In six-dimensional geometry , a pentellated 6-cube is a convex uniform 6-polytope with 5th order truncations of the regular 6-cube .
There are unique 16 degrees of pentellations of the 6-cube with permutations of truncations, cantellations, runcinations, and sterications. The simple pentellated 6-cube is also called an expanded 6-cube , constructed by an expansion operation applied to the regular 6-cube . The highest form, the pentisteriruncicantitruncated 6-cube , is called an omnitruncated 6-cube with all of the nodes ringed. Six of them are better constructed from the 6-orthoplex given at pentellated 6-orthoplex .
Pentellated 6-cube
Pentellated 6-cube
Type
Uniform 6-polytope
Schläfli symbol t0,5 {4,3,3,3,3}
Coxeter-Dynkin diagram File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png
5-faces
4-faces
Cells
Faces
Edges
1920
Vertices
384
Vertex figure 5-cell antiprism
Coxeter group B6 , [4,3,3,3,3]
Properties
convex
Alternate names Pentellated 6-orthoplex
Expanded 6-cube, expanded 6-orthoplex
Small teri-hexeractihexacontitetrapeton (Acronym: stoxog) (Jonathan Bowers)[ 1] Images Pentitruncated 6-cube
Pentitruncated 6-cube
Type
uniform 6-polytope
Schläfli symbol t0,1,5 {4,3,3,3,3}
Coxeter-Dynkin diagrams File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png
5-faces
4-faces
Cells
Faces
Edges
8640
Vertices
1920
Vertex figure
Coxeter groups B6 , [4,3,3,3,3]
Properties
convex
Alternate names Teritruncated hexeract (Acronym: tacog) (Jonathan Bowers)[ 2] Images Penticantellated 6-cube
Penticantellated 6-cube
Type
uniform 6-polytope
Schläfli symbol t0,2,5 {4,3,3,3,3}
Coxeter-Dynkin diagrams File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png
5-faces
4-faces
Cells
Faces
Edges
21120
Vertices
3840
Vertex figure
Coxeter groups B6 , [4,3,3,3,3]
Properties
convex
Alternate names Terirhombated hexeract (Acronym: topag) (Jonathan Bowers)[ 3] Images Penticantitruncated 6-cube
Penticantitruncated 6-cube
Type
uniform 6-polytope
Schläfli symbol t0,1,2,5 {4,3,3,3,3}
Coxeter-Dynkin diagrams File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png
5-faces
4-faces
Cells
Faces
Edges
30720
Vertices
7680
Vertex figure
Coxeter groups B6 , [4,3,3,3,3]
Properties
convex
Alternate names Terigreatorhombated hexeract (Acronym: togrix) (Jonathan Bowers)[ 4] Images Pentiruncitruncated 6-cube
Pentiruncitruncated 6-cube
Type
uniform 6-polytope
Schläfli symbol t0,1,3,5 {4,3,3,3,3}
Coxeter-Dynkin diagrams File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png
5-faces
4-faces
Cells
Faces
Edges
151840
Vertices
11520
Vertex figure
Coxeter groups B6 , [4,3,3,3,3]
Properties
convex
Alternate names Tericellirhombated hexacontitetrapeton (Acronym: tocrag) (Jonathan Bowers)[ 5] Images Pentiruncicantellated 6-cube
Pentiruncicantellated 6-cube
Type
uniform 6-polytope
Schläfli symbol t0,2,3,5 {4,3,3,3,3}
Coxeter-Dynkin diagrams File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png
5-faces
4-faces
Cells
Faces
Edges
46080
Vertices
11520
Vertex figure
Coxeter groups B6 , [4,3,3,3,3]
Properties
convex
Alternate names Teriprismatorhombi-hexeractihexacontitetrapeton (Acronym: tiprixog) (Jonathan Bowers)[ 6] Images Pentiruncicantitruncated 6-cube
Pentiruncicantitruncated 6-cube
Type
uniform 6-polytope
Schläfli symbol t0,1,2,3,5 {4,3,3,3,3}
Coxeter-Dynkin diagrams File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png
5-faces
4-faces
Cells
Faces
Edges
80640
Vertices
23040
Vertex figure
Coxeter groups B6 , [4,3,3,3,3]
Properties
convex
Alternate names Terigreatoprismated hexeract (Acronym: tagpox) (Jonathan Bowers)[ 7] Images Pentisteritruncated 6-cube
Pentisteritruncated 6-cube
Type
uniform 6-polytope
Schläfli symbol t0,1,4,5 {4,3,3,3,3}
Coxeter-Dynkin diagrams File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png
5-faces
4-faces
Cells
Faces
Edges
30720
Vertices
7680
Vertex figure
Coxeter groups B6 , [4,3,3,3,3]
Properties
convex
Alternate names Tericellitrunki-hexeractihexacontitetrapeton (Acronym: tactaxog) (Jonathan Bowers)[ 8] Images Pentistericantitruncated 6-cube
Pentistericantitruncated 6-cube
Type
uniform 6-polytope
Schläfli symbol t0,1,2,4,5 {4,3,3,3,3}
Coxeter-Dynkin diagrams File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png
5-faces
4-faces
Cells
Faces
Edges
80640
Vertices
23040
Vertex figure
Coxeter groups B6 , [4,3,3,3,3]
Properties
convex
Alternate names Tericelligreatorhombated hexeract (Acronym: tocagrax) (Jonathan Bowers)[ 9] Images Omnitruncated 6-cube
Omnitruncated 6-cube
Type
Uniform 6-polytope
Schläfli symbol t0,1,2,3,4,5 {35 }
Coxeter-Dynkin diagrams File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png
5-faces
728:t0,1,2,3,4 {3,3,3,4} File:5-cube t01234.svg 0,1,2,3 {3,3,4} File:Complete graph K2.svg File:4-cube t0123.svg 0,1,2 {3,4} File:2-simplex t01.svg File:3-cube t012 B2.svg 0,1,2 {3,3} File:Regular polygon 8 annotated.svg File:3-simplex t012.svg 0,1,2,3 {33 } File:Complete graph K2.svg File:4-simplex t0123.svg t0,1,2,3,4 {34 } File:5-simplex t01234.svg
4-faces
14168
Cells
72960
Faces
151680
Edges
138240
Vertices
46080
Vertex figure irregular 5-simplex
Coxeter group B6 , [4,3,3,3,3]
Properties
convex , isogonal
The omnitruncated 6-cube has 5040 vertices , 15120 edges , 16800 faces (4200 hexagons and 1260 squares ), 8400 cells , 1806 4-faces, and 126 5-faces. With 5040 vertices, it is the largest of 35 uniform 6-polytopes generated from the regular 6-cube .
Alternate names Pentisteriruncicantitruncated 6-cube or 6-orthoplex (omnitruncation for 6-polytopes)
Omnitruncated hexeract
Great teri-hexeractihexacontitetrapeton (Acronym: gotaxog) (Jonathan Bowers)[ 10] Images Full snub 6-cube The full snub 6-cube or omnisnub 6-cube , defined as an alternation of the omnitruncated 6-cube is not uniform, but it can be given Coxeter diagram File:CDel node h.png File:CDel 4.png File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 3.png File:CDel node h.png symmetry [4,3,3,3,3]+ , and constructed from 12 snub 5-cubes , 64 snub 5-simplexes , 60 snub tesseract antiprisms, 192 snub 5-cell antiprisms, 160 3-sr{4,3} duoantiprisms, 240 4-s{3,4} duoantiprisms, and 23040 irregular 5-simplexes filling the gaps at the deleted vertices.
Related polytopes These polytopes are from a set of 63 uniform 6-polytopes generated from the B6 Coxeter plane , including the regular 6-cube or 6-orthoplex .
B6 polytopes
File:6-cube t5.svg β6
File:6-cube t4.svg t1 β6
File:6-cube t3.svg t2 β6
File:6-cube t2.svg t2 γ6
File:6-cube t1.svg t1 γ6
File:6-cube t0.svg γ6
File:6-cube t45.svg t0,1 β6
File:6-cube t35.svg t0,2 β6
File:6-cube t34.svg t1,2 β6
File:6-cube t25.svg t0,3 β6
File:6-cube t24.svg t1,3 β6
File:6-cube t23.svg t2,3 γ6
File:6-cube t15.svg t0,4 β6
File:6-cube t14.svg t1,4 γ6
File:6-cube t13.svg t1,3 γ6
File:6-cube t12.svg t1,2 γ6
File:6-cube t05.svg t0,5 γ6
File:6-cube t04.svg t0,4 γ6
File:6-cube t03.svg t0,3 γ6
File:6-cube t02.svg t0,2 γ6
File:6-cube t01.svg t0,1 γ6
File:6-cube t345.svg t0,1,2 β6
File:6-cube t245.svg t0,1,3 β6
File:6-cube t235.svg t0,2,3 β6
File:6-cube t234.svg t1,2,3 β6
File:6-cube t145.svg t0,1,4 β6
File:6-cube t135.svg t0,2,4 β6
File:6-cube t134.svg t1,2,4 β6
File:6-cube t125.svg t0,3,4 β6
File:6-cube t124.svg t1,2,4 γ6
File:6-cube t123.svg t1,2,3 γ6
File:6-cube t045.svg t0,1,5 β6
File:6-cube t035.svg t0,2,5 β6
File:6-cube t034.svg t0,3,4 γ6
File:6-cube t025.svg t0,2,5 γ6
File:6-cube t024.svg t0,2,4 γ6
File:6-cube t023.svg t0,2,3 γ6
File:6-cube t015.svg t0,1,5 γ6
File:6-cube t014.svg t0,1,4 γ6
File:6-cube t013.svg t0,1,3 γ6
File:6-cube t012.svg t0,1,2 γ6
File:6-cube t2345.svg t0,1,2,3 β6
File:6-cube t1345.svg t0,1,2,4 β6
File:6-cube t1245.svg t0,1,3,4 β6
File:6-cube t1235.svg t0,2,3,4 β6
File:6-cube t1234.svg t1,2,3,4 γ6
File:6-cube t0345.svg t0,1,2,5 β6
File:6-cube t0245.svg t0,1,3,5 β6
File:6-cube t0235.svg t0,2,3,5 γ6
File:6-cube t0234.svg t0,2,3,4 γ6
File:6-cube t0145.svg t0,1,4,5 γ6
File:6-cube t0135.svg t0,1,3,5 γ6
File:6-cube t0134.svg t0,1,3,4 γ6
File:6-cube t0125.svg t0,1,2,5 γ6
File:6-cube t0124.svg t0,1,2,4 γ6
File:6-cube t0123.svg t0,1,2,3 γ6
File:6-cube t12345.svg t0,1,2,3,4 β6
File:6-cube t02345.svg t0,1,2,3,5 β6
File:6-cube t01345.svg t0,1,2,4,5 β6
File:6-cube t01245.svg t0,1,2,4,5 γ6
File:6-cube t01235.svg t0,1,2,3,5 γ6
File:6-cube t01234.svg t0,1,2,3,4 γ6
File:6-cube t012345.svg t0,1,2,3,4,5 γ6
Notes
↑ Klitzing, (x4o3o3o3o3x - stoxog)
↑ Klitzing, (x4x3o3o3o3x - tacog)
↑ Klitzing, (x4o3x3o3o3x - topag)
↑ Klitzing, (x4x3x3o3o3x - togrix)
↑ Klitzing, (x4x3o3x3o3x - tocrag)
↑ Klitzing, (x4o3x3x3o3x - tiprixog)
↑ Klitzing, (x4x3x3o3x3x - tagpox)
↑ Klitzing, (x4x3o3o3x3x - tactaxog)
↑ Klitzing, (x4x3x3o3x3x - tocagrax)
↑ Klitzing, (x4x3x3x3x3x - gotaxog)
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 , 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 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. "6D uniform polytopes (polypeta)" . External links
