Rectified tesseract

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Rectified tesseract
Schlegel half-solid rectified 8-cell.png
Schlegel diagram
Centered on cuboctahedron
tetrahedral cells shown
Type Uniform 4-polytope
Schläfli symbol r{4,3,3} =
Coxeter-Dynkin diagrams CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel nodes 11.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.png = CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Cells 24 8 (
16 (3.3.3)Tetrahedron.png
Faces 88 64 {3}
24 {4}
Edges 96
Vertices 32
Vertex figure Rectified 8-cell verf.pngCantellated demitesseract verf.png
(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

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 CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png construction, called a runcic tesseract.

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.


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:


orthographic projections
Coxeter plane B4 B3 / D4 / A2 B2 / D3
Graph 4-cube t1.svg 4-cube t1 B3.svg 4-cube t1 B2.svg
Dihedral symmetry [8] [6] [4]
Coxeter plane F4 A3
Graph 4-cube t1 F4.svg 4-cube t1 A3.svg
Dihedral symmetry [12/3] [4]
Rectified tesseract1.png
Rectified tesseract2.png
16 tetrahedral cells


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[edit]

  • 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[edit]

Runcic cubic polytopes[edit]

Tesseract polytopes[edit]


Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / E9 / E10 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds