5-simplex honeycomb

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5-simplex honeycomb
(No image)
Type Uniform honeycomb
Family Simplectic honeycomb
Schläfli symbol {3[6]}
Coxeter–Dynkin diagrams CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png
5-face types {34} 5-simplex t0.svg, t1{34} 5-simplex t1.svg
t2{34} 5-simplex t2.svg
4-face types {33} 4-simplex t0.svg, t1{33} 4-simplex t1.svg
Cell types {3,3} 3-simplex t0.svg, t1{3,3} 3-simplex t1.svg
Face types {3} 2-simplex t0.svg
Vertex figure t0,4{34} 5-simplex t04.svg
Coxeter groups {\tilde{A}}_5×2, <[3[6]]>
Properties vertex-transitive

In five-dimensional Euclidean geometry, the 5-simplex honeycomb or hexateric honeycomb is a space-filling tessellation (or honeycomb or pentacomb). Each vertex is shared by 12 5-simplexes, 30 rectified 5-simplexes, and 20 birectified 5-simplexes. These facet types occur in proportions of 2:2:1 respectively in the whole honeycomb.

A5 lattice[edit]

This vertex arrangement is called the A5 lattice or 5-simplex lattice. The 30 vertices of the stericated 5-simplex vertex figure represent the 30 roots of the {\tilde{A}}_5 Coxeter group.[1] It is the 5-dimensional case of a simplectic honeycomb.

The A2
5
lattice can be constructed as the union of two A5 lattices:

CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node 1.png

The A3
5
is the union of three A5 lattices:

CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10lru.pngCDel split2.pngCDel node.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 01lr.pngCDel split2.pngCDel node.png.

The A*
5
lattice (also called A6
5
) is the union of six A5 lattices, and is the dual vertex arrangement to the omnitruncated 5-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 5-simplex.

CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel node.pngCDel split1.pngCDel nodes 10lur.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel node.pngCDel split1.pngCDel nodes 01lr.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10lru.pngCDel split2.pngCDel node.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 01lr.pngCDel split2.pngCDel node.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node 1.png = dual of CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel split2.pngCDel node 1.png

Related polytopes and honeycombs[edit]

This honeycomb is one of 12 unique uniform honeycombs[2] constructed by the {\tilde{A}}_5 Coxeter group. The extended symmetry of the hexagonal diagram of the {\tilde{A}}_5 Coxeter group allows for isomorphisms that map diagram nodes (mirrors) on to each other. So the various 12 honeycombs represent higher symmetries based on the ring arrangement symmetry in the diagrams:

Hexagon
symmetry
Extended
symmetry
Extended
diagram
Extended
order
Diagrams
a1 [3[6]] CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png ×1 CDel node 1.pngCDel split1.pngCDel nodes 10lur.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node 1.png
d2 <[3[6]]> CDel node c1.pngCDel split1.pngCDel nodeab c2.pngCDel 3ab.pngCDel nodeab c3.pngCDel split2.pngCDel node c4.png ×2 CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png1, CDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png, CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png, CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node 1.png, CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel split2.pngCDel node.png
p2 [[3[6]]] CDel node c3.pngCDel split1.pngCDel nodeab c1-2.pngCDel 3ab.pngCDel nodeab c1-2.pngCDel split2.pngCDel node c4.png ×2 CDel node.pngCDel split1.pngCDel nodes 10lur.pngCDel 3ab.pngCDel nodes 10lru.pngCDel split2.pngCDel node.png2, CDel node 1.pngCDel split1.pngCDel nodes 10lur.pngCDel 3ab.pngCDel nodes 10lru.pngCDel split2.pngCDel node 1.png
i4 [2[3[6]]] CDel node c3.pngCDel split1.pngCDel nodeab c1-2.pngCDel 3ab.pngCDel nodeab c1-2.pngCDel split2.pngCDel node c3.png ×4 CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node 1.png, CDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel split2.pngCDel node.png
d6 [3[3[6]]] CDel node c1.pngCDel split1.pngCDel nodeab c2.pngCDel 3ab.pngCDel nodeab c1.pngCDel split2.pngCDel node c2.png ×6 CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.pngCDel split2.pngCDel node.png
r12 [6[3[6]]] CDel node c1.pngCDel split1.pngCDel nodeab c1.pngCDel 3ab.pngCDel nodeab c1.pngCDel split2.pngCDel node c1.png ×12 CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel split2.pngCDel node 1.png3

Projection by folding[edit]

The 5-simplex honeycomb can be projected into the 3-dimensional cubic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:

{\tilde{A}}_5 CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png
{\tilde{C}}_3 CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png

See also[edit]

Regular and uniform honeycombs in 5-space:

Notes[edit]

References[edit]

  • 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 [2]
    • (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]