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 polychoron
Schläfli symbol t1{3,3,3}
Coxeter-Dynkin diagram CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.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 polychoron 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.

It is one of three semiregular polychora 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.

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.

Alternate names[edit]

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:

\left(\sqrt{\frac{2}{5}},\   \frac{2}{\sqrt{6}},\  \frac{2}{\sqrt{3}},\  0   \right)
\left(\sqrt{\frac{2}{5}},\   \frac{2}{\sqrt{6}},\  \frac{-1}{\sqrt{3}},\ \pm1\right)
\left(\sqrt{\frac{2}{5}},\   \frac{-2}{\sqrt{6}},\ \frac{1}{\sqrt{3}},\  \pm1\right)
\left(\sqrt{\frac{2}{5}},\   \frac{-2}{\sqrt{6}},\ \frac{-2}{\sqrt{3}},\ 0   \right)
\left(\frac{-3}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\  \frac{1}{\sqrt{3}},\  \pm1\right)
\left(\frac{-3}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\  \frac{-2}{\sqrt{3}},\ 0   \right)
\left(\frac{-3}{\sqrt{10}},\ -\sqrt{\frac{3}{2}},\ 0,\                   0   \right)

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 polychora[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 polychora 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} t0,1{3,3,3} t1{3,3,3} t0,2{3,3,3} t1,2{3,3,3} t0,1,2{3,3,3} t0,3{3,3,3} t0,1,3{3,3,3} t0,1,2,3{3,3,3}
Coxeter-Dynkin
diagram
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.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 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 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 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). 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
Coxeter
group
E3=A2×A1 E4=A4 E5=D5 E6 E7 E8 E9 = {\tilde{E}}_{8} = E8+ E10 = E8++
Coxeter
diagram
CDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png
Symmetry
(order)
[3-1,2,1]
(12)
[30,2,1]
(120)
[31,2,1]
(192)
[32,2,1]
(51,840)
[33,2,1]
(2,903,040)
[34,2,1]
(696,729,600)
[35,2,1]
(∞)
[36,2,1]
(∞)
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[edit]

References[edit]

  • T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
  • 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)

External links[edit]