Login
Login

Truncated 7-orthoplexes#Bitruncated 7-orthoplex

{{Short description|7-polytope}}

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

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

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

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

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

align=center valign=top

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

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

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

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

colspan=4|Orthogonal projections in B7 Coxeter plane

In seven-dimensional geometry, a truncated 7-orthoplex is a convex uniform 7-polytope, being a truncation of the regular 7-orthoplex.

There are 6 truncations of the 7-orthoplex. Vertices of the truncation 7-orthoplex are located as pairs on the edge of the 7-orthoplex. Vertices of the bitruncated 7-orthoplex are located on the triangular faces of the 7-orthoplex. Vertices of the tritruncated 7-orthoplex are located inside the tetrahedral cells of the 7-orthoplex. The final three truncations are best expressed relative to the 7-cube.

{{TOC left}}

{{-}}

Truncated 7-orthoplex

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

!bgcolor=#e7dcc3 colspan=2|Truncated 7-orthoplex

bgcolor=#e7dcc3|Typeuniform 7-polytope
bgcolor=#e7dcc3|Schläfli symbolt{35,4}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node_1|3|node_1|3|node|3|node|3|node|3|node|4|node}}

{{CDD|node_1|3|node_1|3|node|3|node|3|node|split1|nodes}}

bgcolor=#e7dcc3|6-faces
bgcolor=#e7dcc3|5-faces
bgcolor=#e7dcc3|4-faces
bgcolor=#e7dcc3|Cells3920
bgcolor=#e7dcc3|Faces2520
bgcolor=#e7dcc3|Edges924
bgcolor=#e7dcc3|Vertices168
bgcolor=#e7dcc3|Vertex figure( )v{3,3,4}
bgcolor=#e7dcc3|Coxeter groupsB7, [35,4]
D7, [34,1,1]
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names=

  • Truncated heptacross
  • Truncated hecatonicosoctaexon (Jonathan Bowers)Klitzing, (x3x3o3o3o3o4o - tez)

= Coordinates =

Cartesian coordinates for the vertices of a truncated 7-orthoplex, centered at the origin, are all 168 vertices are sign (4) and coordinate (42) permutations of

: (±2,±1,0,0,0,0,0)

= Images =

{{7-cube Coxeter plane graphs|t56|150}}

= Construction =

There are two Coxeter groups associated with the truncated 7-orthoplex, one with the C7 or [4,35] Coxeter group, and a lower symmetry with the D7 or [34,1,1] Coxeter group.

Bitruncated 7-orthoplex

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

!bgcolor=#e7dcc3 colspan=2|Bitruncated 7-orthoplex

bgcolor=#e7dcc3|Typeuniform 7-polytope
bgcolor=#e7dcc3|Schläfli symbol2t{35,4}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node|3|node_1|3|node_1|3|node|3|node|3|node|4|node}}

{{CDD|node|3|node_1|3|node_1|3|node|3|node|split1|nodes}}

bgcolor=#e7dcc3|6-faces
bgcolor=#e7dcc3|5-faces
bgcolor=#e7dcc3|4-faces
bgcolor=#e7dcc3|Cells
bgcolor=#e7dcc3|Faces
bgcolor=#e7dcc3|Edges4200
bgcolor=#e7dcc3|Vertices840
bgcolor=#e7dcc3|Vertex figure{ }v{3,3,4}
bgcolor=#e7dcc3|Coxeter groupsB7, [35,4]
D7, [34,1,1]
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names=

  • Bitruncated heptacross
  • Bitruncated hecatonicosoctaexon (Jonathan Bowers)Klitzing, (o3x3x3o3o3o4o - botaz)

= Coordinates =

Cartesian coordinates for the vertices of a bitruncated 7-orthoplex, centered at the origin, are all sign and coordinate permutations of

: (±2,±2,±1,0,0,0,0)

=Images=

{{7-cube Coxeter plane graphs|t45|150}}

Tritruncated 7-orthoplex

The tritruncated 7-orthoplex can tessellation space in the quadritruncated 7-cubic honeycomb.

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

!bgcolor=#e7dcc3 colspan=2|Tritruncated 7-orthoplex

bgcolor=#e7dcc3|Typeuniform 7-polytope
bgcolor=#e7dcc3|Schläfli symbol3t{35,4}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node|3|node|3|node_1|3|node_1|3|node|3|node|4|node}}

{{CDD|node|3|node|3|node_1|3|node_1|3|node|split1|nodes}}

bgcolor=#e7dcc3|6-faces
bgcolor=#e7dcc3|5-faces
bgcolor=#e7dcc3|4-faces
bgcolor=#e7dcc3|Cells
bgcolor=#e7dcc3|Faces
bgcolor=#e7dcc3|Edges10080
bgcolor=#e7dcc3|Vertices2240
bgcolor=#e7dcc3|Vertex figure{3}v{3,4}
bgcolor=#e7dcc3|Coxeter groupsB7, [35,4]
D7, [34,1,1]
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names=

  • Tritruncated heptacross
  • Tritruncated hecatonicosoctaexon (Jonathan Bowers)Klitzing, (o3o3x3x3o3o4o - totaz)

= Coordinates =

Cartesian coordinates for the vertices of a tritruncated 7-orthoplex, centered at the origin, are all sign and coordinate permutations of

: (±2,±2,±2,±1,0,0,0)

=Images=

{{7-cube Coxeter plane graphs|t34|150}}

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|polyexa.htm|7D|uniform polytopes (polyexa)}} x3x3o3o3o3o4o - tez, o3x3x3o3o3o4o - botaz, o3o3x3x3o3o4o - totaz

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:7-polytopes