• M. A. Alkhadra

The Beauty of Fractals

Mathematics is often described as beautiful, and I personally derive aesthetic pleasure from this subject. At some point during my undergraduate education, I became interested in abstract mathematics; I was motivated to write on fractals, however, because of my exposure to Prof. John Rogers’ work on stretchable electronics with designs inspired by fractals.

In mathematics, a fractal is an abstract object that exhibits a self-similar pattern at increasingly small scales (that is, the object exhibits an infinitely repeating pattern as it is magnified). Two notable fractals are the Koch snowflake (Figure 1), defined by mathematician H. von Koch in 1904, and the Mandelbrot set (Figure 2), defined and drawn by mathematicians R. W. Brooks and P. Matelski in 1978. The Koch snowflake is a mathematical curve and one of the earliest fractals to have been described. Both the Koch snowflake and Mandelbrot set exhibit self-similarity that repeats infinitely as the curves are magnified.

Figure 1. Self-similarity is depicted in the Mandelbrot set by zooming in on a special point called the Feigenbaum point at (−1.401..., 0).
Figure 2. The Koch snowflake is a curve that begins with an equilateral triangle in which the middle third of every line segment is replaced with a pair of line segments that form an equilateral bump. Iteration (7 steps shown in the animation) produces a fractal that exhibits infinitely repeating self-similarity as it is magnified.

The origin of fractals has been traced throughout the years as a formal path of mathematical publications that started in the 17th century with the concept of recursion and recurrence relations, by which each term of a sequence is defined as a function of the preceding term. Recurrence relations then underwent increasingly rigorous mathematical treatment, which, in the 19th century, produced the study of functions that are continuous but not differentiable. (A function is described as non-differentiable at some point if, for example, the function makes a sharp corner at that point; think of the V-shaped absolute value function f(x) = |x| at its vertex.) In 1872, K. Weierstrass—a pioneer in modern calculus and mathematical analysis—presented the first definition of a function with a graph that would today be considered a fractal, though this term was not coined until much later. This so-called Weierstrass function (Figure 3) is an example of a pathological function, as it has the non-intuitive property of being continuous everywhere but differentiable nowhere.

Figure 3. The Weierstrass function, like fractals, exhibits self-similarity: every magnification resembles the global plot shown over the interval [−2, 2]. This function is continuous everywhere because (crudely) the graph is represented by a single, unbroken curve, though the function is differentiable nowhere because it makes a sharp corner at every point.

Modern engineering applications of fractals include the patterning of thin films of rigid electronic materials in deterministic fractal motifs to enable unusual mechanics with important implications in the design of stretchable devices. Pattern designs based on fractals allow structures to better accommodate elastic strain (i.e., mechanical deformation) along the desired dimension(s). Fractal-based architectures thus represent one possible approach to interface traditionally rigid electronic materials with soft and curved biological surfaces such as skin and tissue.

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