10-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 4-polytope (4-dimensional uniform polytope) formed as the truncation of the regular 5-cell.

There are two degrees of truncations, including a bitruncation.

Truncated 5-cell[]

Truncated 5-cell
Schlegel half-solid truncated pentachoron.png
Schlegel diagram
(tetrahedron cells visible)
Type Uniform 4-polytope
Schläfli symbol t0,1{3,3,3}
t{3,3,3}
Coxeter 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
Equilateral-triangular pyramid
Symmetry group A4, [3,3,3], order 120
Properties convex, isogonal
Uniform index 2 3 4

The truncated 5-cell, truncated pentachoron 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 of its edge length. This transforms the 5 tetrahedral cells into truncated tetrahedra, and introduces 5 new tetrahedral cells positioned near the original vertices.

Structure[]

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

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.[1]

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

Projections[]

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

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[]

Coordinates[]

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

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.

Related polytopes[]

The convex hull of the truncated 5-cell and its dual (assuming that they are congruent) is a nonuniform polychoron composed of 60 cells: 10 tetrahedra, 20 octahedra (as triangular antiprisms), 30 tetrahedra (as tetragonal disphenoids), and 40 vertices. Its vertex figure is a hexakis triangular cupola.

Bitruncatodecachoron vertex figure.png
Vertex figure

Bitruncated 5-cell[]

Bitruncated 5-cell
Schlegel half-solid bitruncated 5-cell.png
Schlegel diagram with alternate cells hidden.
Type Uniform 4-polytope
Schläfli symbol t1,2{3,3,3}
2t{3,3,3}
Coxeter diagram CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
or CDel branch 11.pngCDel 3ab.pngCDel nodes.png or CDel branch 11.pngCDel 3ab-cross.pngCDel nodes.png
Cells 10 (3.6.6) Truncated tetrahedron.png
Faces 40 20 {3}
20 {6}
Edges 60
Vertices 30
Vertex figure Bitruncated 5-cell vertex figure.png
({ }v{ })
dual polytope Disphenoidal 30-cell
Symmetry group Aut(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 4-polytope, composed of 10 cells in the shape of truncated tetrahedra.

Topologically, under its highest symmetry, [[3,3,3]], there is only one geometrical form, containing 10 uniform truncated tetrahedra. The hexagons are always regular because of the polychoron's inversion symmetry, of which the regular hexagon is the only such case among ditrigons (an isogonal hexagon with 3-fold symmetry).

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

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 a 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 -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 4-polytopes which are cell-transitive. The other is the bitruncated 24-cell, which is composed of 48 truncated cubes.

Symmetry[]

This 4-polytope has a higher extended pentachoric symmetry (2×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[]

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]] = [10] [4] [[3]] = [6]
Decachoron stereographic (hexagon).png
stereographic projection of spherical 4-polytope
(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:

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, centered on the origin are all 20 permutations of (-1,-1,0,1,1).

Related polytopes[]

The bitruncated 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.

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.svg 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
Vertex figure ( )v( ) Octahedron vertfig.png
{ }×{ }
Bitruncated 5-cell verf.png
{ }v{ }
Birectified hexateron verf.png
{3}×{3}
Tritruncated 6-simplex verf.png
{3}v{3}
{3,3}x{3,3} Quadritruncated 8-simplex verf.png
{3,3}v{3,3}
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

Related regular skew polyhedron[]

A 3D net for {6,4|3}, with pairs of yellow triangles folded together into 4D and removed

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 30 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.

Disphenoidal 30-cell[]

Disphenoidal 30-cell
Type perfect[2] polychoron
Symbol f1,2A4[2]
Coxeter CDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.png
Cells 30 congruent tetragonal disphenoids
Faces 60 congruent isosceles
  (2 short edges)
Edges 40 20 of length
20 of length
Vertices 10
Vertex figure Disphenoidal 30-cell vertex figure.png
(Triakis tetrahedron)
Dual Bitruncated 5-cell
Coxeter group Aut(A4), [[3,3,3]], order 240
Orbit vector (1, 2, 1, 1)
Properties convex, isochoric

The disphenoidal 30-cell is the dual of the bitruncated 5-cell. It is a 4-dimensional polytope (or polychoron) derived from the 5-cell. It is the convex hull of two 5-cells in opposite orientations.

Being the dual of a uniform polychoron, it is cell-transitive, consisting of 30 congruent tetragonal disphenoids. In addition, it is vertex-transitive under the group Aut(A4).

Related polytopes[]

These polytope are from a set of 9 uniform 4-polytope 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

References[]

Specific
  1. ^ Klitzing, Richard. "x3x4o3o - tip".
  2. ^ a b On Perfect 4-Polytopes Gabor Gévay Contributions to Algebra and Geometry Volume 43 (2002), No. 1, 243-259 ] Table 2, page 252
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 polychoron Pentachoron 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