“Runcinated 5-orthoplexes”的意思、由来-开放百科全书

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Runcinated 5-orthoplexes

释义

Runcinated 5-orthoplex
Alternate names Coordinates Images
Runcitruncated 5-orthoplex
Alternate names Coordinates Images
Runcicantellated 5-orthoplex
Alternate names Coordinates Images
Runcicantitruncated 5-orthoplex
Alternate names Coordinates Images Snub 5-demicube
Related polytopes
Notes
References
External links

Orthogonal projections in B₅ Coxeter plane
5-orthoplex {{CDD>node_1\|3\|node\|3\|node\|3\|node\|4\|node}}	Runcinated 5-orthoplex {{CDD>node_1\|3\|node\|3\|node\|3\|node_1\|4\|node}}	Runcinated 5-cube {{CDD>node\|3\|node_1\|3\|node\|3\|node\|4\|node_1}}
Runcitruncated 5-orthoplex {{CDD>node_1\|3\|node_1\|3\|node\|3\|node_1\|4\|node}}	Runcicantellated 5-orthoplex {{CDD>node_1\|3\|node\|3\|node_1\|3\|node_1\|4\|node}}	Runcicantitruncated 5-orthoplex {{CDD>node_1\|3\|node_1\|3\|node_1\|3\|node_1\|3\|node}}
Runcitruncated 5-cube {{CDD>node\|3\|node_1\|3\|node\|3\|node_1\|4\|node_1}}	Runcicantellated 5-cube {{CDD>node\|3\|node_1\|3\|node_1\|3\|node\|4\|node_1}}	Runcicantitruncated 5-cube {{CDD>node\|3\|node_1\|3\|node_1\|3\|node_1\|4\|node_1}}

In five-dimensional geometry, a runcinated 5-orthoplex is a convex uniform 5-polytope with 3rd order truncation (runcination) of the regular 5-orthoplex.

There are 8 runcinations of the 5-orthoplex with permutations of truncations, and cantellations. Four are more simply constructed relative to the 5-cube.

Runcinated 5-orthoplex

Runcinated 5-orthoplex
Type	Uniform 5-polytope
Schläfli symbol	t_0,3{3,3,3,4}
Coxeter-Dynkin diagram	{{CDD	3\|node	node\|3\|node_1\|4\|node}} {{CDD	3\|node	node\|split1\|nodes_11}
4-faces	162
Cells	1200
Faces	2160
Edges	1440
Vertices	320
Vertex figure
Coxeter group	B₅ [4,3,3,3] D₅ [3^2,1,1]
Properties	convex

Alternate names

Runcinated pentacross
Small prismated triacontiditeron (Acronym: spat) (Jonathan Bowers)^[1]

Coordinates

The vertices of the can be made in 5-space, as permutations and sign combinations of:

(0,1,1,1,2)

Images

Runcitruncated 5-orthoplex

Runcitruncated 5-orthoplex
Type	uniform 5-polytope
Schläfli symbol	t_0,1,3{3,3,3,4} t_0,1,3{3,3^1,1}
Coxeter-Dynkin diagrams	node\|4\|node_1\|3\|node\|3\|node_1\|3\|node_1}} {{CDD\|nodes_11\|split2\|node\|3\|node_1\|3\|node_1}}
4-faces	162
Cells	1440
Faces	3680
Edges	3360
Vertices	960
Vertex figure
Coxeter groups	B₅, [3,3,3,4] D₅, [3^2,1,1]
Properties	convex

Alternate names

Runcitruncated pentacross
Prismatotruncated triacontiditeron (Acronym: pattit) (Jonathan Bowers)^[2]

Coordinates

Cartesian coordinates for the vertices of a runcitruncated 5-orthoplex, centered at the origin, are all 80 vertices are sign (4) and coordinate (20) permutations of

(±3,±2,±1,±1,0)

Images

Runcicantellated 5-orthoplex

Runcicantellated 5-orthoplex
Type	Uniform 5-polytope
Schläfli symbol	t_0,2,3{3,3,3,4} t_0,2,3{3,3,3^1,1}
Coxeter-Dynkin diagram	{{CDD	3\|node	node_1\|3\|node_1\|4\|node}} {{CDD	3\|node	node_1\|split1\|nodes_11}
4-faces	162
Cells	1200
Faces	2960
Edges	2880
Vertices	960
Vertex figure
Coxeter group	B₅ [4,3,3,3] D₅ [3^2,1,1]
Properties	convex

Alternate names

Runcicantellated pentacross
Prismatorhombated triacontiditeron (Acronym: pirt) (Jonathan Bowers)^[3]

Coordinates

The vertices of the runcicantellated 5-orthoplex can be made in 5-space, as permutations and sign combinations of:

(0,1,2,2,3)

Images

Runcicantitruncated 5-orthoplex

Runcicantitruncated 5-orthoplex
Type	Uniform 5-polytope
Schläfli symbol	t_0,1,2,3{3,3,3,4}
Coxeter-Dynkin diagram	node\|4\|node_1\|3\|node_1\|3\|node_1\|3\|node_1}} {{CDD\|nodes_11\|split2\|node_1\|3\|node_1\|3\|node_1}}
4-faces	162
Cells	1440
Faces	4160
Edges	4800
Vertices	1920
Vertex figure	Irregular 5-cell
Coxeter groups	B₅ [4,3,3,3] D₅ [3^2,1,1]
Properties	convex, isogonal

Alternate names

Runcicantitruncated pentacross
Great prismated triacontiditeron (gippit) (Jonathan Bowers)^[4]

Coordinates

The Cartesian coordinates of the vertices of a runcicantitruncated tesseract having an edge length of {{radic|2}} are given by all permutations of coordinates and sign of:

Images

Snub 5-demicube

The snub 5-demicube defined as an alternation of the omnitruncated 5-demicube is not uniform, but it can be given Coxeter diagram {{CDD|nodes_hh|split2|node_h|3|node_h|3|node_h}} or {{CDD|node|4|node_h|3|node_h|3|node_h|3|node_h}} and symmetry [3^2,1,1]⁺ or [4,(3,3,3)⁺], and constructed from 32 snub 5-cells, 80 alternated 6-6 duoprisms, 40 icosahedral prisms, 10 snub 24-cells, and 960 irregular tetrahedrons filling the gaps at the deleted vertices.

Related polytopes

This polytope is one of 31 uniform 5-polytopes generated from the regular 5-cube or 5-orthoplex.

Notes

1. ^Klitzing, (x3o3o3x4o - spat)
2. ^Klitzing, (x3x3o3x4o - pattit)
3. ^Klitzing, (x3o3x3x4o - pirt)
4. ^Klitzing, (x3x3x3x4o - gippit)

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}}
- (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)}} x3o3o3x4o - spat, x3x3o3x4o - pattit, x3o3x3x4o - pirt, x3x3x3x4o - gippit

External links

{{PolyCell | urlname = glossary.html#simplex| title = Glossary for hyperspace}}
Polytopes of Various Dimensions, Jonathan Bowers
- Runcinated uniform polytera (spid), Jonathan Bowers
Multi-dimensional Glossary

1 : 5-polytopes

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