Tiles from projections of the root and weight lattices of An.

Abstract

The root lattice A_n and its dual weight lattice A_n* acquire remarkable quasicrystallographic significance through their projections onto the Coxeter plane, where (n + 1)-fold symmetry emerges naturally. The primary objective of this work is to present a general technique for projection of the Voronoi tessellation of the weight lattice A_n* and apply it for the lattice A₄*. The projection of the Voronoi tessellation of the weight lattice A₄* produces a totally different tiling scheme than the tiling obtained from the Voronoi cell projection of the lattice A₄ which leads to the Penrose tiling with thick and thin rhombuses. The Voronoi cell of the weight lattice A₄* is a permutohedron of order 5, comprising 2D faces of regular hexagons and squares in 4D but which project into two types of hexagons and two types of rhombuses with edges of two lengths in proportion to the golden ratio. A detailed structure of the permutohedron with its 3D and 2D facets has been worked out. A centrally symmetric tessellation of the Coxeter plane exhibiting fivefold symmetry has been constructed using four distinct tile types.