Dodecahedral Structures with Mosseri-Sadoc Tiles

TL;DR

This paper explores the projection and tiling of 3D facets derived from D6 lattice cells into three-dimensional space, revealing new dodecahedral structures with golden ratio edges and their inflation properties.

Contribution

It introduces a novel classification of Mosseri-Sadoc tetrahedral tiles and demonstrates how these can be used to tile 3D space with dodecahedral structures exhibiting golden ratio scaling.

Findings

3D space can be tiled with composite tiles with inflation factor tau

Dodecahedra with edges of length 1 and tau naturally occur in inflation sequences

Inflated structures display 5-fold, 3-fold, and 2-fold symmetries

Abstract

3D-facets of the Delone cells representing the deep and shallow holes of the root lattice D6 which tile the six-dimensional Euclidean space in an alternating order are projected into three-dimensional space. They are classified into six Mosseri-Sadoc tetrahedral tiles of edge lengths 1 and golden ratio (tau) with faces normal to the 5-fold and 3-fold axes. The icosahedron, dodecahedron and icosidodecahedron whose vertices are obtained from the fundamental weights of the icosahedral group are dissected in terms of six tetrahedra. A set of four tiles are composed out of six fundamental tiles, faces of which, are normal to the 5-fold axes of the icosahedral group. It is shown that the 3D-Euclidean space can be tiled face-to-face with maximal face coverage by the composite tiles with an inflation factor tau generated by an inflation matrix. We note that dodecahedra with edge lengths of 1 and tau naturally occur already in the second and third order of the inflations. The 3D patches displaying 5-fold, 3-fold and 2-fold symmetries are obtained in the inflated dodecahedral structures with edge lengths tau to the power n with n equals 3 or greater than 3. The planar tiling of the faces of the composite tiles follow the edge-to-edge matching of the Robinson triangles.