Cantellated 5-cubes
In six-dimensional geometry , a cantellated 5-cube is a convex uniform 5-polytope , being a cantellation of the regular 5-cube .
There are 6 unique cantellation for the 5-cube, including truncations. Half of them are more easily constructed from the dual 5-orthoplex
Cantellated 5-cube
Cantellated 5-cube
Type
Uniform 5-polytope
Schläfli symbol
rr{4,3,3,3} =
r
{
4
3
,
3
,
3
}
{\displaystyle r\left\{{\begin{array}{l}4\\3,3,3\end{array}}\right\}}
Coxeter-Dynkin diagram
4-faces
122
10
Cells
680
40
Faces
1520
80
Edges
1280
320+960
Vertices
320
Vertex figure
Coxeter group
B5 [4,3,3,3]
Properties
convex , uniform
Alternate names
Small rhombated penteract (Acronym: sirn) (Jonathan Bowers)
Coordinates
The Cartesian coordinates of the vertices of a cantellated 5-cube having edge length 2 are all permutations of:
(
± ± -->
1
,
± ± -->
1
,
± ± -->
(
1
+
2
)
,
± ± -->
(
1
+
2
)
,
± ± -->
(
1
+
2
)
)
{\displaystyle \left(\pm 1,\ \pm 1,\ \pm (1+{\sqrt {2}}),\ \pm (1+{\sqrt {2}}),\ \pm (1+{\sqrt {2}})\right)}
Images
Bicantellated 5-cube
Bicantellated 5-cube
Type
Uniform 5-polytope
Schläfli symbols
2rr{4,3,3,3} =
r
{
3
,
4
3
,
3
}
{\displaystyle r\left\{{\begin{array}{l}3,4\\3,3\end{array}}\right\}}
2,1,1 } =
r
{
3
,
3
3
3
}
{\displaystyle r\left\{{\begin{array}{l}3,3\\3\\3\end{array}}\right\}}
Coxeter-Dynkin diagrams
4-faces
122
10
Cells
840
40
Faces
2160
240
Edges
1920
960+960
Vertices
480
Vertex figure
Coxeter groups
B5 , [3,3,3,4]5 , [32,1,1 ]
Properties
convex , uniform
In five-dimensional geometry , a bicantellated 5-cube is a uniform 5-polytope .
Alternate names
Bicantellated penteract, bicantellated 5-orthoplex, or bicantellated pentacross
Small birhombated penteractitriacontiditeron (Acronym: sibrant) (Jonathan Bowers)
Coordinates
The Cartesian coordinates of the vertices of a bicantellated 5-cube having edge length 2 are all permutations of:
(0,1,1,2,2)
Images
Cantitruncated 5-cube
Cantitruncated 5-cube
Type
Uniform 5-polytope
Schläfli symbol
tr{4,3,3,3} =
t
{
4
3
,
3
,
3
}
{\displaystyle t\left\{{\begin{array}{l}4\\3,3,3\end{array}}\right\}}
Coxeter-Dynkin
4-faces
122
10
Cells
680
40
Faces
1520
80
Edges
1600
320+320+960
Vertices
640
Vertex figure
Coxeter group
B5 [4,3,3,3]
Properties
convex , uniform
Alternate names
Tricantitruncated 5-orthoplex / tricantitruncated pentacross
Great rhombated penteract (girn) (Jonathan Bowers)
Coordinates
The Cartesian coordinates of the vertices of a cantitruncated 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
+
2
2
)
{\displaystyle \left(1,\ 1+{\sqrt {2}},\ 1+2{\sqrt {2}},\ 1+2{\sqrt {2}},\ 1+2{\sqrt {2}}\right)}
Images
It is third in a series of cantitruncated hypercubes:
Bicantitruncated 5-cube
Bicantitruncated 5-cube
Type
uniform 5-polytope
Schläfli symbol
2tr{3,3,3,4} =
t
{
3
,
4
3
,
3
}
{\displaystyle t\left\{{\begin{array}{l}3,4\\3,3\end{array}}\right\}}
2,1,1 } =
t
{
3
,
3
3
3
}
{\displaystyle t\left\{{\begin{array}{l}3,3\\3\\3\end{array}}\right\}}
Coxeter-Dynkin diagrams
4-faces
122
10
Cells
840
40
Faces
2160
240
Edges
2400
960+480+960
Vertices
960
Vertex figure
Coxeter groups
B5 , [3,3,3,4]5 , [32,1,1 ]
Properties
convex , uniform
Alternate names
Bicantitruncated penteract
Bicantitruncated pentacross
Great birhombated penteractitriacontiditeron (Acronym: gibrant) (Jonathan Bowers)
Coordinates
Cartesian coordinates for the vertices of a bicantitruncated 5-cube, centered at the origin, are all sign and coordinate permutations of
(±3,±3,±2,±1,0)
Images
These polytopes are from a set of 31 uniform 5-polytopes generated from the regular 5-cube or 5-orthoplex .
B5 polytopes
β5
t1 β5
t2 γ5
t1 γ5
γ5
t0,1 β5
t0,2 β5
t1,2 β5
t0,3 β5
t1,3 γ5
t1,2 γ5
t0,4 γ5
t0,3 γ5
t0,2 γ5
t0,1 γ5
t0,1,2 β5
t0,1,3 β5
t0,2,3 β5
t1,2,3 γ5
t0,1,4 β5
t0,2,4 γ5
t0,2,3 γ5
t0,1,4 γ5
t0,1,3 γ5
t0,1,2 γ5
t0,1,2,3 β5
t0,1,2,4 β5
t0,1,3,4 γ5
t0,1,2,4 γ5
t0,1,2,3 γ5
t0,1,2,3,4 γ5
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)" .
External links
