A Fractal Crystal Comprised of Cubes and Some Related Fractal Arrangements of other Platonic Solids

Robert W. Fathauer, Hank Kaczmarski and and Nicholas Duchnowski
Bridges Leeuwarden: Mathematics, Music, Art, Architecture, Culture (2008)
Pages 289–296


A simple iterative arrangement of cubes leads to a visually rich and complex fractal “crystal” with an overall regular-octahedron convex hull and infinitely many “facets”. Each facet is essentially a Sierpinski triangle, and the vertex of a cube just touches the center of each triangular hole. This fractal crystal is constructed by starting with a first generation cube and placing a half-scale cube on the center of each face. The second-generation cubes have the same orientation as the first-generation cube. Third-generation cubes again scaled by half are placed on each unoccupied face of a second-generation cube. This process is continued ad infinitum to form the fractal crystal. Some related constructions created using other Platonic Solids are described as well.