Rectified 24-cell honeycomb

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Rectified 24-cell honeycomb
(No image)
Type Uniform 4-honeycomb
Schläfli symbol r{3,4,3,3}
rr{3,3,4,3}
r2r{4,3,3,4}
r2r{4,3,31,1}
Coxeter-Dynkin diagrams

CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel nodes 11.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png = CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel nodes 11.pngCDel split2.pngCDel node.pngCDel split1.pngCDel nodes 11.png = CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel 3g.pngCDel node g.png
CDel nodes 11.pngCDel split2.pngCDel node.pngCDel split1.pngCDel nodes 11.png = CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node h0.png

4-face type Tesseract Schlegel wireframe 8-cell.png
Rectified 24-cell Schlegel half-solid cantellated 16-cell.png
Cell type Cube Hexahedron.png
Cuboctahedron Cuboctahedron.png
Face type Square
Triangle
Vertex figure Rectified 24-cell honeycomb verf.png
Tetrahedral prism
Coxeter groups , [3,4,3,3]
, [4,3,3,4]
, [4,3,31,1]
, [31,1,1,1]
Properties Vertex transitive

In four-dimensional Euclidean geometry, the rectified 24-cell honeycomb is a uniform space-filling honeycomb. It is constructed by a rectification of the regular 24-cell honeycomb, containing tesseract and rectified 24-cell cells.

Alternate names[]

  • Rectified icositetrachoric tetracomb
  • Rectified icositetrachoric honeycomb
  • Cantellated 16-cell honeycomb
  • Bicantellated tesseractic honeycomb

Symmetry constructions[]

There are five different symmetry constructions of this tessellation. Each symmetry can be represented by different arrangements of colored rectified 24-cell and tesseract facets. The tetrahedral prism vertex figure contains 4 rectified 24-cells capped by two opposite tesseracts.

Coxeter group Coxeter
diagram
Facets Vertex figure Vertex
figure
symmetry
(order)

= [3,4,3,3]
CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 4: CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
1: CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Rectified 24-cell honeycomb verf.png CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png, [3,3,2]
(48)
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png 3: CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
1: CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
1: CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
Rectified 24-cell honeycomb F4b verf.png CDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png, [3,2]
(12)

= [4,3,3,4]
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png 2,2: CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
1: CDel node.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.png
Bicantellated tesseractic honeycomb verf.png CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png, [2,2]
(8)

= [31,1,3,4]
CDel nodes 11.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png 1,1: CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
2: CDel nodes 11.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.png
1: CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.png
Rectified 24-cell honeycomb B4 verf.png CDel node.pngCDel 2.pngCDel node.png, [2]
(4)

= [31,1,1,1]
CDel nodes 11.pngCDel split2.pngCDel node.pngCDel split1.pngCDel nodes 11.png 1,1,1,1:
CDel nodes 11.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.png
1: CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png
Rectified 24-cell honeycomb D4 verf.png CDel node.png, []
(2)

See also[]

Regular and uniform honeycombs in 4-space:

References[]

  • Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p. 296, Table II: Regular honeycombs
  • 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 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 93
  • Klitzing, Richard. "4D Euclidean tesselations"., o3o3o4x3o, o4x3o3x4o - ricot - O93
Space Family / /
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
E10 Uniform 10-honeycomb {3[11]} δ11 11 11
En-1 Uniform (n-1)-honeycomb {3[n]} δn n n 1k22k1k21
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