Some Hyperbolic Fractal Tilings

The concepts of fractal tiling and hyperbolic tiling are combined to create novel fractal surfaces in Euclidean threespace. Paper folding and computer modeling are used to create these constructs. We show examples using triangular, trapezoidal, and dart-shaped prototiles. Smaller tiles deflect out of the plane of adjacent larger tiles, resulting in nonplanar surfaces, the shapes of which are dependent on the rules governing the sign of the deflections as well as their magnitudes. These constructs are an intriguing blend of organic and geometric character and in some cases bear marked resemblance to natural leaves. Fractal Tilings and Hyperbolic Tilings We have previously described a variety of fractal tilings, in which tiles are adjacent to larger and smaller tiles that are similar [2-3, 6]. Typically, the starting point is a small group of first-generation tiles of a single type, about which smaller tiles are arranging in edge-to-edge fashion according to some matching rule. These fractal tilings contain singular points and are of finite extent in the Euclidean plane, with boundaries that are fractal curves. Hyperbolic tilings are commonly depicted using the Poincaré disk model of the hyperbolic plane [1], and they satisfy the condition that the sum of the angles meeting at each vertex exceeds 360°. They can also be constructed in Euclidean three-space, in which case a rumpled and chaotic surface results [4]. Hyperbolic surfaces of limited extent have been created using crocheting and knitting, often with esthetically-pleasing results [9]. We report here on fractal tilings in which the sum of the angles meeting at a vertex is greater than 360 degrees; i.e., hyperbolic fractal tilings. These have been constructed in Euclidian three-space using both computer modeling and paper models. These structures are two-dimensional surfaces of finite extent living in three-dimensional space and do not conform to any rigorous definition of the hyperbolic plane. They are constructed of flat two-dimensional tiles, but they don’t tile any particular surface. The goal of this work is to determine what sort of novel and esthetically-pleasing fractal surfaces can be constructed using this technique. Of particular interest are geometric structures that mimic biological structures. All of the images, with the exception of Figures 1 and 7, were generated in Mathematica. Folded Two-dimensional Fractal Tilings Perhaps the simplest and most insightful starting point for exploring this topic is to experiment with folding some two-dimensional fractal tilings. A simple fractal tiling is shown in Figure 1, for which each tile is an isosceles right triangle [6]. A tiling in which every tile is similar is referred to as a singleprototile tiling, and all of the constructs described here are of this type. The long edges of two secondgeneration tiles are matched to the short edges of the first-generation tile along the entire length of each Proceedings of Bridges 2014: Mathematics, Music, Art, Architecture, Culture