Icosahedral Tiling with Dodecahedral Structures

TL;DR

This paper presents a novel 3D icosahedral tiling using dodecahedral structures derived from tetrahedral tiles related to the D6 lattice, extending aperiodic tilings into three dimensions with golden ratio-based dodecahedra.

Contribution

It introduces a new 3D tiling method based on dissected icosahedral and dodecahedral structures from D6 lattice facets, generalizing 2D aperiodic tilings to three dimensions.

Findings

Tiling space with composite tiles generated by an inflation matrix with factor τ.

Construction of dodecahedra with edges of length 1 and τ from tetrahedral tiles.

Extension of 2D Robinson triangle tilings to 3D icosahedral structures.

Abstract

Icosahedron and dodecahedron can be dissected into tetrahedral tiles projected from 3D-facets of the Delone polytopes representing the deep and shallow holes of the root lattice D_6. The six fundamental tiles of tetrahedra of edge lengths 1 and \tau are assembled into four composite tiles whose faces are normal to the 5-fold axes of the icosahedral group. The 3D Euclidean space is tiled face-to-face by the composite tiles with an inflation factor \tau generated by an inflation matrix. The aperiodic tiling is a generalization of the Tubingen triangular tiling in 2-dimensions for the faces of the tiles are made of Robinson triangles. Certain combinations of the tiles constitute dodecahedra with edge lengths of 1 and the golden ratio \tau=(1+\sqrt(5))/2.