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} =
2r{3,31,1}
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 [31,1,1], 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{3,31,1}, 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:

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

Runcic cubic polytopes
Runcic n-cubes
n45678
[1+,4,3n-2]
= [3,3n-3,1]		 [1+,4,32]
= [3,31,1]		 [1+,4,33]
= [3,32,1]		 [1+,4,34]
= [3,33,1]		 [1+,4,35]
= [3,34,1]		 [1+,4,36]
= [3,35,1]
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}		 t0,1{4,3,3}
t{4,3,3}		 t0,2{4,3,3}
rr{4,3,3}		 t0,3{4,3,3} t1,2{4,3,3}
2t{4,3,3}		 t0,1,2{4,3,3}
tr{4,3,3}		 t0,1,3{4,3,3} t0,1,2,3{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}		 t0,1{3,3,4}
t{3,3,4}		 t0,2{3,3,4}
rr{3,3,4}		 t0,3{3,3,4} t1,2{3,3,4}
2t{3,3,4}		 t0,1,2{3,3,4}
tr{3,3,4}		 t0,1,3{3,3,4} t0,1,2,3{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 An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
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 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
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