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} 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 3.3.3.3 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]

Structure[edit]

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]

The birectified 5-cell can be seen as the intersection of two regular 5-cells in dual positions. CDel branch 11.pngCDel 3ab.pngCDel nodes.png = CDel branch.pngCDel 3ab.pngCDel nodes 10l.pngCDel branch.pngCDel 3ab.pngCDel nodes 01l.png.

Semiregular polytope[edit]

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]

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

Alternate names[edit]

  • 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)

Images[edit]

orthographic projections
Ak
Coxeter plane
A4 A3 A2
Graph 4-simplex t1.svg 4-simplex t1 A3.svg 4-simplex t1 A2.svg
Dihedral symmetry [5] [4] [3]
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.

Coordinates[edit]

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[edit]

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
Schläfli
symbol
{3,3,3}
3r{3,3,3}
t{3,3,3}
2t{3,3,3}
r{3,3,3}
2r{3,3,3}
rr{3,3,3}
r2r{3,3,3}
2t{3,3,3} tr{3,3,3}
t2r{3,3,3}
t0,3{3,3,3} t0,1,3{3,3,3}
t0,2,3{3,3,3}
t0,1,2,3{3,3,3}
Coxeter
diagram
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Schlegel
diagram
Schlegel wireframe 5-cell.png Schlegel half-solid truncated pentachoron.png Schlegel half-solid rectified 5-cell.png Schlegel half-solid cantellated 5-cell.png Schlegel half-solid bitruncated 5-cell.png Schlegel half-solid cantitruncated 5-cell.png Schlegel half-solid runcinated 5-cell.png Schlegel half-solid runcitruncated 5-cell.png Schlegel half-solid omnitruncated 5-cell.png
A4
Coxeter plane
Graph
4-simplex t0.svg 4-simplex t01.svg 4-simplex t1.svg 4-simplex t02.svg 4-simplex t12.svg 4-simplex t012.svg 4-simplex t03.svg 4-simplex t013.svg 4-simplex t0123.svg
A3 Coxeter plane
Graph
4-simplex t0 A3.svg 4-simplex t01 A3.svg 4-simplex t1 A3.svg 4-simplex t02 A3.svg 4-simplex t12 A3.svg 4-simplex t012 A3.svg 4-simplex t03 A3.svg 4-simplex t013 A3.svg 4-simplex t0123 A3.svg
A2 Coxeter plane
Graph
4-simplex t0 A2.svg 4-simplex t01 A2.svg 4-simplex t1 A2.svg 4-simplex t02 A2.svg 4-simplex t12 A2.svg 4-simplex t012 A2.svg 4-simplex t03 A2.svg 4-simplex t013 A2.svg 4-simplex t0123 A2.svg

Related polytopes and honeycombs[edit]

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.

Isotopics polytopes[edit]

Isotopic uniform truncated simplices
Dim. 2 3 4 5 6 7 8
Name
Coxeter
Hexagon
CDel branch 11.png = CDel node 1.pngCDel 6.pngCDel node.png
t{3} = {6}
Octahedron
CDel node 1.pngCDel split1.pngCDel nodes.png = CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
r{3,3} = {31,1} = {3,4}
Decachoron
CDel branch 11.pngCDel 3ab.pngCDel nodes.png
2t{33}
Dodecateron
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png
2r{34} = {32,2}
Tetradecapeton
CDel branch 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png
3t{35}
Hexadecaexon
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png
3r{36} = {33,3}
Octadecazetton
CDel branch 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png
4t{37}
Images Truncated triangle.png 3-cube t2.svgUniform polyhedron-33-t1.png 4-simplex t12.svgSchlegel half-solid bitruncated 5-cell.png 5-simplex t2.svg5-simplex t2 A4.svg 6-simplex t23.svg6-simplex t23 A5.svg 7-simplex t3.svg7-simplex t3 A5.svg 8-simplex t34.svg8-simplex t34 A7.svg
Facets {3} Regular polygon 3 annotated.svg t{3,3} Uniform polyhedron-33-t01.png r{3,3,3} Schlegel half-solid rectified 5-cell.png 2t{3,3,3,3} 5-simplex t12.svg 2r{3,3,3,3,3} 6-simplex t2.svg 3t{3,3,3,3,3,3} 7-simplex t23.svg
As
intersecting
dual
simplexes
Regular hexagon as intersection of two triangles.png
CDel branch 10.pngCDel branch 01.png
Stellated octahedron A4 A5 skew.png
CDel node.pngCDel split1.pngCDel nodes 10lu.pngCDel node.pngCDel split1.pngCDel nodes 01ld.png
Compound dual 5-cells and bitruncated 5-cell intersection A4 coxeter plane.png
CDel branch.pngCDel 3ab.pngCDel nodes 10l.pngCDel branch.pngCDel 3ab.pngCDel nodes 01l.png
Dual 5-simplex intersection graph a5.pngDual 5-simplex intersection graph a4.png
CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10l.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 01l.png
CDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10l.pngCDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 01l.png CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10l.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 01l.png CDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10l.pngCDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 01l.png

See also[edit]

Notes[edit]

  1. ^ Conway, 2008
  2. ^ Eppstein, David; Kuperberg, Greg; Ziegler, Günter M. (2003), "Fat 4-polytopes and fatter 3-spheres", in Bezdek, Andras, Discrete Geometry: In honor of W. Kuperberg's 60th birthday, Pure and Applied Mathematics 253, Marcel Dekker, pp. 239–265, arXiv:math.CO/0204007 .
  3. ^ 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 .
  4. ^ Gosset, 1900

References[edit]

  • 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[edit]

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 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