Three-dimensional analogs of the planar Penrose tilings and quasicrystals (Q1823471)

This review paraphrases the author's introduction. \textit{R. M. Schectman} et al [Phys. Rev. Letter 53, 1951-1953 (1984)] discovered alloys with a structure intermediate between crystalline and amorphous: quasicrystals which exhibit long-range orientational order but no translational symmetry. Since fivefold and even icosahedral symmetry is observed, it has been conjectured that the so called ``golden rhombohedra'' yield a geometric explanation, in analogy to the ``Penrose pieces'' in the plane. Guided by the idea that the long-range order of the quasicrystals must stem from local conditions, the author searched for subfamilies of a certain collection of 15 tetrahedra (derived from the Platonic icosahedron) which become aperiodic when subject to appropriate matching conditions - as do the Penrose pieces. He was successful in the search. In this paper he gives a semi-axiomatic approach to the subject, describes a realization and concludes with open problems.