Login
Login

Stericated 6-simplexes#Stericantitruncated 6-simplex

class=wikitable align=right width=480 style="margin-left:1em;"
align=center valign=top

|160px
6-simplex
{{CDD|node_1|3|node|3|node|3|node|3|node|3|node}}

|160px
Stericated 6-simplex
{{CDD|node_1|3|node|3|node|3|node|3|node_1|3|node}}

|160px
Steritruncated 6-simplex
{{CDD|node_1|3|node_1|3|node|3|node|3|node_1|3|node}}

align=center valign=top

|160px
Stericantellated 6-simplex
{{CDD|node_1|3|node|3|node_1|3|node|3|node_1|3|node}}

|160px
Stericantitruncated 6-simplex
{{CDD|node_1|3|node_1|3|node_1|3|node|3|node_1|3|node}}

|160px
Steriruncinated 6-simplex
{{CDD|node_1|3|node|3|node|3|node_1|3|node_1|3|node}}

align=center valign=top

|160px
Steriruncitruncated 6-simplex
{{CDD|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node}}

|160px
Steriruncicantellated 6-simplex
{{CDD|node_1|3|node|3|node_1|3|node_1|3|node_1|3|node}}

|160px
Steriruncicantitruncated 6-simplex
{{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node}}

colspan=3|Orthogonal projections in A6 Coxeter plane

In six-dimensional geometry, a stericated 6-simplex is a convex uniform 6-polytope with 4th order truncations (sterication) of the regular 6-simplex.

There are 8 unique sterications for the 6-simplex with permutations of truncations, cantellations, and runcinations.

{{-}}

Stericated 6-simplex

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

!bgcolor=#e7dcc3 colspan=2|Stericated 6-simplex

bgcolor=#e7dcc3|Typeuniform 6-polytope
bgcolor=#e7dcc3|Schläfli symbolt0,4{3,3,3,3,3}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node_1|3|node|3|node|3|node_1|3|node|3|node}}
bgcolor=#e7dcc3|5-faces105
bgcolor=#e7dcc3|4-faces700
bgcolor=#e7dcc3|Cells1470
bgcolor=#e7dcc3|Faces1400
bgcolor=#e7dcc3|Edges630
bgcolor=#e7dcc3|Vertices105
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupA6, [35], order 5040
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Small cellated heptapeton (Acronym: scal) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/scal.htm (x3o3o3o3x3o - scal)]}}

= Coordinates =

The vertices of the stericated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,1,1,1,2). This construction is based on facets of the stericated 7-orthoplex.

= Images =

{{6-simplex Coxeter plane graphs|t04|150}}

Steritruncated 6-simplex

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

!bgcolor=#e7dcc3 colspan=2|Steritruncated 6-simplex

bgcolor=#e7dcc3|Typeuniform 6-polytope
bgcolor=#e7dcc3|Schläfli symbolt0,1,4{3,3,3,3,3}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node_1|3|node_1|3|node|3|node|3|node_1|3|node}}
bgcolor=#e7dcc3|5-faces105
bgcolor=#e7dcc3|4-faces945
bgcolor=#e7dcc3|Cells2940
bgcolor=#e7dcc3|Faces3780
bgcolor=#e7dcc3|Edges2100
bgcolor=#e7dcc3|Vertices420
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupA6, [35], order 5040
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Cellitruncated heptapeton (Acronym: catal) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/catal.htm (x3x3o3o3x3o - catal)]}}

= Coordinates =

The vertices of the steritruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,1,1,2,3). This construction is based on facets of the steritruncated 7-orthoplex.

= Images =

{{6-simplex Coxeter plane graphs|t014|150}}

Stericantellated 6-simplex

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

!bgcolor=#e7dcc3 colspan=2|Stericantellated 6-simplex

bgcolor=#e7dcc3|Typeuniform 6-polytope
bgcolor=#e7dcc3|Schläfli symbolt0,2,4{3,3,3,3,3}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node_1|3|node|3|node_1|3|node|3|node_1|3|node}}
bgcolor=#e7dcc3|5-faces105
bgcolor=#e7dcc3|4-faces1050
bgcolor=#e7dcc3|Cells3465
bgcolor=#e7dcc3|Faces5040
bgcolor=#e7dcc3|Edges3150
bgcolor=#e7dcc3|Vertices630
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupA6, [35], order 5040
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Cellirhombated heptapeton (Acronym: cral) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/scral.htm (x3o3x3o3x3o - cral)]}}

= Coordinates =

The vertices of the stericantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,1,2,2,3). This construction is based on facets of the stericantellated 7-orthoplex.

= Images =

{{6-simplex Coxeter plane graphs|t024|150}}

Stericantitruncated 6-simplex

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

!bgcolor=#e7dcc3 colspan=2|stericantitruncated 6-simplex

bgcolor=#e7dcc3|Typeuniform 6-polytope
bgcolor=#e7dcc3|Schläfli symbolt0,1,2,4{3,3,3,3,3}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node_1|3|node_1|3|node_1|3|node|3|node|3|node_1}}
bgcolor=#e7dcc3|5-faces105
bgcolor=#e7dcc3|4-faces1155
bgcolor=#e7dcc3|Cells4410
bgcolor=#e7dcc3|Faces7140
bgcolor=#e7dcc3|Edges5040
bgcolor=#e7dcc3|Vertices1260
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupA6, [35], order 5040
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Celligreatorhombated heptapeton (Acronym: cagral) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/cagral.htm (x3x3x3o3x3o - cagral)]}}

= Coordinates =

The vertices of the stericanttruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,4). This construction is based on facets of the stericantitruncated 7-orthoplex.

= Images =

{{6-simplex Coxeter plane graphs|t0124|150}}

Steriruncinated 6-simplex

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

!bgcolor=#e7dcc3 colspan=2|steriruncinated 6-simplex

bgcolor=#e7dcc3|Typeuniform 6-polytope
bgcolor=#e7dcc3|Schläfli symbolt0,3,4{3,3,3,3,3}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node_1|3|node|3|node|3|node_1|3|node_1|3|node}}
bgcolor=#e7dcc3|5-faces105
bgcolor=#e7dcc3|4-faces700
bgcolor=#e7dcc3|Cells1995
bgcolor=#e7dcc3|Faces2660
bgcolor=#e7dcc3|Edges1680
bgcolor=#e7dcc3|Vertices420
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupA6, [35], order 5040
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Celliprismated heptapeton (Acronym: copal) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/copal.htm (x3o3o3x3x3o - copal)]}}

= Coordinates =

The vertices of the steriruncinated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,2,2,3,3). This construction is based on facets of the steriruncinated 7-orthoplex.

= Images =

{{6-simplex Coxeter plane graphs|t034|150}}

Steriruncitruncated 6-simplex

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

!bgcolor=#e7dcc3 colspan=2|steriruncitruncated 6-simplex

bgcolor=#e7dcc3|Typeuniform 6-polytope
bgcolor=#e7dcc3|Schläfli symbolt0,1,3,4{3,3,3,3,3}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node}}
bgcolor=#e7dcc3|5-faces105
bgcolor=#e7dcc3|4-faces945
bgcolor=#e7dcc3|Cells3360
bgcolor=#e7dcc3|Faces5670
bgcolor=#e7dcc3|Edges4410
bgcolor=#e7dcc3|Vertices1260
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupA6, [35], order 5040
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Celliprismatotruncated heptapeton (Acronym: captal) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/captal.htm (x3x3o3x3x3o - captal)]}}

= Coordinates =

The vertices of the steriruncittruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,4). This construction is based on facets of the steriruncitruncated 7-orthoplex.

= Images =

{{6-simplex Coxeter plane graphs|t0134|150}}

Steriruncicantellated 6-simplex

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

!bgcolor=#e7dcc3 colspan=2|steriruncicantellated 6-simplex

bgcolor=#e7dcc3|Typeuniform 6-polytope
bgcolor=#e7dcc3|Schläfli symbolt0,2,3,4{3,3,3,3,3}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node_1|3|node|3|node_1|3|node_1|3|node_1|3|node}}
bgcolor=#e7dcc3|5-faces105
bgcolor=#e7dcc3|4-faces1050
bgcolor=#e7dcc3|Cells3675
bgcolor=#e7dcc3|Faces5880
bgcolor=#e7dcc3|Edges4410
bgcolor=#e7dcc3|Vertices1260
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupA6, [35], order 5040
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Bistericantitruncated 6-simplex as t1,2,3,5{3,3,3,3,3}
  • Celliprismatorhombated heptapeton (Acronym: copril) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/copril.htm (x3o3x3x3x3o - copril)]}}

= Coordinates =

The vertices of the steriruncitcantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,4). This construction is based on facets of the steriruncicantellated 7-orthoplex.

= Images =

{{6-simplex Coxeter plane graphs|t0234|150}}

Steriruncicantitruncated 6-simplex

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

!bgcolor=#e7dcc3 colspan=2|Steriuncicantitruncated 6-simplex

bgcolor=#e7dcc3|Typeuniform 6-polytope
bgcolor=#e7dcc3|Schläfli symbolt0,1,2,3,4{3,3,3,3,3}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node}}
bgcolor=#e7dcc3|5-faces105
bgcolor=#e7dcc3|4-faces1155
bgcolor=#e7dcc3|Cells4620
bgcolor=#e7dcc3|Faces8610
bgcolor=#e7dcc3|Edges7560
bgcolor=#e7dcc3|Vertices2520
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupA6, [35], order 5040
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Great cellated heptapeton (Acronym: gacal) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/gacal.htm (x3x3x3x3x3o - gacal)]}}

= Coordinates =

The vertices of the steriruncicantittruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,2,3,4,5). This construction is based on facets of the steriruncicantitruncated 7-orthoplex.

= Images =

{{6-simplex Coxeter plane graphs|t01234|150}}

Related uniform 6-polytopes

The truncated 6-simplex is one of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A6 Coxeter plane orthographic projections.

{{Heptapeton family}}

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]
  • (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.
  • {{KlitzingPolytopes|polypeta.htm|6D uniform polytopes (polypeta) with acronyms}} {{sfn whitelist| CITEREFKlitzing}}

External links

  • [https://web.archive.org/web/20070310205351/http://members.cox.net/hedrondude/topes.htm Polytopes of Various Dimensions]
  • [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]

{{Polytopes}}

Category:6-polytopes