Omnitruncated simplicial honeycomb

From HandWiki

In geometry an omnitruncated simplicial honeycomb or omnitruncated n-simplex honeycomb is an n-dimensional uniform tessellation, based on the symmetry of the [math]\displaystyle{ {\tilde{A}}_n }[/math] affine Coxeter group. Each is composed of omnitruncated simplex facets. The vertex figure for each is an irregular n-simplex. The facets of an omnitruncated simplicial honeycomb are called permutahedra and can be positioned in n+1 space with integral coordinates, permutations of the whole numbers (0,1,..,n).

n [math]\displaystyle{ {\tilde{A}}_{1+} }[/math] Image Tessellation Facets Vertex figure Facets per vertex figure Vertices per vertex figure
1 [math]\displaystyle{ {\tilde{A}}_1 }[/math] Uniform apeirogon.png Apeirogon
CDel node 1.pngCDel infin.pngCDel node 1.png
Line segment Line segment 1 2
2 [math]\displaystyle{ {\tilde{A}}_2 }[/math] Uniform tiling 333-t012.png Hexagonal tiling
CDel node 1.pngCDel split1.pngCDel branch 11.png
2-simplex t01.svg
hexagon
Equilateral triangle
Hexagonal tiling vertfig.png
3 hexagons 3
3 [math]\displaystyle{ {\tilde{A}}_3 }[/math] Bitruncated cubic honeycomb2.png Bitruncated cubic honeycomb
CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel split2.pngCDel node 1.png
3-cube t12 B2.svg
Truncated octahedron
irr. tetrahedron
Omnitruncated 3-simplex honeycomb verf.png
4 truncated octahedron 4
4 [math]\displaystyle{ {\tilde{A}}_4 }[/math] Omnitruncated 4-simplex honeycomb
CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel branch 11.png
4-simplex t0123.svg
Omnitruncated 4-simplex
irr. 5-cell
Omnitruncated 4-simplex honeycomb verf.png
5 omnitruncated 4-simplex 5
5 [math]\displaystyle{ {\tilde{A}}_5 }[/math] Omnitruncated 5-simplex honeycomb
CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel split2.pngCDel node 1.png
5-simplex t01234.svg
Omnitruncated 5-simplex
irr. 5-simplex
Omnitruncated 5-simplex honeycomb verf.png
6 omnitruncated 5-simplex 6
6 [math]\displaystyle{ {\tilde{A}}_6 }[/math] Omnitruncated 6-simplex honeycomb
CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch 11.png
6-simplex t012345.svg
Omnitruncated 6-simplex
irr. 6-simplex
Omnitruncated 6-simplex honeycomb verf.png
7 omnitruncated 6-simplex 7
7 [math]\displaystyle{ {\tilde{A}}_7 }[/math] Omnitruncated 7-simplex honeycomb
CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel split2.pngCDel node 1.png
7-simplex t0123456.svg
Omnitruncated 7-simplex
irr. 7-simplex
Omnitruncated 7-simplex honeycomb verf.png
8 omnitruncated 7-simplex 8
8 [math]\displaystyle{ {\tilde{A}}_8 }[/math] Omnitruncated 8-simplex honeycomb
CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch 11.png
8-simplex t01234567 A7.svg
Omnitruncated 8-simplex
irr. 8-simplex
Omnitruncated 8-simplex honeycomb verf.png
9 omnitruncated 8-simplex 9

Projection by folding

The (2n-1)-simplex honeycombs can be projected into the n-dimensional omnitruncated hypercubic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:

[math]\displaystyle{ {\tilde{A}}_3 }[/math] CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel split2.pngCDel node 1.png [math]\displaystyle{ {\tilde{A}}_5 }[/math] CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel split2.pngCDel node 1.png [math]\displaystyle{ {\tilde{A}}_7 }[/math] CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel split2.pngCDel node 1.png [math]\displaystyle{ {\tilde{A}}_9 }[/math] CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel split2.pngCDel node 1.png ...
[math]\displaystyle{ {\tilde{C}}_2 }[/math] CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.png [math]\displaystyle{ {\tilde{C}}_3 }[/math] CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png [math]\displaystyle{ {\tilde{C}}_4 }[/math] CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png [math]\displaystyle{ {\tilde{C}}_5 }[/math] CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png ...

See also

References

  • George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
  • Branko Grünbaum, Uniform tilings of 3-space. Geombinatorics 4(1994), 49 - 56.
  • Norman Johnson Uniform Polytopes, Manuscript (1991)
  • Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
  • 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]
Fundamental convex regular and uniform honeycombs in dimensions 2-9
Space Family [math]\displaystyle{ {\tilde{A}}_{n-1} }[/math] [math]\displaystyle{ {\tilde{C}}_{n-1} }[/math] [math]\displaystyle{ {\tilde{B}}_{n-1} }[/math] [math]\displaystyle{ {\tilde{D}}_{n-1} }[/math] [math]\displaystyle{ {\tilde{G}}_2 }[/math] / [math]\displaystyle{ {\tilde{F}}_4 }[/math] / [math]\displaystyle{ {\tilde{E}}_{n-1} }[/math]
E2 Uniform tiling {3[3]} δ3 3 3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 4 4
E4 Uniform 4-honeycomb {3[5]} δ5 5 5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 6 6
E6 Uniform 6-honeycomb {3[7]} δ7 7 7 222
E7 Uniform 7-honeycomb {3[8]} δ8 8 8 133331
E8 Uniform 8-honeycomb {3[9]} δ9 9 9 152251521
E9 Uniform 9-honeycomb {3[10]} δ10 10 10
En-1 Uniform (n-1)-honeycomb {3[n]} δn n n 1k22k1k21