Math and art have had an active relationship for centuries. Think of perspective geometry and Renaissance art, higher-dimensional geometry and cubism, how the Alhambra walls and Klemscott press margins use patterns of wallpaper and frieze groups, the tilings both Euclidean and hyperbolic that appear in Escher’s designs, and on and on. The Art of the Equation museum exhibit is a recent inversion: math as art. The beauty of (some) math purely as marks on a board.

Fractals, shapes that are self-similar, that is, made up of small copies of themselves, may provide another bridge between math and art. Here’s why. Aesthetically interesting images share two features: some element of familiarity and some element of novelty. The simplest fractals may appear interesting — for a while — because their repetition across scales provides familiarity: a Sierpinski gasket is made of smaller gaskets. But the gasket’s novelty is fleeting. As soon as we understand that each little triangular bit is an exact shrunken copy of the whole gasket, this fractal holds no more surprises.

a Sierpinski gasket

(That isn’t quite true: the gasket does have some interesting mathematical properties, but once decoded it is visually dull.)

Even slight variations on this exact repetition across scales can hide the pattern, mixing novelty in with familiarity. Here are two examples, obtained by imposing minor variations, just cutting out some pieces and then all the smaller copies of those pieces. Do these look interesting? In the first image below, collections of vertical and diagonal lines repeat across many scales, but not always where or how you expect. The second image below is more wiggly, but do you see parts of three steep diagonal lines pointing up and parts of two less-steep diagonal lines pointing down? Do you see echoes of some of these lines? What’s the pattern?
What’s hiding it?

Collections of vertical and diagonal lines repeat across many scales, but not always where or how you expect

Do you see parts of three steep diagonal lines pointing up and parts of two less-steep diagonal lines pointing down?

Are these pictures art? By themselves, no. They are computer-generated with simple software running a small set of instructions, but they might provide a beginning, a background, a hint on which art might be built.

The composer Charles Wuorinen described his nesting method for constructing self-similar pitch classes and intervals as the preparation for his composing. With this background, Wuorinen writes that he can follow his intuition and still produce a piece with the large-scale coherence he wants because of the patterns built in by the nesting method.

Could these fractals pictured above (not the gasket!) be a superstructure for making art? Print them and paint them, or cut and fold along some lines or curves, or build sculptures on top of them, or illuminate them with lights whose colors correspond to the complexity of the image. Interesting fractals might be a basis for interesting art.

Or let physics do some of the work. Put blobs of fairly viscous paint between two pieces of stiff paper. Apply pressure to the top piece, then pull the papers apart. Here’s an example of what happens.

In the early twentieth century, Max Ernst and other artists used this method to add a dream-like quality to some of their paintings. So this isn’t just a toy for preschoolers. Important artists have used this approach.

Are these the only ways fractals and art can work together? Not even close. Use your imagination. Can you find something new?

Michael Frame taught mathematics at Yale University for twenty years.

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