Runcinated 7-orthoplexes

In seven-dimensional geometry, a runcinated 7-orthoplex is a convex uniform 7-polytope with 3rd order truncations (runcination) of the regular 7-orthoplex.

More information Orthogonal projections in B6 Coxeter plane ...

Orthogonal projections in B₆ Coxeter plane
7-orthoplex	Runcinated 7-orthoplex	Biruncinated 7-orthoplex
Runcitruncated 7-orthoplex	Biruncitruncated 7-orthoplex	Runcicantellated 7-orthoplex
Biruncicantellated 7-orthoplex	Runcicantitruncated 7-orthoplex	Biruncicantitruncated 7-orthoplex

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There are 16 unique runcinations of the 7-orthoplex with permutations of truncations, and cantellations. 8 are more simply constructed from the 7-cube.

These polytopes are among 127 uniform 7-polytopes with B₇ symmetry.

Runcinated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_0,3{3⁵,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	23520
Vertices	2240
Vertex figure
Coxeter groups	B₇, [4,3⁵]
Properties	convex

Coxeter plane	B₇ / A₆	B₆ / D₇	B₅ / D₆ / A₄
Graph
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B₄ / D₅	B₃ / D₄ / A₂	B₂ / D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Biruncinated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_1,4{3⁵,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	60480
Vertices	6720
Vertex figure
Coxeter groups	B₇, [4,3⁵]
Properties	convex

Coxeter plane	B₇ / A₆	B₆ / D₇	B₅ / D₆ / A₄
Graph
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B₄ / D₅	B₃ / D₄ / A₂	B₂ / D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Runcitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_0,1,3{3⁵,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	50400
Vertices	6720
Vertex figure
Coxeter groups	B₇, [4,3⁵]
Properties	convex

Runcinated 7-orthoplexes

Runcinated 7-orthoplex

Alternate names

Images

Biruncinated 7-orthoplex

Alternate names

Images

Runcitruncated 7-orthoplex

Alternate names

Images

Biruncitruncated 7-orthoplex

Alternate names

Images

Runcicantellated 7-orthoplex

Alternate names

Images

Biruncicantellated 7-orthoplex

Alternate names

Images

Runcicantitruncated 7-orthoplex

Alternate names

Images

Biruncicantitruncated 7-orthoplex

Alternate names

Images

Notes

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Coxeter plane	B₇ / A₆	B₆ / D₇	B₅ / D₆ / A₄
Graph	too complex
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B₄ / D₅	B₃ / D₄ / A₂	B₂ / D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	$A n$	$B n$	$I 2 (p)$ / $D n$	$E 6$ / $E 7$ / $E 8$ / $F 4$ / $G 2$	$H n$
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	1₂₂ • 2₂₁
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	1₃₂ • 2₃₁ • 3₂₁
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	1₄₂ • 2₄₁ • 4₂₁
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds • Polytope operations