Rectified tesseract

In geometry, the rectified tesseract, rectified 8-cell is a uniform 4-polytope (4-dimensional polytope) bounded by 24 cells: 8 cuboctahedra, and 16 tetrahedra. It has half the vertices of a runcinated tesseract, with its construction, called a runcic tesseract.

Rectified tesseract
Schlegel diagram Centered on cuboctahedron tetrahedral cells shown
Type	Uniform 4-polytope
Schläfli symbol	r{4,3,3} = $\left\{{\begin{array}{l}4\\3,3\end{array}}\right\}$ 2r{3,3^1,1} h₃{4,3,3}
Coxeter-Dynkin diagrams	=
Cells	24	8 (3.4.3.4) 16 (3.3.3)
Faces	88	64 {3} 24 {4}
Edges	96
Vertices	32
Vertex figure	(Elongated equilateral-triangular prism)
Symmetry group	B₄ [3,3,4], order 384 D₄ [3^1,1,1], order 192
Properties	convex, edge-transitive
Uniform index	10 11 12

Net

It has two uniform constructions, as a rectified 8-cell r{4,3,3} and a cantellated demitesseract, rr{3,3^1,1}, the second alternating with two types of tetrahedral cells.

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as tC₈.

Construction

The rectified tesseract may be constructed from the tesseract by truncating its vertices at the midpoints of its edges.

The Cartesian coordinates of the vertices of the rectified tesseract with edge length 2 is given by all permutations of:

(0,\ \pm {\sqrt {2}},\ \pm {\sqrt {2}},\ \pm {\sqrt {2}})

Images

orthographic projections
Coxeter plane	B₄	B₃ / D₄ / A₂	B₂ / D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	F₄	A₃
Graph
Dihedral symmetry	[12/3]	[4]

Wireframe	16 tetrahedral cells

Projections

In the cuboctahedron-first parallel projection of the rectified tesseract into 3-dimensional space, the image has the following layout:

The projection envelope is a cube.
A cuboctahedron is inscribed in this cube, with its vertices lying at the midpoint of the cube's edges. The cuboctahedron is the image of two of the cuboctahedral cells.
The remaining 6 cuboctahedral cells are projected to the square faces of the cube.
The 8 tetrahedral volumes lying at the triangular faces of the central cuboctahedron are the images of the 16 tetrahedral cells, two cells to each image.

Alternative names

Rit (Jonathan Bowers: for rectified tesseract)
Ambotesseract (Neil Sloane & John Horton Conway)
Rectified tesseract/Runcic tesseract (Norman W. Johnson)
- Runcic 4-hypercube/8-cell/octachoron/4-measure polytope/4-regular orthotope
- Rectified 4-hypercube/8-cell/octachoron/4-measure polytope/4-regular orthotope

Related uniform polytopes

Runcic cubic polytopes

Runcic n-cubes
n	4	5	6	7	8
[1⁺,4,3^n-2] = [3,3^n-3,1]	[1⁺,4,3²] = [3,3^1,1]	[1⁺,4,3³] = [3,3^2,1]	[1⁺,4,3⁴] = [3,3^3,1]	[1⁺,4,3⁵] = [3,3^4,1]	[1⁺,4,3⁶] = [3,3^5,1]
Runcic figure
Coxeter	=	=	=	=	=
Schläfli	h₃{4,3²}	h₃{4,3³}	h₃{4,3⁴}	h₃{4,3⁵}	h₃{4,3⁶}

Tesseract polytopes

B4 symmetry polytopes
Name	tesseract	rectified tesseract	truncated tesseract	cantellated tesseract	runcinated tesseract	bitruncated tesseract	cantitruncated tesseract	runcitruncated tesseract	omnitruncated tesseract
Coxeter diagram		=				=
Schläfli symbol	{4,3,3}	t₁{4,3,3} r{4,3,3}	t_0,1{4,3,3} t{4,3,3}	t_0,2{4,3,3} rr{4,3,3}	t_0,3{4,3,3}	t_1,2{4,3,3} 2t{4,3,3}	t_0,1,2{4,3,3} tr{4,3,3}	t_0,1,3{4,3,3}	t_0,1,2,3{4,3,3}
Schlegel diagram
B₄

Name	16-cell	rectified 16-cell	truncated 16-cell	cantellated 16-cell	runcinated 16-cell	bitruncated 16-cell	cantitruncated 16-cell	runcitruncated 16-cell	omnitruncated 16-cell
Coxeter diagram	=	=	=	=		=	=
Schläfli symbol	{3,3,4}	t₁{3,3,4} r{3,3,4}	t_0,1{3,3,4} t{3,3,4}	t_0,2{3,3,4} rr{3,3,4}	t_0,3{3,3,4}	t_1,2{3,3,4} 2t{3,3,4}	t_0,1,2{3,3,4} tr{3,3,4}	t_0,1,3{3,3,4}	t_0,1,2,3{3,3,4}
Schlegel diagram
B₄

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. (1966)
2. Convex uniform polychora based on the tesseract (8-cell) and hexadecachoron (16-cell) - Model 11, George Olshevsky.
Klitzing, Richard. "4D uniform polytopes (polychora) o4x3o3o - rit".

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 4-polytope	5-cell	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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