Fractal Gaskets: Reptiles, Hamiltonian Cycles, and Spatial Development

Robert W. Fathauer
Proceedings of Bridges 2016: Mathematics, Music, Art, Architecture, Education, Culture (2016)
Pages 217–224 Regular Papers


A wide variety of fractal gaskets have been designed from self-replicating tiles. In contrast to the most well-known examples, the Sierpinski carpet and Sierpinski triangle, these gaskets generally have fractal outer boundaries, and the holes in them generally have fractal boundaries. Hamiltonian cycles have been explored that trace out some of these fractal gaskets. Novel solids have been created by spatially developing these gasket fractals over the first several generations. Successive generations are separated in a direction orthogonal to the plane of the gasket, and simple polygons are used to connect the external and internal edges of the gaskets. Since all of the faces in the resulting structures are polygonal, these solids can be described as polyhedra. By varying the spacing between generations, the form of these polyhedra can be varied, creating three-dimensional constructs evocative of architectural forms and geological formations.