Quasiperiodic patterns generated by mixing lattices derived from a dodecahedral star and an icosahedral star

Abstract

A 3D quasiperiodic pattern by projection from an nD lattice can be defined by an orthonormal n × n lattice matrix which produces basis vectors in pattern space with a prescribed arrangement and basis vectors in perpendicular or test space satisfying the quasicrystallographic condition. A 16 × 16 lattice matrix is derived which produces basis vectors in pattern space as a combination or mixing of dually positioned dodecahedral star and icosahedral star. It is shown that the mixed star constitutes a eutactic star. Since the module generated by the mixed eutactic star is totally irrational, patterns generated by projection using the lattice matrix are quasicrystallographic and the equilateral pattern is generated for 1: 1 mixing.