Rectified 8-orthoplexes

In eight-dimensional geometry, a rectified 8-orthoplex is a convex uniform 8-polytope, being a rectification of the regular 8-orthoplex.

Orthogonal projections in A₈ Coxeter plane
8-orthoplex	Rectified 8-orthoplex	Birectified 8-orthoplex	Trirectified 8-orthoplex
Trirectified 8-cube	Birectified 8-cube	Rectified 8-cube	8-cube

There are unique 8 degrees of rectifications, the zeroth being the 8-orthoplex, and the 7th and last being the 8-cube. Vertices of the rectified 8-orthoplex are located at the edge-centers of the 8-orthoplex. Vertices of the birectified 8-orthoplex are located in the triangular face centers of the 8-orthoplex. Vertices of the trirectified 8-orthoplex are located in the tetrahedral cell centers of the 8-orthoplex.

Rectified 8-orthoplex

Rectified 8-orthoplex
Type	uniform 8-polytope
Schläfli symbol	t₁{3,3,3,3,3,3,4}
Coxeter-Dynkin diagrams
7-faces	272
6-faces	3072
5-faces	8960
4-faces	12544
Cells	10080
Faces	4928
Edges	1344
Vertices	112
Vertex figure	6-orthoplex prism
Petrie polygon	hexakaidecagon
Coxeter groups	C₈, [4,3⁶] D₈, [3^5,1,1]
Properties	convex

The rectified 8-orthoplex has 112 vertices. These represent the root vectors of the simple Lie group D₈. The vertices can be seen in 3 hyperplanes, with the 28 vertices rectified 7-simplexs cells on opposite sides, and 56 vertices of an expanded 7-simplex passing through the center. When combined with the 16 vertices of the 8-orthoplex, these vertices represent the 128 root vectors of the B₈ and C₈ simple Lie groups.

Related polytopes

The rectified 8-orthoplex is the vertex figure for the demiocteractic honeycomb.

or

Alternate names

rectified octacross
rectified diacosipentacontahexazetton (Acronym: rek) (Jonathan Bowers)[1]

Construction

There are two Coxeter groups associated with the rectified 8-orthoplex, one with the C₈ or [4,3⁶] Coxeter group, and a lower symmetry with two copies of heptcross facets, alternating, with the D₈ or [3^5,1,1] Coxeter group.

Cartesian coordinates

Cartesian coordinates for the vertices of a rectified 8-orthoplex, centered at the origin, edge length ${\sqrt {2}}$ are all permutations of:

(±1,±1,0,0,0,0,0,0)

Images

orthographic projections
B₈			B₇

[16]			[14]
B₆			B₅

[12]			[10]
B₄		B₃		B₂

[8]		[6]		[4]
A₇		A₅		A₃

[8]		[6]		[4]

Birectified 8-orthoplex

Birectified 8-orthoplex
Type	uniform 8-polytope
Schläfli symbol	t₂{3,3,3,3,3,3,4}
Coxeter-Dynkin diagrams
7-faces	272
6-faces	3184
5-faces	16128
4-faces	34048
Cells	36960
Faces	22400
Edges	6720
Vertices	448
Vertex figure	{3,3,3,4}x{3}
Coxeter groups	C₈, [3,3,3,3,3,3,4] D₈, [3^5,1,1]
Properties	convex

Alternate names

birectified octacross
birectified diacosipentacontahexazetton (Acronym: bark) (Jonathan Bowers)[2]

Cartesian coordinates

Cartesian coordinates for the vertices of a birectified 8-orthoplex, centered at the origin, edge length ${\sqrt {2}}$ are all permutations of:

(±1,±1,±1,0,0,0,0,0)

Images

orthographic projections
B₈			B₇

[16]			[14]
B₆			B₅

[12]			[10]
B₄		B₃		B₂

[8]		[6]		[4]
A₇		A₅		A₃

[8]		[6]		[4]

Trirectified 8-orthoplex

Trirectified 8-orthoplex
Type	uniform 8-polytope
Schläfli symbol	t₃{3,3,3,3,3,3,4}
Coxeter-Dynkin diagrams
7-faces	16+256
6-faces	1024 + 2048 + 112
5-faces	1792 + 7168 + 7168 + 448
4-faces	1792 + 10752 + 21504 + 14336
Cells	8960 + 126880 + 35840
Faces	17920 + 35840
Edges	17920
Vertices	1120
Vertex figure	{3,3,4}x{3,3}
Coxeter groups	C₈, [3,3,3,3,3,3,4] D₈, [3^5,1,1]
Properties	convex

The trirectified 8-orthoplex can tessellate space in the quadrirectified 8-cubic honeycomb.

Alternate names

trirectified octacross
trirectified diacosipentacontahexazetton (acronym: tark) (Jonathan Bowers)[3]

Cartesian coordinates

Cartesian coordinates for the vertices of a trirectified 8-orthoplex, centered at the origin, edge length ${\sqrt {2}}$ are all permutations of:

(±1,±1,±1,±1,0,0,0,0)

Images

orthographic projections
B₈			B₇

[16]			[14]
B₆			B₅

[12]			[10]
B₄		B₃		B₂

[8]		[6]		[4]
A₇		A₅		A₃

[8]		[6]		[4]

Notes

Klitzing, (o3x3o3o3o3o3o4o - rek)
Klitzing, (o3o3x3o3o3o3o4o - bark)
Klitzing, (o3o3o3x3o3o3o4o - tark)

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.
Klitzing, Richard. "8D uniform polytopes (polyzetta)". o3x3o3o3o3o3o4o - rek, o3o3x3o3o3o3o4o - bark, o3o3o3x3o3o3o4o - tark

External links

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	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

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