Geometric models for continuous transitions from quasicrystals to crystals

Abstract

Starting from variable p-vectors half-stars which verify Hadwiger's theorem, the cut-projection method is used here. The strip of projection is projected on a rotatory subspace and a variable tiling is obtained. Two outstanding examples are developed. The first, a continuous evolution from a two-dimensional octagonal quasilattice to two square lattices 45° rotated in between. The second is a continuous evolution from a three-dimensional Penrose tiling to an f.c.c. vertex lattice. Physical applications to quasicrystal-crystal transitions are pointed out.