- Source: Stericated 5-cubes
In five-dimensional geometry, a 5.180.24.3/info/stericated" target="_blank">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 5.180.24.3/info/stericated" target="_blank">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 steriruncicantitruncated 5-cube, is more simply called an omnitruncated 5-cube with all of the nodes ringed.
5.180.24.3/info/stericated" target="_blank">Stericated 5-cube
= Alternate names
=5.180.24.3/info/stericated" target="_blank">Stericated penteract / 5.180.24.3/info/stericated" target="_blank">Stericated 5-orthoplex / 5.180.24.3/info/stericated" target="_blank">Stericated pentacross
Expanded penteract / Expanded 5-orthoplex / Expanded pentacross
Small cellated penteractitriacontaditeron (Acronym: scant) (Jonathan Bowers)
= Coordinates
=The Cartesian coordinates of the vertices of a 5.180.24.3/info/stericated" target="_blank">stericated 5-cube having edge length 2 are all permutations of:
(
±
1
,
±
1
,
±
1
,
±
1
,
±
(
1
+
2
)
)
{\displaystyle \left(\pm 1,\ \pm 1,\ \pm 1,\ \pm 1,\ \pm (1+{\sqrt {2}})\right)}
= Images
=The 5.180.24.3/info/stericated" target="_blank">stericated 5-cube is constructed by a sterication operation applied to the 5-cube.
= Dissections
=The 5.180.24.3/info/stericated" target="_blank">stericated 5-cube can be dissected into two tesseractic cupolae and a runcinated tesseract between them. This dissection can be seen as analogous to the 4D runcinated tesseract being dissected into two cubic cupolae and a central rhombicuboctahedral prism between them, and also the 3D rhombicuboctahedron being dissected into two square cupolae with a central octagonal prism between them.
Steritruncated 5-cube
= Alternate names
=Steritruncated penteract
Celliprismated triacontaditeron (Acronym: capt) (Jonathan Bowers)
= Construction and coordinates
=The Cartesian coordinates of the vertices of a steritruncated 5-cube having edge length 2 are all permutations of:
(
±
1
,
±
(
1
+
2
)
,
±
(
1
+
2
)
,
±
(
1
+
2
)
,
±
(
1
+
2
2
)
)
{\displaystyle \left(\pm 1,\ \pm (1+{\sqrt {2}}),\ \pm (1+{\sqrt {2}}),\ \pm (1+{\sqrt {2}}),\ \pm (1+2{\sqrt {2}})\right)}
= Images
=Stericantellated 5-cube
= Alternate names
=Stericantellated penteract
Stericantellated 5-orthoplex, stericantellated pentacross
Cellirhombated penteractitriacontiditeron (Acronym: carnit) (Jonathan Bowers)
= Coordinates
=The Cartesian coordinates of the vertices of a stericantellated 5-cube having edge length 2 are all permutations of:
(
±
1
,
±
1
,
±
1
,
±
(
1
+
2
)
,
±
(
1
+
2
2
)
)
{\displaystyle \left(\pm 1,\ \pm 1,\ \pm 1,\ \pm (1+{\sqrt {2}}),\ \pm (1+2{\sqrt {2}})\right)}
= Images
=Stericantitruncated 5-cube
= Alternate names
=Stericantitruncated penteract
Steriruncicantellated triacontiditeron / Biruncicantitruncated pentacross
Celligreatorhombated penteract (cogrin) (Jonathan Bowers)
= 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:
(
1
,
1
+
2
,
1
+
2
2
,
1
+
2
2
,
1
+
3
2
)
{\displaystyle \left(1,\ 1+{\sqrt {2}},\ 1+2{\sqrt {2}},\ 1+2{\sqrt {2}},\ 1+3{\sqrt {2}}\right)}
= Images
=Steriruncitruncated 5-cube
= Alternate names
=Steriruncitruncated penteract / Steriruncitruncated 5-orthoplex / Steriruncitruncated pentacross
Celliprismatotruncated penteractitriacontiditeron (captint) (Jonathan Bowers)
= 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:
(
1
,
1
+
2
,
1
+
1
2
,
1
+
2
2
,
1
+
3
2
)
{\displaystyle \left(1,\ 1+{\sqrt {2}},\ 1+1{\sqrt {2}},\ 1+2{\sqrt {2}},\ 1+3{\sqrt {2}}\right)}
= Images
=Steritruncated 5-orthoplex
= Alternate names
=Steritruncated pentacross
Celliprismated penteract (Acronym: cappin) (Jonathan Bowers)
= Coordinates
=Cartesian coordinates for the vertices of a steritruncated 5-orthoplex, centered at the origin, are all permutations of
(
±
1
,
±
1
,
±
1
,
±
1
,
±
(
1
+
2
)
)
{\displaystyle \left(\pm 1,\ \pm 1,\ \pm 1,\ \pm 1,\ \pm (1+{\sqrt {2}})\right)}
= Images
=Stericantitruncated 5-orthoplex
= Alternate names
=Stericantitruncated pentacross
Celligreatorhombated triacontaditeron (cogart) (Jonathan Bowers)
= 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:
(
1
,
1
,
1
+
2
,
1
+
2
2
,
1
+
3
2
)
{\displaystyle \left(1,\ 1,\ 1+{\sqrt {2}},\ 1+2{\sqrt {2}},\ 1+3{\sqrt {2}}\right)}
= Images
=Omnitruncated 5-cube
= 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)
= 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:
(
1
,
1
+
2
,
1
+
2
2
,
1
+
3
2
,
1
+
4
2
)
{\displaystyle \left(1,\ 1+{\sqrt {2}},\ 1+2{\sqrt {2}},\ 1+3{\sqrt {2}},\ 1+4{\sqrt {2}}\right)}
= Images
== 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)". 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
Kata Kunci Pencarian:
- Daftar bentuk matematika
- Stericated 5-cubes
- Steric 5-cubes
- Stericated 6-cubes
- Stericated 5-simplexes
- Stericated 7-cubes
- List of polygons, polyhedra and polytopes
- List of mathematical shapes
- Octagon
- Pentic 6-cubes
- Five-dimensional space
