Rectified 5-cell

Rectified 5-cell
Schlegel half-solid rectified 5-cell.png
Schlegel diagram with the 5 tetrahedral cells shown.
Type Uniform 4-polytope
Schläfli symbol t1{3,3,3} or r{3,3,3}
{32,1} =
Coxeter-Dynkin diagram CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3a.pngCDel nodea.png
Cells 10 5 {3,3} Tetrahedron.png
5 Uniform polyhedron-33-t1.png
Faces 30 {3}
Edges 30
Vertices 10
Vertex figure Rectified 5-cell verf.png
Triangular prism
Symmetry group A4, [3,3,3], order 120
Petrie Polygon Pentagon
Properties convex, isogonal, isotoxal
Uniform index 1 2 3

In four-dimensional geometry, the rectified 5-cell is a uniform 4-polytope composed of 5 regular tetrahedral and 5 regular octahedral cells. Each edge has one tetrahedron and two octahedra. Each vertex has two tetrahedra and three octahedra. In total it has 30 triangle faces, 30 edges, and 10 vertices. Each vertex is surrounded by 3 octahedra and 2 tetrahedra; the vertex figure is a triangular prism.

Topologically, under its highest symmetry, [3,3,3], there is only one geometrical form, containing 5 regular tetrahedra and 5 rectified tetrahedra (which is geometrically the same as a regular octahedron). It is also topologically identical to a tetrahedron-octahedron segmentochoron.[clarification needed]

The vertex figure of the rectified 5-cell is a uniform triangular prism, formed by three octahedra around the sides, and two tetrahedra on the opposite ends.[1]

Despite having the same number of vertices as cells (10) and the same number of edges as faces (30), the rectified 5-cell is not self-dual because the vertex figure (a uniform triangular prism) is not a dual of the polychoron's cells.

Wythoff constructionEdit

Seen in a configuration matrix, all incidence counts between elements are shown. The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time.[2]

A4         k-face fk f0 f1 f2 f3 k-figure Notes
A2A1         ( ) f0 10 6 3 6 3 2 {3}x{ } A4/A2A1 = 5!/3!/2 = 10
A1A1         { } f1 2 30 1 2 2 1 { }v( ) A4/A1A1 = 5!/2/2 = 30
A2A1         {3} f2 3 3 10 * 2 0 { } A4/A2A1 = 5!/3!/2 = 10
A2         3 3 * 20 1 1 A4/A2 = 5!/3! = 20
A3         r{3,3} f3 6 12 4 4 5 * ( ) A4/A3 = 5!/4! = 5
A3         {3,3} 4 6 0 4 * 5


Together with the simplex and 24-cell, this shape and its dual (a polytope with ten vertices and ten triangular bipyramid facets) was one of the first 2-simple 2-simplicial 4-polytopes known. This means that all of its two-dimensional faces, and all of the two-dimensional faces of its dual, are triangles. In 1997, Tom Braden found another dual pair of examples, by gluing two rectified 5-cells together; since then, infinitely many 2-simple 2-simplicial polytopes have been constructed.[3][4]

Semiregular polytopeEdit

It is one of three semiregular 4-polytope made of two or more cells which are Platonic solids, discovered by Thorold Gosset in his 1900 paper. He called it a tetroctahedric for being made of tetrahedron and octahedron cells.[5]

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as tC5.

Alternate namesEdit

  • Tetroctahedric (Thorold Gosset)
  • Dispentachoron
  • Rectified 5-cell (Norman W. Johnson)
  • Rectified 4-simplex
  • Fully truncated 4-simplex
  • Rectified pentachoron (Acronym: rap) (Jonathan Bowers)
  • Ambopentachoron (Neil Sloane & John Horton Conway)
  • (5,2)-hypersimplex (the convex hull of five-dimensional (0,1)-vectors with exactly two ones)


orthographic projections
Coxeter plane
A4 A3 A2
Dihedral symmetry [5] [4] [3]
stereographic projection
(centered on octahedron)
Net (polytope)
  Tetrahedron-centered perspective projection into 3D space, with nearest tetrahedron to the 4D viewpoint rendered in red, and the 4 surrounding octahedra in green. Cells lying on the far side of the polytope have been culled for clarity (although they can be discerned from the edge outlines). The rotation is only of the 3D projection image, in order to show its structure, not a rotation in 4D space.


The Cartesian coordinates of the vertices of an origin-centered rectified 5-cell having edge length 2 are:

More simply, the vertices of the rectified 5-cell can be positioned on a hyperplane in 5-space as permutations of (0,0,0,1,1) or (0,0,1,1,1). These construction can be seen as positive orthant facets of the rectified pentacross or birectified penteract respectively.

Related polytopesEdit

The convex hull of the rectified 5-cell and its dual (assuming that they are congruent) is a nonuniform polychoron composed of 30 cells: 10 tetrahedra, 20 octahedra (as triangular antiprisms), and 20 vertices. Its vertex figure is a triangular bifrustum.

Related 4-polytopesEdit

This polytope is the vertex figure of the 5-demicube, and the edge figure of the uniform 221 polytope.

It is also one of 9 Uniform 4-polytopes constructed from the [3,3,3] Coxeter group.

Name 5-cell truncated 5-cell rectified 5-cell cantellated 5-cell bitruncated 5-cell cantitruncated 5-cell runcinated 5-cell runcitruncated 5-cell omnitruncated 5-cell
2t{3,3,3} tr{3,3,3}
t0,3{3,3,3} t0,1,3{3,3,3}
Coxeter plane
A3 Coxeter plane
A2 Coxeter plane

Related polytopes and honeycombsEdit

The rectified 5-cell is second in a dimensional series of semiregular polytopes. Each progressive uniform polytope is constructed as the vertex figure of the previous polytope. Thorold Gosset identified this series in 1900 as containing all regular polytope facets, containing all simplexes and orthoplexes (tetrahedrons and octahedrons in the case of the rectified 5-cell). The Coxeter symbol for the rectified 5-cell is 021.

Isotopic polytopesEdit

Isotopic uniform truncated simplices
Dim. 2 3 4 5 6 7 8
t{3} = {6}
r{3,3} = {31,1} = {3,4}
2r{34} = {32,2}
3r{36} = {33,3}
Vertex figure ( )v( )  
{ }×{ }
{ }v{ }
Facets {3}   t{3,3}   r{3,3,3}   2t{3,3,3,3}   2r{3,3,3,3,3}   3t{3,3,3,3,3,3}  

See alsoEdit


  1. ^ Conway, 2008
  2. ^ Klitzing, Richard. "o3x4o3o - rap".
  3. ^ Eppstein, David; Kuperberg, Greg; Ziegler, Günter M. (2003), "Fat 4-polytopes and fatter 3-spheres", in Bezdek, Andras (ed.), Discrete Geometry: In honor of W. Kuperberg's 60th birthday, Pure and Applied Mathematics, 253, pp. 239–265, arXiv:math.CO/0204007.
  4. ^ Paffenholz, Andreas; Ziegler, Günter M. (2004), "The Et-construction for lattices, spheres and polytopes", Discrete & Computational Geometry, 32 (4): 601–621, arXiv:math.MG/0304492, doi:10.1007/s00454-004-1140-4, MR 2096750.
  5. ^ Gosset, 1900


  • T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
  • J.H. Conway and M.J.T. Guy: Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, page 38 and 39, 1965
  • 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 [1]
      • (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)
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26)

External linksEdit

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