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Cantellated tesseractic honeycomb

May 10, 2024

Cantellated tesseractic honeycomb
(No image)
Type Uniform 4-honeycomb
Schläfli symbol t0,2{4,3,3,4} or rr{4,3,3,4}
rr{4,3,31,1}
Coxeter-Dynkin diagram
4-face type t0,2{4,3,3}
t1{3,3,4}
{3,4}×{}
Cell type Octahedron
Rhombicuboctahedron
Triangular prism
Face type {3}, {4}
Vertex figure Cubic wedge
Coxeter group = [4,3,3,4]
= [4,3,31,1]
Dual
Properties vertex-transitive

In four-dimensional Euclidean geometry, the cantellated tesseractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 4-space. It is constructed by a cantellation of a tesseractic honeycomb creating cantellated tesseracts, and new 24-cell and octahedral prism facets at the original vertices.

Related honeycombs edit

The [4,3,3,4],          , Coxeter group generates 31 permutations of uniform tessellations, 21 with distinct symmetry and 20 with distinct geometry. The expanded tesseractic honeycomb (also known as the stericated tesseractic honeycomb) is geometrically identical to the tesseractic honeycomb. Three of the symmetric honeycombs are shared in the [3,4,3,3] family. Two alternations (13) and (17), and the quarter tesseractic (2) are repeated in other families.

C4 honeycombs
Extended
symmetry 		Extended
diagram 		Order Honeycombs
[4,3,3,4]:           ×1

          1,           2,           3,           4,
          5,           6,           7,           8,
          9,           10,           11,           12,
          13

[[4,3,3,4]]       ×2           (1),           (2),           (13),           18
          (6),           19,           20
[(3,3)[1+,4,3,3,4,1+]]
↔ [(3,3)[31,1,1,1]]
↔ [3,4,3,3] 		     
↔      
↔           		×6

          14,           15,           16,           17

The [4,3,31,1],        , Coxeter group generates 31 permutations of uniform tessellations, 23 with distinct symmetry and 4 with distinct geometry. There are two alternated forms: the alternations (19) and (24) have the same geometry as the 16-cell honeycomb and snub 24-cell honeycomb respectively.

B4 honeycombs
Extended
symmetry 		Extended
diagram 		Order Honeycombs
[4,3,31,1]:         ×1

        5,         6,         7,         8

<[4,3,31,1]>:
↔[4,3,3,4] 		       
↔           		×2

        9,         10,         11,         12,         13,         14,

        (10),         15,         16,         (13),         17,         18,         19

[3[1+,4,3,31,1]]
↔ [3[3,31,1,1]]
↔ [3,3,4,3] 		       
↔       
↔           		×3

        1,         2,         3,         4

[(3,3)[1+,4,3,31,1]]
↔ [(3,3)[31,1,1,1]]
↔ [3,4,3,3] 		       
↔      
↔           		×12

        20,         21,         22,         23

See also edit

Regular and uniform honeycombs in 4-space:

Notes edit

References edit

  • 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] See p318 [2]
  • George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
  • Klitzing, Richard. "4D Euclidean tesselations#4D". o3x3o *b3o4x, x4o3x3o4o - srittit - O90
  • Conway JH, Sloane NJH (1998). Sphere Packings, Lattices and Groups (3rd ed.). ISBN 0-387-98585-9.
Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space Family           /   /  
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
E10 Uniform 10-honeycomb {3[11]} δ11 hδ11 qδ11
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21

May 10, 2024
cantellated, tesseractic, honeycomb, image, type, uniform, honeycomb, schläfli, symbol, coxeter, dynkin, diagram, face, type, cell, type, octahedron, rhombicuboctahedron, triangular, prism, face, type, vertex, figure, cubic, wedge, coxeter, group, displaystyle. Cantellated tesseractic honeycomb No image Type Uniform 4 honeycomb Schlafli symbol t0 2 4 3 3 4 or rr 4 3 3 4 rr 4 3 31 1 Coxeter Dynkin diagram 4 face type t0 2 4 3 3 t1 3 3 4 3 4 Cell type Octahedron Rhombicuboctahedron Triangular prism Face type 3 4 Vertex figure Cubic wedge Coxeter group C 4 displaystyle tilde C 4 4 3 3 4 B 4 displaystyle tilde B 4 4 3 31 1 Dual Properties vertex transitive In four dimensional Euclidean geometry the cantellated tesseractic honeycomb is a uniform space filling tessellation or honeycomb in Euclidean 4 space It is constructed by a cantellation of a tesseractic honeycomb creating cantellated tesseracts and new 24 cell and octahedral prism facets at the original vertices Contents 1 Related honeycombs 2 See also 3 Notes 4 ReferencesRelated honeycombs editThe 4 3 3 4 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Coxeter group generates 31 permutations of uniform tessellations 21 with distinct symmetry and 20 with distinct geometry The expanded tesseractic honeycomb also known as the stericated tesseractic honeycomb is geometrically identical to the tesseractic honeycomb Three of the symmetric honeycombs are shared in the 3 4 3 3 family Two alternations 13 and 17 and the quarter tesseractic 2 are repeated in other families C4 honeycombs Extendedsymmetry Extendeddiagram Order Honeycombs 4 3 3 4 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 1 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 1 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 2 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 3 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 4 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 5 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 6 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 7 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 8 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 9 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 10 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 11 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 12 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 13 4 3 3 4 nbsp nbsp nbsp nbsp nbsp 2 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 1 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 2 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 13 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 18 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 6 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 19 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 20 3 3 1 4 3 3 4 1 3 3 31 1 1 1 3 4 3 3 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 6 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 14 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 15 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 16 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 17 The 4 3 31 1 nbsp nbsp nbsp nbsp nbsp nbsp nbsp Coxeter group generates 31 permutations of uniform tessellations 23 with distinct symmetry and 4 with distinct geometry There are two alternated forms the alternations 19 and 24 have the same geometry as the 16 cell honeycomb and snub 24 cell honeycomb respectively B4 honeycombs Extendedsymmetry Extendeddiagram Order Honeycombs 4 3 31 1 nbsp nbsp nbsp nbsp nbsp nbsp nbsp 1 nbsp nbsp nbsp nbsp nbsp nbsp nbsp 5 nbsp nbsp nbsp nbsp nbsp nbsp nbsp 6 nbsp nbsp nbsp nbsp nbsp nbsp nbsp 7 nbsp nbsp nbsp nbsp nbsp nbsp nbsp 8 lt 4 3 31 1 gt 4 3 3 4 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 2 nbsp nbsp nbsp nbsp nbsp nbsp nbsp 9 nbsp nbsp nbsp nbsp nbsp nbsp nbsp 10 nbsp nbsp nbsp nbsp nbsp nbsp nbsp 11 nbsp nbsp nbsp nbsp nbsp nbsp nbsp 12 nbsp nbsp nbsp nbsp nbsp nbsp nbsp 13 nbsp nbsp nbsp nbsp nbsp nbsp nbsp 14 nbsp nbsp nbsp nbsp nbsp nbsp nbsp 10 nbsp nbsp nbsp nbsp nbsp nbsp nbsp 15 nbsp nbsp nbsp nbsp nbsp nbsp nbsp 16 nbsp nbsp nbsp nbsp nbsp nbsp nbsp 13 nbsp nbsp nbsp nbsp nbsp nbsp nbsp 17 nbsp nbsp nbsp nbsp nbsp nbsp nbsp 18 nbsp nbsp nbsp nbsp nbsp nbsp nbsp 19 3 1 4 3 31 1 3 3 31 1 1 3 3 4 3 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 3 nbsp nbsp nbsp nbsp nbsp nbsp nbsp 1 nbsp nbsp nbsp nbsp nbsp nbsp nbsp 2 nbsp nbsp nbsp nbsp nbsp nbsp nbsp 3 nbsp nbsp nbsp nbsp nbsp nbsp nbsp 4 3 3 1 4 3 31 1 3 3 31 1 1 1 3 4 3 3 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 12 nbsp nbsp nbsp nbsp nbsp nbsp nbsp 20 nbsp nbsp nbsp nbsp nbsp nbsp nbsp 21 nbsp nbsp nbsp nbsp nbsp nbsp nbsp 22 nbsp nbsp nbsp nbsp nbsp nbsp nbsp 23See also editRegular and uniform honeycombs in 4 space Tesseractic honeycomb Demitesseractic honeycomb 24 cell honeycomb Truncated 24 cell honeycomb Snub 24 cell honeycomb 5 cell honeycomb Truncated 5 cell honeycomb Omnitruncated 5 cell honeycombNotes editReferences editKaleidoscopes 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 See p318 2 George Olshevsky Uniform Panoploid Tetracombs Manuscript 2006 Complete list of 11 convex uniform tilings 28 convex uniform honeycombs and 143 convex uniform tetracombs Klitzing Richard 4D Euclidean tesselations 4D o3x3o b3o4x x4o3x3o4o srittit O90 Conway JH Sloane NJH 1998 Sphere Packings Lattices and Groups 3rd ed ISBN 0 387 98585 9 vteFundamental convex regular and uniform honeycombs in dimensions 2 9 Space Family A n 1 displaystyle tilde A n 1 nbsp C n 1 displaystyle tilde C n 1 nbsp B n 1 displaystyle tilde B n 1 nbsp D n 1 displaystyle tilde D n 1 nbsp G 2 displaystyle tilde G 2 nbsp F 4 displaystyle tilde F 4 nbsp E n 1 displaystyle tilde E n 1 nbsp E2 Uniform tiling 3 3 d3 hd3 qd3 Hexagonal E3 Uniform convex honeycomb 3 4 d4 hd4 qd4 E4 Uniform 4 honeycomb 3 5 d5 hd5 qd5 24 cell honeycomb E5 Uniform 5 honeycomb 3 6 d6 hd6 qd6 E6 Uniform 6 honeycomb 3 7 d7 hd7 qd7 222 E7 Uniform 7 honeycomb 3 8 d8 hd8 qd8 133 331 E8 Uniform 8 honeycomb 3 9 d9 hd9 qd9 152 251 521 E9 Uniform 9 honeycomb 3 10 d10 hd10 qd10 E10 Uniform 10 honeycomb 3 11 d11 hd11 qd11 En 1 Uniform n 1 honeycomb 3 n dn hdn qdn 1k2 2k1 k21 Retrieved from https en wikipedia org w index php title Cantellated tesseractic honeycomb amp oldid 1197261191, wikipedia, wiki, book, books, library,

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