Does it Look Square? Hexagonal Bipyramids, Triangular Antiprismoids, and their Fractals

Hideki Tsuiki
Bridges Donostia: Mathematics, Music, Art, Architecture, Culture (2007)
Pages 277–286


The Sierpinski tetrahedron is two-dimensional with respect to fractal dimensions though it is realized in three-dimensional space, and it has square projections in three orthogonal directions. We study its generalizations and present two-dimensional fractals with many square projections. One is generated from a hexagonal bipyramid which has square projections not in three but in six directions. Another one is generated from an octahedron which we call a triangular antiprismoid. These two polyhedra form a tiling of three-dimensional space, which is a Voronoi tessellation of three-dimensional space with respect to the union of two cubic lattices. We also present other fractals and show a simple object with three square projections obtained as the limit of such fractals.