Quasicrystals and Discrete Geometry

0int 8int abut algebraic angle aperiodic atoms bounded cells centered coronas Ck(P cluster compact Abelian groups conjugate convex corresponding Coxeter group crystal crystalline crystallographic cut-and-project set cyclotomic decagonal defined definition Delone set denote determined edges elements equation ergodic Euclidean Figure finite fractal globally permitted hence hierarchical modular icosahedral icosahedron implies integers invariant isometry isometry that maps isomorphic isosceles trapezoid Laves Lemma locally compact locally compact Abelian Mathematical Meyer set mirror modular loops modular structures modules monohedral tiling Moody mosaics orbit pairwise congruent Patera pentagons permutations Pisot plane polygon polytope possible proof Proposition prototile quasicrystals quasilattices radius rhombs rods root lattice root system Section self-similar semigroup Si(P So(P space species star map subgroup subroot system subset symmetry group T-element tetrahedral Theorem tiling is isohedral topological translation triplet two-dimensional unique vectors vertex of degree vertices Weyl group wreath product