Truncated 5-cell

Truncated 5-cell
Schlegel wireframe 5-cell.png
5-cell
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Schlegel half-solid truncated pentachoron.png
Truncated 5-cell
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Schlegel half-solid bitruncated 5-cell.png
Bitruncated 5-cell
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Schlegel diagrams centered on [3,3] (cells at opposite at [3,3])

In geometry, a truncated 5-cell is a uniform polychoron (4-dimensional uniform polytope) formed as the truncation of the regular 5-cell.

There are two degrees of trunctions, including a bitruncation.

Contents

Truncated 5-cell

Truncated 5-cell
Schlegel half-solid truncated pentachoron.png
Schlegel diagram
(tetrahedron cells visible)
Type Uniform polychoron
Schläfli symbol t0,1{3,3,3}
Coxeter-Dynkin diagram CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Cells 10 5 (3.3.3) Tetrahedron.png
5 (3.6.6) Truncated tetrahedron.png
Faces 30 20 {3}
10 {6}
Edges 40
Vertices 20
Vertex figure Truncated 5-cell verf.png
Irr. tetrahedron
Symmetry group A4, [3,3,3], order 120
Properties convex, isogonal
Uniform index 2 3 4

The truncated 5-cell, truncated pentatope or truncated 4-simplex is bounded by 10 cells: 5 tetrahedra, and 5 truncated tetrahedra. Each vertex is surrounded by 3 truncated tetrahedra and one tetrahedron; the vertex figure is an elongated tetrahedron.

Construction

The truncated 5-cell may be constructed from the 5-cell by truncating its vertices at 1/3 the edge length. This truncates the 5 tetrahedral cells into truncated tetrahedra, and introduces 5 new tetrahedral cells positioned on the original vertices.

Structure

The truncated tetrahedra are joined to each other at their hexagonal faces, and to the tetrahedra at their triangular faces.

Projections

The tetrahedron-first parallel projection of the truncated 5-cell into 3-dimensional space has the following structure:

  • The projection envelope is a truncated tetrahedron.
  • One of the truncated tetrahedral cells project onto the entire envelope.
  • One of the tetrahedral cells project onto a tetrahedron lying at the center of the envelope.
  • Four flattened tetrahedra are joined to the triangular faces of the envelope, and connected to the central tetrahedron via 4 radial edges. These are the images of the remaining 4 tetrahedral cells.
  • Between the central tetrahedron and the 4 hexagonal faces of the envelope are 4 irregular truncated tetrahedral volumes, which are the images of the 4 remaining trucated tetrahedral cells.

This layout of cells in projection is analogous to the layout of faces in the face-first projection of the truncated tetrahedron into 2-dimensional space. The truncated 5-cell is the 4-dimensional analogue of the truncated tetrahedron.

Images

orthographic projections
Ak
Coxeter plane
A4 A3 A2
Graph 4-simplex t01.svg 4-simplex t01 A3.svg 4-simplex t01 A2.svg
Dihedral symmetry [5] [4] [3]

Alternate names

  • Truncated pentatope
  • Truncated 4-simplex
  • Truncated pentachoron (Acronym: tip) (Jonathan Bowers)

Coordinates

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

\left( \frac{3}{\sqrt{10}},\  \sqrt{3 \over 2},\    \pm\sqrt{3},\         \pm1\right)
\left( \frac{3}{\sqrt{10}},\  \sqrt{3 \over 2},\    0,\                   \pm2\right)
\left( \frac{3}{\sqrt{10}},\  \frac{-1}{\sqrt{6}},\ \frac{2}{\sqrt{3}},\  \pm2\right)
\left( \frac{3}{\sqrt{10}},\  \frac{-1}{\sqrt{6}},\ \frac{4}{\sqrt{3}},\  0   \right)
\left( \frac{3}{\sqrt{10}},\  \frac{-5}{\sqrt{6}},\ \frac{1}{\sqrt{3}},\  \pm1\right)
\left( \frac{3}{\sqrt{10}},\  \frac{-5}{\sqrt{6}},\ \frac{-2}{\sqrt{3}},\ 0   \right)
\left( -\sqrt{2 \over 5},\    \sqrt{2 \over 3},\    \frac{2}{\sqrt{3}},\  \pm2\right)
\left( -\sqrt{2 \over 5},\    \sqrt{2 \over 3},\    \frac{-4}{\sqrt{3}},\ 0   \right)
\left( -\sqrt{2 \over 5},\    -\sqrt{6},\           0,\                   0   \right)
\left( \frac{-7}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\  \frac{1}{\sqrt{3}},\  \pm1\right)
\left( \frac{-7}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\  \frac{-2}{\sqrt{3}},\ 0   \right)
\left( \frac{-7}{\sqrt{10}},\ -\sqrt{3 \over 2},\   0,\                   0   \right)

More simply, the vertices of the truncated 5-cell can be constructed on a hyperplane in 5-space as permutations of (0,0,0,1,2) or of (0,1,2,2,2). These coordinates come from positive orthant facets of the truncated pentacross and bitruncated penteract respectively.

Bitruncated 5-cell

Schlegel half-solid bitruncated 5-cell.png
Schlegel diagram with alternate cells hidden.
Bitruncated 5-cell
Type Uniform polychoron
Schläfli symbol t1,2{3,3,3}
Coxeter-Dynkin diagram CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Cells 10 (3.6.6) Truncated tetrahedron.png
Faces 40 20 {3}
20 {6}
Edges 60
Vertices 30
Vertex figure Bitruncated 5-cell verf.png
(Tetragonal disphenoid)
Symmetry group A4, [[3,3,3]], order 240
Properties convex, isogonal isotoxal, isochoric
Uniform index 5 6 7

The bitruncated 5-cell (also called a bitruncated pentachoron, decachoron and 10-cell) is a 4-dimensional polytope, or polychoron, composed of 10 cells in the shape of truncated tetrahedra. Each hexagonal face of the truncated tetrahedra is joined in complementary orientation to the neighboring truncated tetrahedron. Each edge is shared by two hexagons and one triangle. Each vertex is surrounded by 4 truncated tetrahedral cells in a tetragonal disphenoid vertex figure.

The bitruncated 5-cell is the intersection of two pentachora in dual configuration. As such, it is also the intersection of a penteract with the hyperplane that bisects the penteract's long diagonal orthogonally. In this sense it is the 4-dimensional analog of the regular octahedron (intersection of regular tetrahedra in dual configuration / tesseract bisection on long diagonal) and the regular hexagon (equilateral triangles / cube). The 5-dimensional analog is the birectified 5-simplex, and the n-dimensional analog is the polytope whose Coxeter–Dynkin diagram is linear with rings on the middle one or two nodes.

The bitruncated 5-cell is one of the two non-regular uniform polychora which are cell-transitive. The other is the bitruncated 24-cell, which is composed of 48 truncated cubes.

Symmetry

This polychoron has a higher extended pentachoric symmetry (A4, [[3,3,3]]), doubled to order 240, because the element corresponding to any element of the underlying 5-cell can be exchanged with one of those corresponding to an element of its dual.

Alternative names

  • Bitruncated 5-cell (Norman W. Johnson)
  • 10-cell as a cell-transitive 4-polytope
  • Bitruncated pentachoron
  • Bitruncated pentatope
  • Bitruncated 4-simplex
  • Decachoron (Acronym: deca) (Jonathan Bowers)

Images

orthographic projections
Ak
Coxeter plane
A4 A3 A2
Graph 4-simplex t12.svg 4-simplex t12 A3.svg 4-simplex t12 A2.svg
Dihedral symmetry [[5]] [4] [[3]]
Decachoron stereographic (hexagon).png
stereographic projection of spherical polychoron
(centred on a hexagon face)
Bitruncated 5-cell net.png
Net (polytope)

Coordinates

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

\pm\left(\sqrt{\frac{5}{2}},\ \frac{5}{\sqrt{6}},\  \frac{2}{\sqrt{3}},\  0\right)
\pm\left(\sqrt{\frac{5}{2}},\ \frac{5}{\sqrt{6}},\  \frac{-1}{\sqrt{3}},\ \pm1\right)
\pm\left(\sqrt{\frac{5}{2}},\ \frac{1}{\sqrt{6}},\  \frac{4}{\sqrt{3}},\  0\right)
\pm\left(\sqrt{\frac{5}{2}},\ \frac{1}{\sqrt{6}},\  \frac{-2}{\sqrt{3}},\ \pm2\right)
\pm\left(\sqrt{\frac{5}{2}},\ -\sqrt{\frac{3}{2}},\ \pm\sqrt{3},\         \pm1\right)
\pm\left(\sqrt{\frac{5}{2}},\ -\sqrt{\frac{3}{2}},\ 0,\                   \pm2\right)
\pm\left(0,\                  2\sqrt{\frac{2}{3}},\ \frac{4}{\sqrt{3}},\  0\right)
\pm\left(0,\                  2\sqrt{\frac{2}{3}},\ \frac{-2}{\sqrt{3}},\ \pm2\right)

More simply, the vertices of the bitruncated 5-cell can be constructed on a hyperplane in 5-space as permutations of (0,0,1,2,2). These represent positive orthant facets of the bitruncated pentacross.

Another 5-space construction are all 20 permutations of (1,0,0,0,-1).

Related regular skew polyhedron

The regular skew polyhedron, {6,4|3}, exists in 4-space with 4 hexagonal around each vertex, in a zig-zagging nonplanar vertex figure. These hexagonal faces can be seen on the bitruncated 5-cell, using all 60 edges and 20 vertices. The 20 triangular faces of the bitruncated 5-cell can be seen as removed. The dual regular skew polyhedron, {4,6|3}, is similarly related to the square faces of the runcinated 5-cell.

Related polytopes

These polytope are from a set 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

References

  • H.S.M. Coxeter:
    • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
    • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied 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]
  • Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, ISBN 0-486-40919-8 p.88 (Chapter 5: Regular Skew Polyhedra in three and four dimensions and their topological analogues, Proceedings of the London Mathematics Society, Ser. 2, Vol 43, 1937.)
    • Coxeter, H. S. M. Regular Skew Polyhedra in Three and Four Dimensions. Proc. London Math. Soc. 43, 33-62, 1937.
  • Norman Johnson Uniform Polytopes, Manuscript (1991)
    • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
  • Olshevsky, George, Pentachoron at Glossary for Hyperspace.
  • Richard Klitzing, 4D, uniform polytopes (polychora) x3x3o3o - tip, o3x3x3o - deca

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