Rectified 5-cell

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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}
Coxeter-Dynkin diagram Template:CDD
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.

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]


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.[2][3]

Semiregular polytope

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.[4]

Alternate names

  • 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)


Template:4-simplex Coxeter plane graphs

Rectified simplex stereographic.png
stereographic projection
(centered on octahedron)
Rectified 5-cell net.png
Net (polytope)
Rectified 5cell-perspective-tetrahedron-first-01.gif 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 4-polytopes

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. Template:Pentachoron family

Related polytopes and honeycombs

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). In Coxeter's notation the rectified 5-cell is given the symbol 021.

k21 figures in n dimensional
Space Finite Euclidean Hyperbolic
En 3 4 5 6 7 8 9 10
E3=A2A1 E4=A4 E5=D5 E6 E7 E8 E9 = = E8+ E10 = = E8++
Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD
Symmetry [3-1,2,1] [30,2,1] [31,2,1] [32,2,1] [33,2,1] [34,2,1] [35,2,1] [36,2,1]
Order 12 120 192 51,840 2,903,040 696,729,600
Graph Triangular prism.png 4-simplex t1.svg Demipenteract graph ortho.svg E6 graph.svg E7 graph.svg E8 graph.svg - -
Name −121 021 121 221 321 421 521 621

See also


  1. Conway, 2008
  2. {{#invoke:citation/CS1|citation |CitationClass=citation }}.
  3. {{#invoke:citation/CS1|citation |CitationClass=citation }}.
  4. 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 links