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Hexicated 7-orthoplexes#Hexiruncitruncated 7-orthoplex

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!colspan=5|Orthogonal projections in B4 Coxeter plane

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

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

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Hexi-truncated 7-orthoplex
{{CDD|node_1|3|node_1|3|node|3|node|3|node|3|node|4|node_1}}

|120px
Hexi-cantellated 7-orthoplex
{{CDD|node_1|3|node|3|node_1|3|node|3|node|3|node|4|node_1}}

|120px
Hexicanti-truncated 7-orthoplex
{{CDD|node_1|3|node_1|3|node_1|3|node|3|node|3|node|4|node_1}}

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|120px
Hexirunci-truncated 7-orthoplex
{{CDD|node_1|3|node_1|3|node|3|node_1|3|node|3|node|4|node_1}}

|120px
Hexirunci-cantellated 7-orthoplex
{{CDD|node_1|3|node|3|node_1|3|node_1|3|node|3|node|4|node_1}}

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Hexisteri-truncated 7-orthoplex
{{CDD|node_1|3|node_1|3|node|3|node|3|node_1|3|node|4|node_1}}

|120px
Hexiruncicanti-truncated 7-orthoplex
{{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node|4|node_1}}

|120px
Hexistericanti-truncated 7-orthoplex
{{CDD|node_1|3|node_1|3|node_1|3|node|3|node_1|3|node|4|node_1}}

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|120px
Hexisterirunci-truncated 7-orthoplex
{{CDD|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node|4|node_1}}

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Hexipenticanti-truncated 7-orthoplex
{{CDD|node_1|3|node_1|3|node_1|3|node|3|node|3|node_1|4|node_1}}

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Hexisteriruncicanti-truncated 7-orthoplex
{{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node|4|node_1}}

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Hexipentiruncicanti-truncated 7-orthoplex
{{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node_1|4|node_1}}

In seven-dimensional geometry, a hexicated 7-orthoplex (also hexicated 7-cube) is a convex uniform 7-polytope, including 6th-order truncations (hexication) from the regular 7-orthoplex.

There are 32 hexications for the 7-orthoplex, including all permutations of truncations, cantellations, runcinations, sterications, and pentellations. 12 are represented here, while 20 are more easily constructed from the 7-cube.

Hexitruncated 7-orthoplex

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! style="background:#e7dcc3;" colspan="2"|Hexitruncated 7-orthoplex

style="background:#e7dcc3;"|TypeUniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,1,6{35,4
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|3|node_1|3|node|3|node|3|node|3|node|4|node_1}}
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;"|Edges29568
style="background:#e7dcc3;"|Vertices5376
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsB7, [4,35]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names=

  • Petitruncated heptacross

= Images =

{{B7 Coxeter plane graphs|t056|150}}

== Hexicantellated 7-orthoplex ==

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! style="background:#e7dcc3;" colspan="2"|Hexicantellated 7-orthoplex

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,2,6{35,4}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|3|node|3|node_1|3|node|3|node|3|node|4|node_1}}
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;"|Edges94080
style="background:#e7dcc3;"|Vertices13440
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsB7, [4,35]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names=

  • Petirhombated heptacross

= Images =

{{B7 Coxeter plane graphs|t046|150}}

== Hexicantitruncated 7-orthoplex ==

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! style="background:#e7dcc3;" colspan="2"|Hexicantitruncated 7-orthoplex

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,1,2,6{35,4}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|3|node_1|3|node_1|3|node|3|node|3|node|4|node_1}}
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;"|Edges134400
style="background:#e7dcc3;"|Vertices26880
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsB7, [4,35]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names=

  • Petigreatorhombated heptacross

= Images =

{{B7 Coxeter plane graphs|t0456|150}}

== Hexiruncitruncated 7-orthoplex ==

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! style="background:#e7dcc3;" colspan="2"|Hexiruncitruncated 7-orthoplex

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,1,3,6{35,3}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|3|node_1|3|node|3|node_1|3|node|3|node|4|node_1}}
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;"|Edges322560
style="background:#e7dcc3;"|Vertices53760
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsB7, [4,35]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names=

  • Petiprismatotruncated heptacross

= Images =

{{B7 Coxeter plane graphs|t0356|150}}

== Hexiruncicantellated 7-orthoplex ==

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! style="background:#e7dcc3;" colspan="2"|Hexiruncicantellated 7-orthoplex

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,2,3,6{35,4}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|3|node|3|node_1|3|node_1|3|node|3|node|4|node_1}}
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;"|Edges268800
style="background:#e7dcc3;"|Vertices53760
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsB7, [4,35]
style="background:#e7dcc3;"|Propertiesconvex

In seven-dimensional geometry, a hexiruncicantellated 7-orthoplex is a uniform 7-polytope.

= Alternate names=

  • Petiprismatorhombated heptacross

= Images =

{{B7 Coxeter plane graphs|t0346|150}}

== Hexisteritruncated 7-orthoplex ==

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! style="background:#e7dcc3;" colspan="2"|hexisteritruncated 7-orthoplex

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,1,4,6{35,4}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|3|node_1|3|node|3|node|3|node_1|3|node|4|node_1}}
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;"|Edges322560
style="background:#e7dcc3;"|Vertices53760
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsB7, [4,35]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names=

  • Peticellitruncated heptacross

= Images =

{{B7 Coxeter plane graphs|t0256|150}}

== Hexiruncicantitruncated 7-orthoplex ==

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! style="background:#e7dcc3;" colspan="2"|Hexiruncicantitruncated 7-orthoplex

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,1,2,3,6{35,4}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node|4|node_1}}
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;"|Edges483840
style="background:#e7dcc3;"|Vertices107520
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsB7, [4,35]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names=

  • Petigreatoprismated heptacross

= Images =

{{B7 Coxeter plane graphs|t03456|150}}

== Hexistericantitruncated 7-orthoplex ==

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! style="background:#e7dcc3;" colspan="2"|Hexistericantitruncated 7-orthoplex

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,1,2,4,6{35,4}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|3|node_1|3|node_1|3|node|3|node_1|3|node|4|node_1}}
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;"|Edges806400
style="background:#e7dcc3;"|Vertices161280
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsB7, [4,35]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Peticelligreatorhombated heptacross

= Images =

{{B7 Coxeter plane graphs|t03456|150}}

== Hexisteriruncitruncated 7-orthoplex ==

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! style="background:#e7dcc3;" colspan="2"|Hexisteriruncitruncated 7-orthoplex

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,1,3,4,6{35,4}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node|4|node_1}}
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;"|Edges725760
style="background:#e7dcc3;"|Vertices161280
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsB7, [4,35]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names=

  • Peticelliprismatotruncated heptacross

= Images =

{{B7 Coxeter plane graphs|t02356|150|NOB7A6}}

== Hexipenticantitruncated 7-orthoplex ==

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! style="background:#e7dcc3;" colspan="2"|hexipenticantitruncated 7-orthoplex

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,1,2,5,6{35,4}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|3|node_1|3|node_1|3|node|3|node|3|node_1|4|node_1}}
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;"|Edges483840
style="background:#e7dcc3;"|Vertices107520
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsB7, [4,35]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names=

  • Petiterigreatorhombated heptacross

= Images =

{{B7 Coxeter plane graphs|t01456|150}}

== Hexisteriruncicantitruncated 7-orthoplex ==

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! style="background:#e7dcc3;" colspan="2"|Hexisteriruncicantitruncated 7-orthoplex

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,1,2,3,4,6{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|3|node_1}}
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;"|Edges1290240
style="background:#e7dcc3;"|Vertices322560
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsB7, [4,35]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names=

  • Great petacellated heptacross

= Images =

{{B7 Coxeter plane graphs|t012346|150|NOB7A6}}

== Hexipentiruncicantitruncated 7-orthoplex ==

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! style="background:#e7dcc3;" colspan="2"|Hexipentiruncicantitruncated 7-orthoplex

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,1,2,3,5,6{35,3}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node_1|4|node_1}}
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;"|Edges1290240
style="background:#e7dcc3;"|Vertices322560
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsB7, [4,35]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names=

  • Petiterigreatoprismated heptacross

= Images =

{{B7 Coxeter plane graphs|t013456|150|NOB7A6}}

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, {{ISBN|978-0-471-01003-6}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
  • (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, PhD (1966)
  • {{KlitzingPolytopes|polyexa.htm||7D uniform polytopes (polyexa)}}

External links

  • [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions]
  • [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]

{{Polytopes}}

Category:7-polytopes