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Runcinated 5-simplexes#Runcinated 5-simplex

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|150px
5-simplex
{{CDD|node_1|3|node|3|node|3|node|3|node}}

|150px
Runcinated 5-simplex
{{CDD|node_1|3|node|3|node|3|node_1|3|node}}

|150px
Runcitruncated 5-simplex
{{CDD|node_1|3|node_1|3|node|3|node_1|3|node}}

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Birectified 5-simplex
{{CDD|node|3|node|3|node_1|3|node|3|node}}

|150px
Runcicantellated 5-simplex
{{CDD|node_1|3|node|3|node_1|3|node_1|3|node}}

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Runcicantitruncated 5-simplex
{{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node}}

colspan=3|Orthogonal projections in A5 Coxeter plane

In six-dimensional geometry, a runcinated 5-simplex is a convex uniform 5-polytope with 3rd order truncations (Runcination) of the regular 5-simplex.

There are 4 unique runcinations of the 5-simplex with permutations of truncations, and cantellations.

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Runcinated 5-simplex

class="wikitable" align="right" style="margin-left:10px" width="250"
bgcolor=#e7dcc3 align=center colspan=3|Runcinated 5-simplex
bgcolor=#e7dcc3|Type

|colspan=2|Uniform 5-polytope

bgcolor=#e7dcc3|Schläfli symbol

|colspan=2| t0,3{3,3,3,3}

bgcolor=#e7dcc3|Coxeter-Dynkin diagram

|colspan=2|{{CDD

node_1|3|node3|node|3|node_1|3|node}}
bgcolor=#e7dcc3|4-faces

|47

|6 t0,3{3,3,3} 20px
20 {3}×{3}
15 { }×r{3,3}
6 r{3,3,3} 20px

bgcolor=#e7dcc3|Cells

|255

|45 {3,3} 20px
180 { }×{3}
30 r{3,3} 20px

bgcolor=#e7dcc3|Faces

|420

|240 {3} 20px
180 {4}

bgcolor=#e7dcc3|Edges

|colspan=2|270

bgcolor=#e7dcc3|Vertices

|colspan=2|60

bgcolor=#e7dcc3|Vertex figure

|colspan=2|50px

bgcolor=#e7dcc3|Coxeter group

|colspan=2| A5 [3,3,3,3], order 720

bgcolor=#e7dcc3|Properties

|colspan=2|convex

= Alternate names =

  • Runcinated hexateron
  • Small prismated hexateron (Acronym: spix) (Jonathan Bowers)Klitizing, (x3o3o3x3o - spidtix)

= Coordinates =

The vertices of the runcinated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,1,1,1,2) or of (0,1,1,1,2,2), seen as facets of a runcinated 6-orthoplex, or a biruncinated 6-cube respectively.

= Images =

{{5-simplex Coxeter plane graphs|t03|150}}

Runcitruncated 5-simplex

class="wikitable" align="right" style="margin-left:10px" width="280"
bgcolor=#e7dcc3 align=center colspan=3|Runcitruncated 5-simplex
bgcolor=#e7dcc3|Type

|colspan=2|Uniform 5-polytope

bgcolor=#e7dcc3|Schläfli symbol

|colspan=2|t0,1,3{3,3,3,3}

bgcolor=#e7dcc3|Coxeter-Dynkin diagram

|colspan=2|{{CDD

node_1|3|node_13|node|3|node_1|3|node}}
bgcolor=#e7dcc3|4-faces

|47

|6 t0,1,3{3,3,3}
20 {3}×{6}
15 { }×r{3,3}
6 rr{3,3,3}

bgcolor=#e7dcc3|Cells

|315

|

bgcolor=#e7dcc3|Faces

|720

|

bgcolor=#e7dcc3|Edges

|colspan=2|630

bgcolor=#e7dcc3|Vertices

|colspan=2|180

bgcolor=#e7dcc3|Vertex figure

|colspan=2|110px

bgcolor=#e7dcc3|Coxeter group

|colspan=2|A5 [3,3,3,3], order 720

bgcolor=#e7dcc3|Properties

|colspan=2|convex, isogonal

= Alternate names =

  • Runcitruncated hexateron
  • Prismatotruncated hexateron (Acronym: pattix) (Jonathan Bowers)Klitizing, (x3x3o3x3o - pattix)

= Coordinates =

The coordinates can be made in 6-space, as 180 permutations of:

: (0,0,1,1,2,3)

This construction exists as one of 64 orthant facets of the runcitruncated 6-orthoplex.

= Images =

{{5-simplex Coxeter plane graphs|t013|150}}

Runcicantellated 5-simplex

class="wikitable" align="right" style="margin-left:10px" width="280"
bgcolor=#e7dcc3 align=center colspan=3|Runcicantellated 5-simplex
bgcolor=#e7dcc3|Type

|colspan=2|Uniform 5-polytope

bgcolor=#e7dcc3|Schläfli symbol

|colspan=2|t0,2,3{3,3,3,3}

bgcolor=#e7dcc3|Coxeter-Dynkin diagram

|colspan=2|{{CDD

node_1|3|node3|node_1|3|node_1|3|node}}
bgcolor=#e7dcc3|4-faces

|47

|

bgcolor=#e7dcc3|Cells

|255

|

bgcolor=#e7dcc3|Faces

|570

|

bgcolor=#e7dcc3|Edges

|colspan=2|540

bgcolor=#e7dcc3|Vertices

|colspan=2|180

bgcolor=#e7dcc3|Vertex figure

|colspan=2|100px

bgcolor=#e7dcc3|Coxeter group

|colspan=2|A5 [3,3,3,3], order 720

bgcolor=#e7dcc3|Properties

|colspan=2|convex, isogonal

= Alternate names =

  • Runcicantellated hexateron
  • Biruncitruncated 5-simplex/hexateron
  • Prismatorhombated hexateron (Acronym: pirx) (Jonathan Bowers)Klitizing, (x3o3x3x3o - pirx)

= Coordinates =

The coordinates can be made in 6-space, as 180 permutations of:

: (0,0,1,2,2,3)

This construction exists as one of 64 orthant facets of the runcicantellated 6-orthoplex.

= Images =

{{5-simplex Coxeter plane graphs|t023|150}}

Runcicantitruncated 5-simplex

class="wikitable" align="right" style="margin-left:10px" width="280"
bgcolor=#e7dcc3 align=center colspan=3|Runcicantitruncated 5-simplex
bgcolor=#e7dcc3|Type

|colspan=2|Uniform 5-polytope

bgcolor=#e7dcc3|Schläfli symbol

|colspan=2|t0,1,2,3{3,3,3,3}

bgcolor=#e7dcc3|Coxeter-Dynkin diagram

|colspan=2|{{CDD

node_1|3|node_13|node_1|3|node_1|3|node}}
bgcolor=#e7dcc3|4-faces

|47

|6 t0,1,2,3{3,3,3}
20 {3}×{6}
15 {}×t{3,3}
6 tr{3,3,3}

bgcolor=#e7dcc3|Cells

|315

|45 t0,1,2{3,3}
120 { }×{3}
120 { }×{6}
30 t{3,3}

bgcolor=#e7dcc3|Faces

|810

|120 {3}
450 {4}
240 {6}

bgcolor=#e7dcc3|Edges

|colspan=2|900

bgcolor=#e7dcc3|Vertices

|colspan=2|360

bgcolor=#e7dcc3|Vertex figure

|colspan=2|100px
Irregular 5-cell

bgcolor=#e7dcc3|Coxeter group

|colspan=2|A5 [3,3,3,3], order 720

bgcolor=#e7dcc3|Properties

|colspan=2|convex, isogonal

= Alternate names =

  • Runcicantitruncated hexateron
  • Great prismated hexateron (Acronym: gippix) (Jonathan Bowers)Klitizing, (x3x3x3x3o - gippix)

= Coordinates =

The coordinates can be made in 6-space, as 360 permutations of:

: (0,0,1,2,3,4)

This construction exists as one of 64 orthant facets of the runcicantitruncated 6-orthoplex.

= Images =

{{5-simplex Coxeter plane graphs|t0123|150}}

Related uniform 5-polytopes

These polytopes are in a set of 19 uniform 5-polytopes based on the [3,3,3,3] Coxeter group, all shown here in A5 Coxeter plane orthographic projections. (Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple having progressively more vertices)

{{Hexateron 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, {{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, Ph.D.
  • {{KlitzingPolytopes|polytera.htm|5D|uniform polytopes (polytera)}} x3o3o3x3o - spidtix, x3x3o3x3o - pattix, x3o3x3x3o - pirx, x3x3x3x3o - gippix

External links

  • {{PolyCell | urlname = glossary.html#simplex| title = Glossary for hyperspace}}
  • [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions], Jonathan Bowers
  • [http://www.polytope.net/hedrondude/truncates5.htm Runcinated uniform polytera] (spid), Jonathan Bowers
  • [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]

{{Polytopes}}

Category:5-polytopes