a "fixed point" that does not move around when we draw the circle. The compass is stiff so that once we set the angle of the hinge, the points traced out by the pencil are all the same distance from the center.
Another way to describe a circle is Descartes-style, using an algebraic equation. But neither a daisy nor indeed any real flower is really circular. It only appears circular when viewed from a distance. When you take a proper look, you see that the flower is made up of many petals. The shape traced out by these petals is only approximately a circle; in reality it appears much more complicated than a circle. Can mathematics be used to describe the real shape of a daisy? Or how about a flower that is not circular, such as a lilac? Can mathematics describe the shape of a lilac flower?
The question seems to be pointless. After all, what possible benefit is to be gained from giving a mathematical description of a flower? Why would anyone bother to find out? The answer is that it is always a good idea to try to understand nature. Not only because, as human beings, we seem to take satisfaction from understanding our world, but also because we can never be sure when we might need our scientific understanding. After all, when mathematicians of the early nineteenth century first asked themselves, How can we describe the pattern of a knot? no one had any idea that biologists in the late twentieth century would use those mathematical descriptions to help in the fight against viruses. The lesson that history has taught us again and again is that scientific knowledge, which includes mathematical knowledge, generally turns out to be beneficial to us. Of course, it can also have harmful side-effects, such as pollution from chemical factories, just as the benefits of having automobiles come at the price of road accidents. But on balance, the benefits far outweigh the costs.
What might be the benefit of trying to find a mathematical description of, say, a lilac? Well, here is one possibility: it could lead to more accurate weather forecasts. Surprised? Here's how. If you look closely at a lilac, you will notice that a small part of the lilac flower looks much the same as the entire flower. You see the same phenomenon with certain other flowers, with some vegetables, such as broccoli or cauliflower, and with some other plants, such as ferns. Mathematicians refer to the phenomenon whereby a small part looks like the whole as "self-similarity."
What other things do we see that have self-similarity? Clouds, for one. If we had a mathematical way to describe self-similar patterns, we could use it to study clouds. With a good mathematical description of clouds, we could simulate the formation, growth, and movement of clouds on a computer. Using those simulations, maybe we could improve our ability to forecast severe weather, using our forecasts to protect ourselves better from the consequences of a major storm or a tornado. Fanciful? Not at all; researchers have been carrying out just such investigations for some years now. Nature may well turn out to be too unpredictable for us to ever have perfect weather forecasts, but the use of mathematics has already led to better forecasts, which almost certainly has saved lives.
In other words, just as a mathematical study or the patterns or Knots could lead to better techniques to conquer viruses, so too a mathematical study of the self-similar patterns of flowers such as the lilac can lead to better techniques to forecast the weather. This is the way mathematics works.
Przemyslaw Prusinkiewicz is one of a number of researchers who have been trying to find mathematical descriptions of self-similar shapes such as the lilac flower. To obtain the description of a lilac, Prusinkiewicz and his collaborator, Dr. Campbell Davidson, look at the way nature might create the flower's shape. This is the same as arriving at a mathematical description of a circle by looking at how you can draw a circle with a compass. "When I am looking at plants, ' says Prusinkiewicz, "what I find beautiful is their shape and form. But there is another layer of beauty, a hidden beauty. It is not what we see at first sight when we look at a plant; it is the beauty of understanding the mechanisms that bring this form about.'
Just as the mathematical theory of knots was developed long before biologists started to use it to study viruses, so too the mathematics Prusinkiewicz needs to study flowers began many years earlier, in this case at the end of the nineteenth century. One of the key discoveries was made by mathematician Niels Fabian Helge von Koch. He noticed that if you take an equilateral triangle, add a smaller equilateral triangle to the middle third of each side, then repeat the process of adding smaller and smaller triangles to the middle thirds of the sides, eventually you will develop a fascinating shape called the Koch snowflake. (To be precise, the idea is that you delete the middle-third length each time you add a new triangle.)
What the example of the Koch snowflake shows is that a complicated-looking shape can result from the repeated application of a very simple rule. It is the use of the same rule over and over again that results in the self-similarity of the resulting shape. Present-day mathematicians refer to self-similar figures as fractals, a name invented by the mathematician Benoit Mandelbrot in the 1970s. Mandelbrot showed that the repeated application of one particular kind of rule leads to a very important (to mathematicians) fractal shape that now bears his name, the Mandelbrot set. Computer images of the Mandelbrot set have shown that it is a figure of incredible beauty and movies have been quashed that show nothing but different views of this one figure, at different degrees of "magnification."
For Prusinkiewicz, the idea then is to write down rules that, when used over and over again, produce the self-similar shapes he sees in nature, such as the shape of the lilac flower. Mathematicians refer to such a system of repeatable growth rules as an "L-system." The name comes from Aristid Linenmayer, a biologist who in 1968, developed a formal model for describing the development of plants on the cellular level.
For example, a very simple L-system to produce a treelike shape might say that if we start with the top part of any branch, that portion forms two new branches, giving three branches. When we repeat this rule on the new tops, we find we rapidly get a treelike shape.
To generate a lilac on his computer, Prusinkiewicz starts with a very simple L-system to generate the skeleton of the flower. Then, by taking careful measurements of an actual lilac, he refines his L-system so that the figure it produces more closely resembles reality. Using his refined L-system, he then generates the branching structure of the lilac. He then uses the same technique, with a different L-system, to produce the blossoms. And voila! A lilac grows before his eyes. Not a real lilac, produced by nature, but a mathematical lilac, produced on a computer.
For Prusinkiewicz, it is a source of never-ending amazement to see how the seemingly complex shapes of nature can result from very simple rules. "It is very exciting to see that structures which we used to think of as very complex turn out to be very simple in principle," he remarks. "A plant is repeating the same thing over and over again. Since it is doing it in so many places, the plant winds up with a structure that looks complex to us. But it's not really complex; it's just intricate. When you appreciate the beauty of plant form, it comes not only from the static structure, but often also from the process that led to the structure. To a scientist who appreciates the beauty of this flower or leaf, it is an important aspect of understanding to know how these things were evolving over time. I call it the algorithmic beauty of plants. It's a little bit of hidden beauty."
Prusinkiewicz sees his work as creative. He uses mathematics to create (in his case, on a computer) some of the patterns we see in nature. "Creativity is the essence of mathematics, " he says. "Mathematics is not playing with numbers and doing accounting. Mathematics is dealing with ideas in a creative and yet very precise way."
