Repetition and Order: How Space Filling Fractal Curves Exhibit Aesthetics

A space filling curve is a line that can be drawn continuously, without lifting a pen if done on paper.  However, when this curve is drawn infinitely, it completely fills a square without any holes [1].  One example of a curve like this is the Hilbert curve.  To produce the Hilbert curve, take the n step, and using the method of producing the n+1 step, which is detailed within Figure 1., use this same method any number of times to get the desired iteration.  As the iteration number increases, the curve gets longer, and when done infinitely it will completely fill the space.

Figure 1. The process of getting from the first iteration to the second of the Hilbert curve

Figure 2. The first six iterations of the Hilbert curve (Wikipedia)

The Hilbert curve, or any space filling curve, may just be a theoretical way of solving a given problem in mathematics.  The mathematician Peano first discovered the space filling curve in order to disprove that a continuous curve does not have to be enclosed within a region.  Yet, one may argue that these curves do hold aesthetic value, due to their composition of fractal nature.

What gives fractals this sense of aesthetic?  One idea might be that of repetition.  Due to the fact that fractals are self-similar, they do exhibit a form of repetition, producing the same form infinitely, growing smaller with each iteration.  Ellen Levy argues two points for the general concept of repetition in art, which we can apply to repetition by fractals.  Firstly, repetition “…helps defer closure in a work of art by establishing expectations of recurrence while giving pleasure to the viewer” [2].  Secondly, “Active repetition in art can evoke evolutionary processes…” which can be defined as the repetitive aspect of nature.  However, we cannot use this second point to describe fractals because nature is, in fact, fractal.  Gleick states that “Clouds are not spheres. Mountains are not cones. Lightning does not travel in a straight line” [3].  Defining the aesthetics behind fractals by something that is inherently composed of fractal forms creates a cyclical argument.

Instead, fractals may be considered to be art due to the concepts of chaos and order.  On their surface, fractal images may appear to be complex, intricate shapes.  However, there is a great deal of order and pattern to fractals.  This sense of order produces a natural “subconscious curiosity to find relation, symmetry and recurrence” [4, p. 226]. Garousi and Kowsari argued that these fractals of pure order and regularity simultaneously exist with chaos and disorder, much like the outer world we live in [4].  Making sense of the complexity by using this order, we can begin to understand fractals, as well as appreciate them as an art form in this light.

Take, for example, the Koch Snowflake.  The edges exhibit the fractal design, which repeat infinitely.  It may be difficult to understand how the edges work by purely looking at it, but understanding the process of how to create the figure, one may be able to see the underlying structure and how to produce an infinite shape.  In this way, the fractal takes a seemingly complex image and expresses it as a simple formula, step by step, giving it order.

Figure 3. The first four iterations of the Koch snowflake (Wikipedia)

Because of all this, we can begin to understand how a space filling curve can be a form of art.  These fractal curves take an infinite set of points in some multidimensional space, such as a plane or space (even a hyperspace).  It then connects all these points according to a specific curve, repeating a pattern until all points are on this curve.  This set of infinitely is therefore simply expressed by one continuous line, producing order to a concept we cannot conceive.  Fundamentally, these space filling curves contain the same artistic values as other fractals, so that we can conclude these curves are, in fact, a form of art.

References:

[1]  Séébold, Patrice. “Tag-Systems For The Hilbert Curve.” Discrete Mathematics & Theoretical Computer Science (DMTCS) 9.2 (2007): 213-226. Academic Search Complete. Web. 9 Nov. 2016.

[2]  Levy, Ellen K. “Repetition And The Scientific Model In Art.” Art Journal 55.1 (1996): 79. Academic Search Complete. Web. 9 Nov. 2016.

[3]  Liebovitch, Larry S., and Daniela Scheurle. “Two lessons from fractals and chaos.” Complexity 5.4 (2000): 34-43.

[4]  Garousi, Mehrdad, and Masoud Kowsari. “Fractal Art And Postmodern Society.” Journal Of Visual Art Practice 10.3 (2012): 215-229. Academic Search Complete. Web. 9 Nov. 2016.

[5]  Arlinghaus, Sandra Lach. “Fractals Take a Central Place.” Geografiska Annaler. Series B, Human Geography, vol. 67, no. 2, 1985, pp. 83–88. www.jstor.org/stable/490419.