, Volume 118, Issue 2, pp 263-288

Theory of matching rules for the 3-dimensional Penrose tilings

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Abstract

We consider packings of the two Ammann rhombohedra used for tiling the three dimensional space. We define decorations for the facets of the rhombohedra. Using elementary algebraic topology, we prove that any tiling by these rhombohedra with matching decorations is a quasiperiodic Penrose tiling. The proof does not involve any reference to self similarity.

Communicated by A. Jaffe