| Summary: | The orthogonal projections of the Voronoi and Delone cells of root lattice <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>A</mi> <mi>n</mi> </msub> </mrow> </semantics> </math> </inline-formula> 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 <inline-formula> <math display="inline"> <semantics> <mrow> <mn>2</mn> <mi>h</mi> </mrow> </semantics> </math> </inline-formula> involving the Coxeter element of order <inline-formula> <math display="inline"> <semantics> <mrow> <mi>h</mi> <mo>=</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula> of the Coxeter−Weyl group <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>a</mi> <mi>n</mi> </msub> </mrow> </semantics> </math> </inline-formula> play a crucial role for <inline-formula> <math display="inline"> <semantics> <mi>h</mi> </semantics> </math> </inline-formula>-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 <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>A</mi> <mrow> <mn>3</mn> </mrow> </msub> </mrow> </semantics> </math> </inline-formula>, whose Wigner−Seitz cell is the rhombic dodecahedron projects, as expected, onto a square lattice with an <inline-formula> <math display="inline"> <semantics> <mrow> <mi>h</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics> </math> </inline-formula>-fold symmetry.
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