Stericated 5-cubes
![]() 5-cube ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Stericated 5-cube ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Steritruncated 5-cube ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Stericantellated 5-cube ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Steritruncated 5-orthoplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Stericantitruncated 5-cube ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Steriruncitruncated 5-cube ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Stericantitruncated 5-orthoplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Omnitruncated 5-cube ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Orthogonal projections in B5 Coxeter plane |
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In five-dimensional geometry, a stericated 5-cube is a convex uniform 5-polytope with fourth-order truncations (sterication) of the regular 5-cube.
There are eight degrees of sterication for the 5-cube, including permutations of runcination, cantellation, and truncation. The simple stericated 5-cube is also called an expanded 5-cube, with the first and last nodes ringed, for being constructible by an expansion operation applied to the regular 5-cube. The highest form, the sterirunci
Stericated 5-cube
Stericated 5-cube | ||
Type | Uniform 5-polytope | |
Schläfli symbol | 2r2r{4,3,3,3} | |
Coxeter-Dynkin diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4-faces | 242 | |
Cells | 800 | |
Faces | 1040 | |
Edges | 640 | |
Vertices | 160 | |
Vertex figure | ![]() | |
Coxeter group | B5 [4,3,3,3] | |
Properties | convex |
Alternate names
- Stericated penteract / Stericated 5-orthoplex / Stericated pentacross
- Expanded penteract / Expanded 5-orthoplex / Expanded pentacross
- Small cellated penteractitriacontaditeron (Acronym: scant) (Jonathan Bowers)[1]
Coordinates
The Cartesian coordinates of the vertices of a stericated 5-cube having edge length 2 are all permutations of:
- [math]\displaystyle{ \left(\pm1,\ \pm1,\ \pm1,\ \pm1,\ \pm(1+\sqrt{2})\right) }[/math]
Images
The stericated 5-cube is constructed by a sterication operation applied to the 5-cube.
Coxeter plane | B5 | B4 / D5 | B3 / D4 / A2 |
---|---|---|---|
Graph | ![]() |
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Dihedral symmetry | [10] | [8] | [6] |
Coxeter plane | B2 | A3 | |
Graph | ![]() |
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Dihedral symmetry | [4] | [4] |
Steritruncated 5-cube
Steritruncated 5-cube | |
---|---|
Type | uniform 5-polytope |
Schläfli symbol | t0,1,4{4,3,3,3} |
Coxeter-Dynkin diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4-faces | 242 |
Cells | 1600 |
Faces | 2960 |
Edges | 2240 |
Vertices | 640 |
Vertex figure | ![]() |
Coxeter groups | B5, [3,3,3,4] |
Properties | convex |
Alternate names
- Steritruncated penteract
- Celliprismated triacontaditeron (Acronym: capt) (Jonathan Bowers)[2]
Construction and coordinates
The Cartesian coordinates of the vertices of a steritruncated 5-cube having edge length 2 are all permutations of:
- [math]\displaystyle{ \left(\pm1,\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2}),\ \pm(1+2\sqrt{2})\right) }[/math]
Images
Coxeter plane | B5 | B4 / D5 | B3 / D4 / A2 |
---|---|---|---|
Graph | ![]() |
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Dihedral symmetry | [10] | [8] | [6] |
Coxeter plane | B2 | A3 | |
Graph | ![]() |
![]() | |
Dihedral symmetry | [4] | [4] |
Stericantellated 5-cube
Stericantellated 5-cube | ||
Type | Uniform 5-polytope | |
Schläfli symbol | t0,2,4{4,3,3,3} | |
Coxeter-Dynkin diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4-faces | 242 | |
Cells | 2080 | |
Faces | 4720 | |
Edges | 3840 | |
Vertices | 960 | |
Vertex figure | ![]() | |
Coxeter group | B5 [4,3,3,3] | |
Properties | convex |
Alternate names
- Stericantellated penteract
- Stericantellated 5-orthoplex, stericantellated pentacross
- Cellirhombated penteractitriacontiditeron (Acronym: carnit) (Jonathan Bowers)[3]
Coordinates
The Cartesian coordinates of the vertices of a stericantellated 5-cube having edge length 2 are all permutations of:
- [math]\displaystyle{ \left(\pm1,\ \pm1,\ \pm1,\ \pm(1+\sqrt{2}),\ \pm(1+2\sqrt{2})\right) }[/math]
Images
Coxeter plane | B5 | B4 / D5 | B3 / D4 / A2 |
---|---|---|---|
Graph | ![]() |
![]() |
![]() |
Dihedral symmetry | [10] | [8] | [6] |
Coxeter plane | B2 | A3 | |
Graph | ![]() |
![]() | |
Dihedral symmetry | [4] | [4] |
Stericantitruncated 5-cube
Stericantitruncated 5-cube | ||
Type | Uniform 5-polytope | |
Schläfli symbol | t0,1,2,4{4,3,3,3} | |
Coxeter-Dynkin diagram |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4-faces | 242 | |
Cells | 2400 | |
Faces | 6000 | |
Edges | 5760 | |
Vertices | 1920 | |
Vertex figure | ![]() | |
Coxeter group | B5 [4,3,3,3] | |
Properties | convex, isogonal |
Alternate names
- Stericantitruncated penteract
- Steriruncicantellated triacontiditeron / Biruncicantitruncated pentacross
- Celligreatorhombated penteract (cogrin) (Jonathan Bowers)[4]
Coordinates
The Cartesian coordinates of the vertices of a stericantitruncated 5-cube having an edge length of 2 are given by all permutations of coordinates and sign of:
- [math]\displaystyle{ \left(1,\ 1+\sqrt{2},\ 1+2\sqrt{2},\ 1+2\sqrt{2},\ 1+3\sqrt{2}\right) }[/math]
Images
Coxeter plane | B5 | B4 / D5 | B3 / D4 / A2 |
---|---|---|---|
Graph | ![]() |
![]() |
![]() |
Dihedral symmetry | [10] | [8] | [6] |
Coxeter plane | B2 | A3 | |
Graph | ![]() |
![]() | |
Dihedral symmetry | [4] | [4] |
Steriruncitruncated 5-cube
Steriruncitruncated 5-cube | ||
Type | Uniform 5-polytope | |
Schläfli symbol | 2t2r{4,3,3,3} | |
Coxeter-Dynkin diagram |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4-faces | 242 | |
Cells | 2160 | |
Faces | 5760 | |
Edges | 5760 | |
Vertices | 1920 | |
Vertex figure | ![]() | |
Coxeter group | B5 [4,3,3,3] | |
Properties | convex, isogonal |
Alternate names
- Steriruncitruncated penteract / Steriruncitruncated 5-orthoplex / Steriruncitruncated pentacross
- Celliprismatotruncated penteractitriacontiditeron (captint) (Jonathan Bowers)[5]
Coordinates
The Cartesian coordinates of the vertices of a steriruncitruncated penteract having an edge length of 2 are given by all permutations of coordinates and sign of:
- [math]\displaystyle{ \left(1,\ 1+\sqrt{2},\ 1+1\sqrt{2},\ 1+2\sqrt{2},\ 1+3\sqrt{2}\right) }[/math]
Images
Coxeter plane | B5 | B4 / D5 | B3 / D4 / A2 |
---|---|---|---|
Graph | ![]() |
![]() |
![]() |
Dihedral symmetry | [10] | [8] | [6] |
Coxeter plane | B2 | A3 | |
Graph | ![]() |
![]() | |
Dihedral symmetry | [4] | [4] |
Steritruncated 5-orthoplex
Steritruncated 5-orthoplex | |
---|---|
Type | uniform 5-polytope |
Schläfli symbol | t0,1,4{3,3,3,4} |
Coxeter-Dynkin diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4-faces | 242 |
Cells | 1520 |
Faces | 2880 |
Edges | 2240 |
Vertices | 640 |
Vertex figure | ![]() |
Coxeter group | B5, [3,3,3,4] |
Properties | convex |
Alternate names
- Steritruncated pentacross
- Celliprismated penteract (Acronym: cappin) (Jonathan Bowers)[6]
Coordinates
Cartesian coordinates for the vertices of a steritruncated 5-orthoplex, centered at the origin, are all permutations of
- [math]\displaystyle{ \left(\pm1,\ \pm1,\ \pm1,\ \pm1,\ \pm(1+\sqrt{2})\right) }[/math]
Images
Coxeter plane | B5 | B4 / D5 | B3 / D4 / A2 |
---|---|---|---|
Graph | ![]() |
![]() |
![]() |
Dihedral symmetry | [10] | [8] | [6] |
Coxeter plane | B2 | A3 | |
Graph | ![]() |
![]() | |
Dihedral symmetry | [4] | [4] |
Stericantitruncated 5-orthoplex
Stericantitruncated 5-orthoplex | ||
Type | Uniform 5-polytope | |
Schläfli symbol | t0,2,3,4{4,3,3,3} | |
Coxeter-Dynkin diagram |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4-faces | 242 | |
Cells | 2320 | |
Faces | 5920 | |
Edges | 5760 | |
Vertices | 1920 | |
Vertex figure | ![]() | |
Coxeter group | B5 [4,3,3,3] | |
Properties | convex, isogonal |
Alternate names
- Stericantitruncated pentacross
- Celligreatorhombated triacontaditeron (cogart) (Jonathan Bowers)[7]
Coordinates
The Cartesian coordinates of the vertices of a stericantitruncated 5-orthoplex having an edge length of 2 are given by all permutations of coordinates and sign of:
- [math]\displaystyle{ \left(1,\ 1,\ 1+\sqrt{2},\ 1+2\sqrt{2},\ 1+3\sqrt{2}\right) }[/math]
Images
Coxeter plane | B5 | B4 / D5 | B3 / D4 / A2 |
---|---|---|---|
Graph | ![]() |
![]() |
![]() |
Dihedral symmetry | [10] | [8] | [6] |
Coxeter plane | B2 | A3 | |
Graph | ![]() |
![]() | |
Dihedral symmetry | [4] | [4] |
Omnitruncated 5-cube
Omnitruncated 5-cube | ||
Type | Uniform 5-polytope | |
Schläfli symbol | tr2r{4,3,3,3} | |
Coxeter-Dynkin diagram |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4-faces | 242 | |
Cells | 2640 | |
Faces | 8160 | |
Edges | 9600 | |
Vertices | 3840 | |
Vertex figure | ![]() irr. {3,3,3} | |
Coxeter group | B5 [4,3,3,3] | |
Properties | convex, isogonal |
Alternate names
- Steriruncicantitruncated 5-cube (Full expansion of omnitruncation for 5-polytopes by Johnson)
- Omnitruncated penteract
- Omnitruncated triacontiditeron / omnitruncated pentacross
- Great cellated penteractitriacontiditeron (Jonathan Bowers)[8]
Coordinates
The Cartesian coordinates of the vertices of an omnitruncated 5-cube having an edge length of 2 are given by all permutations of coordinates and sign of:
- [math]\displaystyle{ \left(1,\ 1+\sqrt{2},\ 1+2\sqrt{2},\ 1+3\sqrt{2},\ 1+4\sqrt{2}\right) }[/math]
Images
Coxeter plane | B5 | B4 / D5 | B3 / D4 / A2 |
---|---|---|---|
Graph | ![]() |
![]() |
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Dihedral symmetry | [10] | [8] | [6] |
Coxeter plane | B2 | A3 | |
Graph | ![]() |
![]() | |
Dihedral symmetry | [4] | [4] |
Full snub 5-cube
The full snub 5-cube or omnisnub 5-cube, defined as an alternation of the omnitruncated 5-cube is not uniform, but it can be given Coxeter diagram and symmetry [4,3,3,3]+, and constructed from 10 snub tesseracts, 32 snub 5-cells, 40 snub cubic antiprisms, 80 snub tetrahedral antiprisms, 80 3-4 duoantiprisms, and 1920 irregular 5-cells filling the gaps at the deleted vertices.
Related polytopes
This polytope is one of 31 uniform 5-polytopes generated from the regular 5-cube or 5-orthoplex.
Notes
References
- H.S.M. Coxeter:
- H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
- Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied 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]
- (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
- Norman Johnson Uniform Polytopes, Manuscript (1991)
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
- Klitzing, Richard. "5D uniform polytopes (polytera)". https://bendwavy.org/klitzing/dimensions/polytera.htm. x3o3o3o4x - scan, x3o3o3x4x - capt, x3o3x3o4x - carnit, x3o3x3x4x - cogrin, x3x3o3x4x - captint, x3x3x3x4x - gacnet, x3x3x3o4x - cogart
External links
- Glossary for hyperspace, George Olshevsky.
- Polytopes of Various Dimensions, Jonathan Bowers
- Multi-dimensional Glossary
Fundamental convex regular and uniform polytopes in dimensions 2–10
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Family | An | Bn | I2(p) / Dn | E6 / E7 / E8 / F4 / G2 | Hn | |||||||
Regular polygon | Triangle | Square | p-gon | Hexagon | Pentagon | |||||||
Uniform polyhedron | Tetrahedron | Octahedron • Cube | Demicube | Dodecahedron • Icosahedron | ||||||||
Uniform 4-polytope | 5-cell | 16-cell • Tesseract | Demitesseract | 24-cell | 120-cell • 600-cell | |||||||
Uniform 5-polytope | 5-simplex | 5-orthoplex • 5-cube | 5-demicube | |||||||||
Uniform 6-polytope | 6-simplex | 6-orthoplex • 6-cube | 6-demicube | 122 • 221 | ||||||||
Uniform 7-polytope | 7-simplex | 7-orthoplex • 7-cube | 7-demicube | 132 • 231 • 321 | ||||||||
Uniform 8-polytope | 8-simplex | 8-orthoplex • 8-cube | 8-demicube | 142 • 241 • 421 | ||||||||
Uniform 9-polytope | 9-simplex | 9-orthoplex • 9-cube | 9-demicube | |||||||||
Uniform 10-polytope | 10-simplex | 10-orthoplex • 10-cube | 10-demicube | |||||||||
Uniform n-polytope | n-simplex | n-orthoplex • n-cube | n-demicube | 1k2 • 2k1 • k21 | n-pentagonal polytope | |||||||
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds |
![]() | Original source: https://en.wikipedia.org/wiki/Stericated 5-cubes.
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