On the Structure of a Quasicrystal —Three-Dimensional Penrose Transformation—

Abstract

A 3-D Penrose transformation is found. The expansion rates, τ³ in linear dimension and τ⁹ in volume, are by far the larger than τ and τ³ respectively ever believed. The transformation decides only the skeleton uniquely, leaving some other degrees of freedom undecided. It strikingly differs from the deterministic 2-D case. Many other properties are common with the 2-D case. This transformation is helpful in understanding the quasi-crystal structure about which most of discussions so far were based on some incomplete or wrong knowledge. The beautiful structure of the transformation inspires to speculate the possibility of providing many theoretical tools to discuss various fundamental problems. Some details of the transformation is given in the form of an instruction for the model construction in Appendix.