Dodecahedral structures with Mosseri-Sadoc tiles

Area of Science:

• Geometry
• Crystallography
• Tiling Theory

Background:

• The study investigates the projection of 6D Euclidean space tiling facets into 3D.
• Focuses on the root lattice D6 and its Delone cells.

Purpose of the Study:

• Classify projected 3D facets into specific tetrahedral tiles.
• Demonstrate tiling of 3D Euclidean space using composite tiles derived from these tetrahedra.
• Analyze the symmetries and inflation properties of the resulting structures.

Main Methods:

• Projection of 6D Delone cells of D6 lattice into 3D space.
• Classification of projected facets into Mosseri-Sadoc tetrahedral tiles.
• Dissection of icosahedral group related polyhedra (icosahedron, dodecahedron, icosidodecahedron) into tetrahedra.
• Composition of fundamental tiles into larger composite tiles.
• Analysis of tiling properties, including face-to-face coverage and inflation factors.

Main Results:

• Identified six types of Mosseri-Sadoc tetrahedral tiles with edge lengths 1 and the golden ratio τ.
• Demonstrated that 3D Euclidean space can be tiled face-to-face with maximal coverage using composite tiles.
• Observed dodecahedra with edge lengths 1 and τ in early inflation orders (n=2, 3).
• Generated 3D patches with fivefold, threefold, and twofold symmetries from inflated dodecahedral structures (edge lengths τ^n, n ≥ 3).
• Planar tiling of composite tile faces follows Robinson triangle edge-to-edge matching.

Conclusions:

• The composite tiles derived from D6 lattice projections effectively tile 3D Euclidean space.
• The inflation process generates structures with characteristic symmetries and polyhedral forms.
• The findings connect higher-dimensional lattices to 3D tiling phenomena with applications in geometry and materials science.