# Douglas McKenna

Like other areas where constraint and freedom conflict, the tension between symmetry and asymmetry in mathematically defined or algorithmically constructed objects is where art lies. This interplay is important when trying to find a balance between platonic and aesthetic beauty in any satisfying mathematical art. Even more gratifying is when one's aesthetic explorations of a constrained combinatorial space lead to new mathematical discoveries.

Space-filling curve constructions are threaded tilings (essentially special Hamiltonian paths on grid graphs and their duals) that visually evince this symmetry vs. asymmetry tension. These self-avoiding paths in turn are composable elements of algebraic structures called monoids. Much of my "mathemaesthetic" work combines research into these monoidal patterns and their constraints, followed by algorithmic and/or visual play with the results.

The basis of this piece is a compound Lissajous figure called the Pas de Deux curve, made calligraphic by widening the path as a function of distance from the center. The resulting sweep is then drawn directly in PostScript using over 10,000 thin, fixed-width line segments connected into a simple loop using a (then-novel) fill algorithm the artist developed for pen-plotters in the early 1980s. An explanation of the math and history is in the 2011 Bridges paper, "From Lissajous to Pas de Deux to Tattoo: The Graphic Life of a Beautiful Loop". This loop was determined after much experimentation with frequencies, amplitudes, and phases, with an eye towards symmetry, asymmetry, and grace.

The light-colored trace over the black background creates a 1960s-era black-light shading effect that gives a mathematically two-dimensional figure a very three-dimensional, dynamic, and mysterious look.

This graphically fanciful piece is based on a space-filling curve construction devised by the artist. It starts out with 120 x 120 = 14,400 small square tiles, threaded to form a Hamiltonian path in their dual adjacency graph, much the same way the Hilbert Curve construction linearizes its initial four square tiles. Groups of consecutive, co-linear segments are coalesced, with corners smoothed. The underlying tiling is thrown away. The still self-avoiding result divides the blue and white world into spiral, square, and fingering forms that harmoniously reference each other. The layout and patterns are largely a result of underlying combinatorial constraints, as opposed to purely aesthetic choices on the part of the artist. The mathematics here isn't just a tool or medium. Here, a mathematical constraint system evinces itself as art.