Abstract
The orthogonal projections of the Voronoi and Delone cells of root lattice A n onto the Coxeter plane display various rhombic and triangular prototiles including thick and thin rhombi of Penrose, Amman–Beenker tiles, Robinson triangles, and Danzer triangles to name a few. We point out that the symmetries representing the dihedral subgroup of order 2 h involving the Coxeter element of order h = n + 1 of the Coxeter–Weyl group a n play a crucial role for h -fold symmetric tilings of the Coxeter plane. After setting the general scheme we give samples of patches with 4-, 5-, 6-, 7-, 8-, and 12-fold symmetries. The face centered cubic (f.c.c.) lattice described by the root lattice A 3 , whose Wigner–Seitz cell is the rhombic dodecahedron projects, as expected, onto a square lattice with an h = 4 -fold symmetry.
Highlights
Discovery of a fivefold symmetric material [1] has led to growing interest in quasicrystallography.For a review, see for instance [2,3,4]
We show that the 2D facets of the Delone cells of a given lattice An are equilateral triangles; when projected into the Coxeter plane they lead to various triangles, which tile the plane in an aperiodic manner
In references [14,39] we have introduced an equivalent definition of the Coxeter plane through the eigenvalues and eigenvectors of the Cartan matrix of the root system of the lattice An
Summary
Discovery of a fivefold symmetric material [1] has led to growing interest in quasicrystallography. Projections of the 2D facets of the Voronoi polytope of An lead to (n + 1)-symmetric tilings of the Coxeter plane by a number of rhombi with different interior angles. We show that the 2D facets of the Delone cells of a given lattice An are equilateral triangles; when projected into the Coxeter plane they lead to various triangles, which tile the plane in an aperiodic manner. The paper displays the lists of the rhombic and triangular prototiles in Tables 1 and 2, originating from the projections of the Voronoi and Delone cells of the lattice An onto the Coxeter plane, respectively, and illustrates some patches of the aperiodic tilings.
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