For the love of broccoli
Fractals are ubiquitous in nature and mathematics
There is broccoli at MIT’s dining halls almost every week. The broccoli is usually steamed or roasted. Although I miss the garlic, ginger, oil, onions, coriander seed, chili, methi, and masala that go well with broccoli, I enjoy eating it bland. I should instead say that I love staring at its details. When broccoli is cooked well with spices, either the details get destroyed or the spices obscure the details. On the other hand, with steamed, unseasoned broccoli, I can look at its hierarchy of stalks that are self-similar at all levels and detailed heads to the point that my food gets cold.
Whoever made the first broccoli must have tried to sculpt the details at an even smaller scale to show the existence of a fractal but failed to do so because of limited time. I feel good about the creator because they never had to worry about the uniqueness part of the solution. Nevertheless, they seemed to have tried hard to prove a fractal’s existence in cauliflower, Queen Anne’s lace, ferns, mountains, and coastlines to no avail. Well, a physicist could argue that broccoli is close enough to be a fractal: a stalk resembles the whole broccoli. A smaller stalk resembles a bigger stalk and so on. But for a mathematician, it does not have details smaller than the leaves that look like the whole broccoli. So, it does not qualify as a rigorous fractal.
“Fractal” must be a cousin of “fracture,” as fractals are broken in some sense. Consider a fractal called a middle third Cantor set. Take an interval from 0 to 1 in the real number line. Delete the middle third of the line (excluding its endpoints). We should get two lines: one from 0 to 1/3 and the other from 2/3 to 1. With each of these lines, delete their middle third and continue the process forever. What is left after this iteration is the Cantor set. It is fractal because of its self-similarity at all levels. And it is constructed by breaking lines. However, not all fractals are broken.
Fractal could be a foe of “fair,” since they are rough. Even the close-enough fractal, broccoli, is rough. If you have taken 18.100B (Real Analysis), you must know about the nowhere-differentiable function. For those who don’t, when you zoom in enough on a differentiable function, say x squared at any point, you will start to see a straight line. But for a nowhere differentiable function, the function always has some roughness and a straight line never appears if you keep zooming in. In fact, the zoomed portion always has details that resemble the whole function. The complex details account for the roughness.
Fractal should be a mutant of “fraction,” as fractals have “fractional” dimensions. Take a square whose sides are a unit length. It is made up of 2^2 squares with sides of 1/2 length. On another note, it is also made up of 4^2 squares with sides of 1/4 length. Similarly, a cube whose sides are a unit length is made up of 2^3 cubes of sides of 1/2 length. Or we could also say it is made up of 4^3 cubes with sides of 1/4 length. Note that the exponents in both cases carry the information of the dimension of objects.
Mathematically, suppose a set is composed of n sets that are a scaled down (by r units) version of the original set. Then the dimension of the set is d = ln(n) / ln(r). Now consider the Cantor set. The essence of the construction of the Cantor set is in taking out the middle third of lines. Therefore, the points that we get after a deletion procedure starting with [0, 1] are similar to those starting with [0, 1/3] and [2/3, 1]. So, the new Cantor set consists of two copies of a scaled down (by three units) version of the original Cantor set. Therefore, its dimension has to be ln(2) / ln(3) ≈ 0.63 which is definitely not an integer.
As of now, you might be thinking that fractals are akin to “fiction.” They sound like mathematical fiction. After all, the examples in the real world that I gave are definitely not fractals because they are not infinitely broken and self-similar. Nor do they have fractional dimensions. Nevertheless, the creator of broccoli was successful in creating other fractals in the real world. It took an MIT physicist, Ed Lorenz, and an MIT computer scientist, Ellen Fetter, to figure out that the equation that models the convection of atmosphere can have fractals. Further, fractals occur in chemical reactions (see Strogatz) and in population modeling (such as in the logistic map). Fractals are not fiction, so fractals must have an affair with “factual.”
Actually, fractal has traits of “frenzy,” as they usually appear in a chaotic environment. In the model that Ed Lorenz studied, fractals appear when the convection is chaotic. Even in a chemical reaction, the fractals appear in a chaotic reaction.
I am certain that fractal is a forebear of “fabulous.” “Freakish” sounds closer to fractal, though. Regardless, fractals are so fantastic that I have fun staring at them.
They have a rich underlying mathematical beauty. The weird fractional dimension and self-similarity are only some aspects of it. One can do a whole lot of analysis on fractals. For instance, one can define a Laplacian on it. Recall that the standard Laplacian is used in modeling the heat equation. In a loose sense, it models the evolution of the temperature of a solid. The Laplacian involves two derivatives, and derivatives are defined on a smooth domain. But fractals as we have seen are not smooth. So, it takes some effort to define Laplacian (see Kigami). After we develop a Laplacian, we can model how the temperature of a hot fractal will evolve. One could also generalize the ideas like the uncertainty principle to fractals (see Dyatlov).
I get excited about all these things just by looking at the close-enough fractal, broccoli. More specifically, bland broccoli. Thus, I would like to thank the chefs at MIT’s dining halls for not adding any spices to broccoli.
For beautiful pictures of fractals:
Barnsley, M. F. (1988) Fractals Everywhere (Academic Press, Orlando, FL).
Feder, J. (1988) Fractals (Plenum, New York).
Mandelbrot, B. B. (1982) The Fractal Geometry of Nature (Freeman, San Francisco).
Peitgen, H.-O., and Richter, P. H. (1986) The Beauty of Fractals (Springer, New York).
Schroeder, M. (1991) Fractals, Chaos, Power Laws (Freeman, New York).
For advanced math:
Dyatlov, S. (2019) An Introduction to Fractal Uncertainty Principle. https://arxiv.org/abs/1903.02599
Kigami, J. (2001) Analysis on Fractals (Cambridge University Press, Cambridge, UK).
Strogatz, S. H. (2015) Nonlinear Dynamics and Chaos with Applications to Physics, Biology, Chemistry, and Engineering (CRC Press, New York).
The photo associated with this article was created by Wikipedia user IkamusumeFan and was not altered in any way.