Simplicial honeycomb
| Triangular tiling | Tetrahedral-octahedral honeycomb |
With red and yellow equilateral triangles | With cyan and yellow tetrahedra, and red rectified tetrahedra |
In geometry, the simplicial honeycomb is a dimensional infinite series of honeycombs, based on the affine Coxeter group symmetry. It is represented by a Coxeter-Dynkin diagram as a cyclic graph of nodes with one node ringed. It is composed of -simplex facets, along with all rectified -simplices. It can be thought of as an -dimensional hypercubic honeycomb that has been subdivided along all hyperplanes, then stretched along its main diagonal until the simplices on the ends of the hypercubes become regular. The vertex figure of an -simplex honeycomb is an expanded -simplex.
In 2 dimensions, the honeycomb represents the triangular tiling, with Coxeter graph filling the plane with alternately colored triangles. In 3 dimensions it represents the tetrahedral-octahedral honeycomb, with Coxeter graph filling space with alternately tetrahedral and octahedral cells. In 4 dimensions it is called the 5-cell honeycomb, with Coxeter graph, with 5-cell and rectified 5-cell facets. In 5 dimensions it is called the 5-simplex honeycomb, with Coxeter graph, filling space by 5-simplex, rectified 5-simplex, and birectified 5-simplex facets. In 6 dimensions it is called the 6-simplex honeycomb, with Coxeter graph, filling space by 6-simplex, rectified 6-simplex, and birectified 6-simplex facets.
By dimension
Projection by folding
The -simplex honeycombs and 2n-simplex honeycombs can be projected into the n-dimensional hypercubic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:| ... | ||||||||||
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