AKLT Model

4-simplex honeycomb
(No image)
Type	Uniform 4-honeycomb
Family	Simplectic honeycomb
Schläfli symbol	{3[5]}
Coxeter diagram	Template:CDD
4-face types	{3,3,3} t1{3,3,3}
Cell types	{3,3} t1{3,3}
Face types	{3}
Vertex figure	t0,3{3,3,3}
Symmetry	${\tilde {A}}_{4}$×2, [[3[5]]]
Properties	vertex-transitive

In four-dimensional Euclidean geometry, the 4-simplex honeycomb, 5-cell honeycomb or pentachoric-dispentachoric honeycomb is a space-filling tessellation honeycomb. It is composed of 5-cells and rectified 5-cells facets in a ratio of 1:1.

Cells of the vertex figure are ten tetrahedrons and 20 triangular prisms, corresponding to the ten 5-cells and 20 rectified 5-cells that meet at each vertex. All the vertices lie in parallel realms in which they form alternated cubic honeycombs, the tetrahedra being either tops of the rectified 5-cell or the bases of the 5-cell, and the octahedra being the bottoms of the rectified 5-cell.

A4 lattice

This vertex arrangement is called the A4 lattice, or 4-simplex lattice. The 20 vertices of its vertex figure, the runcinated 5-cell represent the 20 roots of the ${\tilde {A}}_{4}$ Coxeter group. It is the 4-dimensional case of a simplectic honeycomb.

The ATemplate:Sup sub lattice is the union of five A₄ lattices, and is the dual to the omnitruncated 5-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 5-cell

Template:CDD + Template:CDD + Template:CDD + Template:CDD + Template:CDD = dual of Template:CDD

Alternate names

• Cyclopentachoric tetracomb
• Pentachoric-dispentachoric tetracomb

Related polytopes and honeycombs

The tops of the 5-cells in this honeycomb adjoin the bases of the 5-cells, and vice versa, in adjacent laminae; but alternating laminae may be inverted so that the tops of the rectified 5-cells adjoin the tops of the rectified 5-cells and the bases of the 5-cells adjoin the bases of other 5-cells. This inversion results in another non-Wythoffian uniform convex honeycomb. Octahedral prisms and tetrahedral prisms may be inserted in between alternated laminae as well, resulting in two more non-Wythoffian elongated uniform honeycombs.

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Projection by folding

The 5-cell honeycomb can be projected into the 2-dimensional square tiling by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:

${\tilde {A}}_{3}$	Template:CDD
${\tilde {C}}_{2}$	Template:CDD

See also

Regular and uniform honeycombs in 4-space:

• Tesseractic honeycomb
• 16-cell honeycomb
• 24-cell honeycomb
• Truncated 24-cell honeycomb
• Snub 24-cell honeycomb
• Truncated 5-cell honeycomb
• Omnitruncated 5-cell honeycomb

Notes

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References

• Norman Johnson Uniform Polytopes, Manuscript (1991)
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
  • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)
  • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs) Model 134
• Template:KlitzingPolytopes, x3o3o3o3o3*a - cypit - O134
• Affine Coxeter group Wa(A4), Quaternions, and Decagonal Quasicrystals Mehmet Koca, Nazife O. Koca, Ramazan Koc (2013) [2]

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