Iterative Arrangements of Polyhedra—Relationships to Classical Fractals and Haüy Constructions

Robert W. Fathauer
Proceedings of Bridges 2013: Mathematics, Music, Art, Architecture, Culture (2013)
Pages 175–182

Abstract

This paper exhibits and explains esthetically-pleasing constructions using scaled-down polyhedra that have been iteratively arranged on the faces of a starting polyhedron. Sierpinski triangles usually arise when half-scale polyhedra are iteratively arranged on three faces meeting at a vertex. In contrast, a regular array results when half- scale polyhedra are iteratively arranged on four faces meeting at a vertex. The convex hulls of such constructs are the duals of the starting polyhedra for a variety of polyhedra. These arrangements can be thought of as generalized Haüy constructions using a scaling factor less than one. One half is shown to be a special number for such scalings. When arrangements are made about vertices with five faces, a scaling factor of the square of the Golden mean results in a fractal that can be described as a Sierpinski pentagon.

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