Tiles from projections of the root and weight lattices of $A_n$

Abstract

Main purpose of this work is to introduce a general technique of projection of the Voronoi tessellation of the weight lattice $A_n^\ast$ and apply it for the lattice $A_4^\ast$. The projection of the Voronoi tessellation of the weight lattice $A_4^\ast$ produces a totally different tiling scheme than the tiling obtained from the Voronoi cell projection of the lattice $A_4$. The 2D faces of the Voronoi cell of the lattice $A_4^\ast$ are of two types: regular hexagons and squares in 4-dimensions but project into two types of hexagons and two types of rhombuses with edges of two lengths in proportion to golden ratio. The mathematical technique employed is also useful for the projections of the root lattice $A_n$. A convenient set of linearly dependent and non-orthogonal $\left(n+1\right)$ vectors $k_i$ is introduced. The simple roots and the fundamental weights are defined as $α_i=k_i-k_{i+1},\left(i=1,2,\ldots,n\right) ,ω_i=k_1+k_2+\ldots+k_i$, respectively. When the vectors $k_i$ are defined in an orthogonal basis, the first two components of $k_i$ determine the Coxeter plane. Projection of the Delone cells of $A_n$ and $A_n^\ast$ on the Coxeter plane displays the same type of tiles and tilings but the Voronoi cell projection of these lattices yields different tiles and tilings. Vertices of the Voronoi cell $V(0)$ of $A_n$ is the union of the orbits of the weight vectors $W(a_n){(ω}_1)\cup W\left(a_n\right)(ω_2)\cup\ldots\cup W\left(a_n\right)(ω_n)$ and the 2D faces are the rhombuses. The Voronoi cell ${V(0)}^\ast$ of $A_n^\ast$ is the permutohedron of order $(n+1)$ and its vertices are the permutations of the vectors ${k}_i$ of the vertex $\frac{1}{n+1}[\left(n+1\right)k_1+nk_2+\ldots+k_{n+1}]$. It has regular hexagons and squares as 2D faces in $n$-dimensions.