Pentellated 7-cubes#Pentiruncicantellated 7-cube

In seven-dimensional geometry, a pentellated 7-cube is a convex uniform 7-polytope with 5th order truncations (pentellation) of the regular 7-cube. There are 32 unique {{not a typo|pentellations}} of the 7-cube with permutations of truncations, cantellations, runcinations, and {{not a typo|sterications}}. 16 are more simply constructed relative to the 7-orthoplex.

class=wikitable style="float:right; margin-left:8px; width:480px"
style="text-align:center;"

|120px
7-cube
{{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node}}

|120px
Pentellated 7-cube
{{CDD|node_1|4|node|3|node|3|node|3|node|3|node_1|3|node}}

|120px
Pentitruncated 7-cube
{{CDD|node_1|4|node_1|3|node|3|node|3|node|3|node_1|3|node}}

|120px
Penticantellated 7-cube
{{CDD|node_1|4|node|3|node_1|3|node|3|node|3|node_1|3|node}}

style="text-align:center;"

|120px
Penticantitruncated 7-cube
{{CDD|node_1|4|node_1|3|node_1|3|node|3|node|3|node_1|3|node}}

|120px
Pentiruncinated 7-cube
{{CDD|node_1|4|node|3|node|3|node_1|3|node|3|node_1|3|node}}

|120px
Pentiruncitruncated 7-cube
{{CDD|node_1|4|node_1|3|node|3|node_1|3|node|3|node_1|3|node}}

|120px
Pentiruncicantellated 7-cube
{{CDD|node_1|4|node|3|node_1|3|node_1|3|node|3|node_1|3|node}}

style="text-align:center;"

|120px
Pentiruncicantitruncated 7-cube
{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node|3|node_1|3|node}}

|120px
Pentistericated 7-cube
{{CDD|node_1|4|node|3|node|3|node|3|node_1|3|node_1|3|node}}

|120px
Pentisteritruncated 7-cube
{{CDD|node_1|4|node_1|3|node|3|node|3|node_1|3|node_1|3|node}}

|120px
Pentistericantellated 7-cube
{{CDD|node_1|4|node|3|node_1|3|node|3|node_1|3|node_1|3|node}}

style="text-align:center;"

|120px
Pentistericantitruncated 7-cube
{{CDD|node_1|4|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node}}

|120px
Pentisteriruncinated 7-cube
{{CDD|node_1|4|node|3|node|3|node_1|3|node_1|3|node_1|3|node}}

|120px
Pentisteriruncitruncated 7-cube
{{CDD|node_1|4|node_1|3|node|3|node_1|3|node_1|3|node_1|3|node}}

|120px
Pentisteriruncicantellated 7-cube
{{CDD|node_1|4|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node}}

style="text-align:center;"

|120px
Pentisteriruncicantitruncated 7-cube
{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node}}

{{-}}

Pentellated 7-cube

class="wikitable" style="margin-left:10px; float:right; width:250px;"

! style="background:#e7dcc3;" colspan="2"|Pentellated 7-cube

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,5{4,35}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|4|node|3|node|3|node|3|node|3|node_1|3|node}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges
style="background:#e7dcc3;"|Vertices
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsB7, [4,35]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Small {{not a typo|terated}} hepteract (acronym: stesa) (Jonathan Bowers)Klitzing, (x4o3o3o3o3x3o - stesa)

= Images =

{{B7 Coxeter plane graphs|t05|150}}

Pentitruncated 7-cube

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! style="background:#e7dcc3;" colspan="2"|pentitruncated 7-cube

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,1,5{4,35}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|4|node_1|3|node|3|node|3|node|3|node_1|3|node}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges
style="background:#e7dcc3;"|Vertices
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsB7, [4,35]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Teritruncated hepteract (acronym: tetsa) (Jonathan Bowers)Klitzing, (x4x3o3o3o3x3o - tetsa)

= Images =

{{B7 Coxeter plane graphs|t015|150}}

Penticantellated 7-cube

class="wikitable" style="margin-left:10px; float:right; width:250px;"

! style="background:#e7dcc3;" colspan="2"|Penticantellated 7-cube

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,2,5{4,35}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|4|node|3|node_1|3|node|3|node|3|node_1|3|node}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges
style="background:#e7dcc3;"|Vertices
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsB7, [4,35]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Terirhombated hepteract (acronym: tersa) (Jonathan Bowers)Klitzing, (x4o3x3o3o3x3o - tersa)

= Images =

{{B7 Coxeter plane graphs|t025|150}}

Penticantitruncated 7-cube

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! style="background:#e7dcc3;" colspan="2"|penticantitruncated 7-cube

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,1,2,5{4,35}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|4|node_1|3|node_1|3|node|3|node|3|node_1|3|node}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges
style="background:#e7dcc3;"|Vertices
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsB7, [4,35]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Terigreatorhombated hepteract (acronym: togresa) (Jonathan Bowers)Klitzing, (x4x3x3o3o3x3o - togresa)

= Images =

{{B7 Coxeter plane graphs|t0125|150}}

Pentiruncinated 7-cube

class="wikitable" style="margin-left:10px; float:right; width:250px;"

! style="background:#e7dcc3;" colspan="2"|pentiruncinated 7-cube

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,3,5{4,35}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|4|node|3|node|3|node_1|3|node|3|node_1|3|node}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges
style="background:#e7dcc3;"|Vertices
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsB7, [4,35]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Teriprismated hepteract (acronym: tapsa) (Jonathan Bowers)Klitzing, (x4o3o3x3o3x3o - tapsa)

= Images =

{{B7 Coxeter plane graphs|t035|150}}

Pentiruncitruncated 7-cube

class="wikitable" style="margin-left:10px; float:right; width:250px;"

! style="background:#e7dcc3;" colspan="2"|pentiruncitruncated 7-cube

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,1,3,5{4,35}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|4|node_1|3|node|3|node_1|3|node|3|node_1|3|node}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges
style="background:#e7dcc3;"|Vertices
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsB7, [4,35]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Teriprismatotruncated hepteract (acronym: toptasa) (Jonathan Bowers)Klitzing, (x4x3o3x3o3x3o - toptasa)

= Images =

{{B7 Coxeter plane graphs|t0135|150}}

Pentiruncicantellated 7-cube

class="wikitable" style="margin-left:10px; float:right; width:250px;"

! style="background:#e7dcc3;" colspan="2"|pentiruncicantellated 7-cube

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,2,3,5{4,35}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|4|node|3|node_1|3|node_1|3|node|3|node_1|3|node}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges
style="background:#e7dcc3;"|Vertices
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsB7, [4,35]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Teriprismatorhombated hepteract (acronym: topresa) (Jonathan Bowers)Klitzing, (x4o3x3x3o3x3o - topresa)

= Images =

{{B7 Coxeter plane graphs|t0235|180}}

Pentiruncicantitruncated 7-cube

class="wikitable" style="margin-left:10px; float:right; width:250px;"

! style="background:#e7dcc3;" colspan="2"|pentiruncicantitruncated 7-cube

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,1,2,3,5{4,35}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node|3|node_1|3|node}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges
style="background:#e7dcc3;"|Vertices
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsB7, [4,35]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Terigreatoprismated hepteract (acronym: togapsa) (Jonathan Bowers)Klitzing, (x4x3x3x3o3x3o - togapsa)

= Images =

{{B7 Coxeter plane graphs|t01235|150|NOB7A6|NOA5orA3}}

Pentistericated 7-cube

class="wikitable" style="margin-left:10px; float:right; width:250px;"

! style="background:#e7dcc3;" colspan="2"|pentistericated 7-cube

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,4,5{4,35}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|4|node|3|node|3|node|3|node_1|3|node_1|3|node}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges
style="background:#e7dcc3;"|Vertices
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsB7, [4,35]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Tericellated hepteract (acronym: tacosa) (Jonathan Bowers)Klitzing, (x4o3o3o3x3x3o - tacosa)

= Images =

{{B7 Coxeter plane graphs|t045|150}}

Pentisteritruncated 7-cube

class="wikitable" style="margin-left:10px; float:right; width:250px;"

! style="background:#e7dcc3;" colspan="2"|pentisteritruncated 7-cube

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,1,4,5{4,35}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|4|node_1|3|node|3|node|3|node_1|3|node_1|3|node}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges
style="background:#e7dcc3;"|Vertices
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsB7, [4,35]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Tericellitruncated hepteract (acronym: tecatsa) (Jonathan Bowers)Klitzing, (x4x3o3o3x3x3o - tecatsa)

= Images =

{{B7 Coxeter plane graphs|t0145|150}}

Pentistericantellated 7-cube

class="wikitable" style="margin-left:10px; float:right; width:250px;"

! style="background:#e7dcc3;" colspan="2"|pentistericantellated 7-cube

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,2,4,5{4,35}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|4|node|3|node_1|3|node|3|node_1|3|node_1|3|node}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges
style="background:#e7dcc3;"|Vertices
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsB7, [4,35]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Tericellirhombated hepteract (acronym: tecresa) (Jonathan Bowers)Klitzing, (x4o3x3o3x3x3o - tecresa)

= Images =

{{B7 Coxeter plane graphs|t0245|150}}

Pentistericantitruncated 7-cube

class="wikitable" style="margin-left:10px; float:right; width:250px;"

! style="background:#e7dcc3;" colspan="2"|pentistericantitruncated 7-cube

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,1,2,4,5{4,35}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|4|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges
style="background:#e7dcc3;"|Vertices
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsB7, [4,35]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Tericelligreatorhombated hepteract (acronym: tecgresa) (Jonathan Bowers)Klitzing, (x4x3x3o3x3x3o - tecgresa)

= Images =

{{B7 Coxeter plane graphs|t01245|150|NOB7A6}}

Pentisteriruncinated 7-cube

class="wikitable" style="margin-left:10px; float:right; width:250px;"

! style="background:#e7dcc3;" colspan="2"|Pentisteriruncinated 7-cube

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,3,4,5{4,35}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|4|node|3|node|3|node_1|3|node_1|3|node_1|3|node}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges
style="background:#e7dcc3;"|Vertices
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsB7, [4,35]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Bipenticantitruncated 7-cube as t1,2,3,6{4,35}
  • Tericelliprismated hepteract (acronym: tecpasa) (Jonathan Bowers)Klitzing, (x4o3o3x3x3x3o - tecpasa)

= Images =

{{B7 Coxeter plane graphs|t0345|150}}

Pentisteriruncitruncated 7-cube

class="wikitable" style="margin-left:10px; float:right; width:250px;"

! style="background:#e7dcc3;" colspan="2"|pentisteriruncitruncated 7-cube

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,1,3,4,5{4,35}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|4|node_1|3|node|3|node_1|3|node_1|3|node_1|3|node}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges40320
style="background:#e7dcc3;"|Vertices10080
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsB7, [4,35]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Tericelliprismatotruncated hepteract (acronym: tecpetsa) (Jonathan Bowers)Klitzing, (x4x3o3x3x3x3o - tecpetsa)

= Images =

{{B7 Coxeter plane graphs|t01345|150|NOB7A6}}

Pentisteriruncicantellated 7-cube

class="wikitable" style="margin-left:10px; float:right; width:250px;"

! style="background:#e7dcc3;" colspan="2"|pentisteriruncicantellated 7-cube

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,2,3,4,5{4,35}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|4|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges40320
style="background:#e7dcc3;"|Vertices10080
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsB7, [4,35]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Bipentiruncicantitruncated 7-cube as t1,2,3,4,6{4,35}
  • Tericelliprismatorhombated hepteract (acronym: tocpresa) (Jonathan Bowers)Klitzing, (x4o3x3x3x3x3o - tocpresa)

= Images =

{{B7 Coxeter plane graphs|t02345|150|NOB7A6}}

Pentisteriruncicantitruncated 7-cube

class="wikitable" style="margin-left:10px; float:right; width:250px;"

! style="background:#e7dcc3;" colspan="2"|pentisteriruncicantitruncated 7-cube

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,1,2,3,4,5{4,35}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges
style="background:#e7dcc3;"|Vertices
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsB7, [4,35]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Great {{not a typo|terated}} hepteract (acronym: gotesa) (Jonathan Bowers)Klitzing, (x4x3x3x3x3x3o - gotesa)

= Images =

{{B7 Coxeter plane graphs|t012345|150|NOB7A6}}

Related polytopes

These polytopes are a part of a set of 127 uniform 7-polytopes with B7 symmetry.

Notes

{{reflist}}

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, [https://www.wiley.com/en-us/Kaleidoscopes-p-9780471010036 wiley.com], {{isbn|978-0-471-01003-6}}
  • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10]
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  • {{KlitzingPolytopes|polyexa.htm|7D uniform polytopes (polyexa) with acronyms}}