Pentellated 6-orthoplexes
In six-dimensional geometry , a pentellated 6-orthoplex is a convex uniform 6-polytope with 5th order truncations of the regular 6-orthoplex .
There are unique 16 degrees of pentellations of the 6-orthoplex with permutations of truncations, cantellations, runcinations, and sterications. Ten are shown, with the other 6 more easily constructed as a pentellated 6-cube . The simple pentellated 6-orthoplex (Same as pentellated 5-cube) is also called an expanded 6-orthoplex , constructed by an expansion operation applied to the regular 6-orthoplex . The highest form, the pentisteriruncicantitruncated 6-orthoplex , is called an omnitruncated 6-orthoplex with all of the nodes ringed.
Pentitruncated 6-orthoplex
Pentitruncated 6-orthoplex
Type
uniform 6-polytope
Schläfli symbol t0,1,5 {3,3,3,3,4}
Coxeter-Dynkin diagrams
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 hexacontatetrapeton (Acronym: tacox) (Jonathan Bowers)[ 1]
Images
Penticantellated 6-orthoplex
Penticantellated 6-orthoplex
Type
uniform 6-polytope
Schläfli symbol t0,2,5 {3,3,3,3,4}
Coxeter-Dynkin diagrams
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 hexacontitetrapeton (Acronym: tapox) (Jonathan Bowers)[ 2]
Images
Penticantitruncated 6-orthoplex
Penticantitruncated 6-orthoplex
Type
uniform 6-polytope
Schläfli symbol t0,1,2,5 {3,3,3,3,4}
Coxeter-Dynkin diagrams
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 hexacontitetrapeton (Acronym: togrig) (Jonathan Bowers)[ 3]
Images
Pentiruncitruncated 6-orthoplex
Pentiruncitruncated 6-orthoplex
Type
uniform 6-polytope
Schläfli symbol t0,1,3,5 {3,3,3,3,4}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges
51840
Vertices
11520
Vertex figure
Coxeter groups B6 , [4,3,3,3,3]
Properties
convex
Alternate names
Teriprismatotruncated hexacontitetrapeton (Acronym: tocrax) (Jonathan Bowers)[ 4]
Images
Pentiruncicantitruncated 6-orthoplex
Pentiruncicantitruncated 6-orthoplex
Type
uniform 6-polytope
Schläfli symbol t0,1,2,3,5 {3,3,3,3,4}
Coxeter-Dynkin diagrams
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 hexacontitetrapeton (Acronym: tagpog) (Jonathan Bowers)[ 5]
Images
Pentistericantitruncated 6-orthoplex
Pentistericantitruncated 6-orthoplex
Type
uniform 6-polytope
Schläfli symbol t0,1,2,4,5 {3,3,3,3,4}
Coxeter-Dynkin diagrams
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 hexacontitetrapeton (Acronym: tecagorg) (Jonathan Bowers)[ 6]
Images
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
β6
t1 β6
t2 β6
t2 γ6
t1 γ6
γ6
t0,1 β6
t0,2 β6
t1,2 β6
t0,3 β6
t1,3 β6
t2,3 γ6
t0,4 β6
t1,4 γ6
t1,3 γ6
t1,2 γ6
t0,5 γ6
t0,4 γ6
t0,3 γ6
t0,2 γ6
t0,1 γ6
t0,1,2 β6
t0,1,3 β6
t0,2,3 β6
t1,2,3 β6
t0,1,4 β6
t0,2,4 β6
t1,2,4 β6
t0,3,4 β6
t1,2,4 γ6
t1,2,3 γ6
t0,1,5 β6
t0,2,5 β6
t0,3,4 γ6
t0,2,5 γ6
t0,2,4 γ6
t0,2,3 γ6
t0,1,5 γ6
t0,1,4 γ6
t0,1,3 γ6
t0,1,2 γ6
t0,1,2,3 β6
t0,1,2,4 β6
t0,1,3,4 β6
t0,2,3,4 β6
t1,2,3,4 γ6
t0,1,2,5 β6
t0,1,3,5 β6
t0,2,3,5 γ6
t0,2,3,4 γ6
t0,1,4,5 γ6
t0,1,3,5 γ6
t0,1,3,4 γ6
t0,1,2,5 γ6
t0,1,2,4 γ6
t0,1,2,3 γ6
t0,1,2,3,4 β6
t0,1,2,3,5 β6
t0,1,2,4,5 β6
t0,1,2,4,5 γ6
t0,1,2,3,5 γ6
t0,1,2,3,4 γ6
t0,1,2,3,4,5 γ6
Notes
^ Klitzing, (x4o3o3o3x3x - tacox)
^ Klitzing, (x4o3o3x3o3x - tapox)
^ Klitzing, (x4o3o3x3x3x - togrig)
^ Klitzing, (x4o3x3o3x3x - tocrax)
^ Klitzing, (x4x3o3x3x3x - tagpog)
^ Klitzing, (x4x3o3x3x3x - tecagorg)
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
