Three-dimensional analogs of the planar penrose tilings and quasicrystals

In 1984 Shechtman et al. [lo] discovered alloys with a novel kind of structure, intermediate between crystalline and amorphous. These alloys, called qruzskrystuk [8], exhibit long-range orientational order but no translational symmetry. Since fivefold and even icosahedral symmetry is observed, several authors conjectured that the so-called “golden rhombohedra” yield a geometric explanation, in analogy to the “Penrose pieces” (see [3,9]) in the plane. N.G. de Bruijn and others [l, 5,6,7, 1111 developed the “projection method”, using cubic lattices in higher dimensions to obtain nonperiodic tilings with golden rhombohedra. Guided by the idea that the long-range order of the quasicrystals must stem from some focal conditions, I have sought families of prototiles which become aperiodic when subject to appropriate matching conditions as do the Penrose pieces. I restricted my search to subfamilies 9 of a certain family #:={A,B,C ,..., P} of fifteentetrahedra, which are derived from the Platonic icosahedron (see below), and succeeded in finding a four-member family S1. To my knowledge, the work of Katz [4] is the only attempt to give matching conditions for the golden rhombohedra which force aperiodicity. His proof is along completely different lines, using algebraic topology in place of inflation/deflation. In fact, his family consists of 22 prototiles (14 congruent to one and 8 to the other golden rhombohedron); they differ in their decorations (matching conditions). In this note I first give a semi-axiomatic approach to the subject, and then describe a realization.