Runcinated 5-cell

Orthogonal projections in A₄ Coxeter plane
File:4-simplex t0.svg 5-cell File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	File:4-simplex t03.svg Runcinated 5-cell File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png
File:4-simplex t013.svg Runcitruncated 5-cell File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png	File:4-simplex t0123.svg Omnitruncated 5-cell (Runcicantitruncated 5-cell) File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png

In four-dimensional geometry, a runcinated 5-cell is a convex uniform 4-polytope, being a runcination (a 3rd order truncation, up to face-planing) of the regular 5-cell.

There are 3 unique degrees of runcinations of the 5-cell, including with permutations, truncations, and cantellations.

Runcinated 5-cell

Runcinated 5-cell
File:Schlegel half-solid runcinated 5-cell.png Schlegel diagram with half of the tetrahedral cells visible.
Type	Uniform 4-polytope
Schläfli symbol	t_0,3{3,3,3}
Coxeter diagram	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png
Cells	30	10 (3.3.3) File:Tetrahedron.png 20 (3.4.4) File:Triangular prism.png
Faces	70	40 {3} 30 {4}
Edges	60
Vertices	20
Vertex figure	File:Runcinated 5-cell verf.svg (Elongated equilateral-triangular antiprism)
Symmetry group	Aut(A₄), [[3,3,3]], order 240
Properties	convex, isogonal isotoxal
Uniform index	4 5 6

The runcinated 5-cell or small prismatodecachoron is constructed by expanding the cells of a 5-cell radially and filling in the gaps with triangular prisms (which are the face prisms and edge figures) and tetrahedra (cells of the dual 5-cell). It consists of 10 tetrahedra and 20 triangular prisms. The 10 tetrahedra correspond with the cells of a 5-cell and its dual.

Topologically, under its highest symmetry, [[3,3,3]], there is only one geometrical form, containing 10 tetrahedra and 20 uniform triangular prisms. The rectangles are always squares because the two pairs of edges correspond to the edges of the two sets of 5 regular tetrahedra each in dual orientation, which are made equal under extended symmetry.

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

Alternative names

Runcinated 5-cell (Norman Johnson)
Runcinated pentachoron
Runcinated 4-simplex
Expanded 5-cell/4-simplex/pentachoron
Small prismatodecachoron (Acronym: Spid) (Jonathan Bowers)

Structure

Two of the ten tetrahedral cells meet at each vertex. The triangular prisms lie between them, joined to them by their triangular faces and to each other by their square faces. Each triangular prism is joined to its neighbouring triangular prisms in anti orientation (i.e., if edges A and B in the shared square face are joined to the triangular faces of one prism, then it is the other two edges that are joined to the triangular faces of the other prism); thus each pair of adjacent prisms, if rotated into the same hyperplane, would form a gyrobifastigium.

Configuration

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]

File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png	f_k	f₀	f₁		f₂			f₃
File:CDel node x.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node x.png	f₀	20	3	3	3	6	3	1	3	3	1
File:CDel node 1.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node x.png	f₁	2	30	*	2	2	0	1	2	1	0
File:CDel node x.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node 1.png	f₁	2	*	30	0	2	2	0	1	2	1
File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node x.png	f₂	3	3	0	20	*	*	1	1	0	0
File:CDel node 1.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node 1.png		4	2	2	*	30	*	0	1	1	0
File:CDel node x.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png		3	0	3	*	*	20	0	0	1	1
File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node x.png	f₃	4	6	0	4	0	0	5	*	*	*
File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node 1.png		6	6	3	2	3	0	*	10	*	*
File:CDel node 1.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png		6	3	6	0	3	2	*	*	10	*
File:CDel node x.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png		4	0	6	0	0	4	*	*	*	5

Dissection

The runcinated 5-cell can be dissected by a central cuboctahedron into two tetrahedral cupola. This dissection is analogous to the 3D cuboctahedron being dissected by a central hexagon into two triangular cupola.

File:4D Tetrahedral Cupola-perspective-cuboctahedron-first.png

Images

orthographic projections
A_k Coxeter plane	A₄	A₃	A₂
Graph	File:4-simplex t03.svg	File:4-simplex t03 A3.svg	File:4-simplex t03 A2.svg
Dihedral symmetry	[[5]] = [10]	[4]	[[3]] = [6]

File:Runcinated pentatope.png View inside of a 3-sphere projection Schlegel diagram with its 10 tetrahedral cells	File:Small prismatodecachoron net.png Net

Coordinates

The Cartesian coordinates of the vertices of an origin-centered runcinated 5-cell with edge length 2 are:

$\pm (\sqrt{\frac{5}{2}}, \frac{1}{\sqrt{6}}, \frac{1}{\sqrt{3}}, \pm 1)$ $\pm (\sqrt{\frac{5}{2}}, \frac{1}{\sqrt{6}}, \frac{- 2}{\sqrt{3}}, 0)$ $\pm (\sqrt{\frac{5}{2}}, - \sqrt{\frac{3}{2}}, 0, 0)$	$\pm (0, 2 \sqrt{\frac{2}{3}}, \frac{1}{\sqrt{3}}, \pm 1)$ $\pm (0, 2 \sqrt{\frac{2}{3}}, \frac{- 2}{\sqrt{3}}, 0)$ $(0, 0, \pm \sqrt{3}, \pm 1)$ $(0, 0, 0, \pm 2)$

An alternate simpler set of coordinates can be made in 5-space, as 20 permutations of:

(0,1,1,1,2)

This construction exists as one of 32 orthant facets of the runcinated 5-orthoplex.

A second construction in 5-space, from the center of a rectified 5-orthoplex is given by coordinate permutations of:

(1,-1,0,0,0)

Root vectors

Its 20 vertices represent the root vectors of the simple Lie group A₄. It is also the vertex figure for the 5-cell honeycomb in 4-space.

Cross-sections

The maximal cross-section of the runcinated 5-cell with a 3-dimensional hyperplane is a cuboctahedron. This cross-section divides the runcinated 5-cell into two tetrahedral hypercupolae consisting of 5 tetrahedra and 10 triangular prisms each.

Projections

The tetrahedron-first orthographic projection of the runcinated 5-cell into 3-dimensional space has a cuboctahedral envelope. The structure of this projection is as follows:

The cuboctahedral envelope is divided internally as follows:

Four flattened tetrahedra join 4 of the triangular faces of the cuboctahedron to a central tetrahedron. These are the images of 5 of the tetrahedral cells.
The 6 square faces of the cuboctahedron are joined to the edges of the central tetrahedron via distorted triangular prisms. These are the images of 6 of the triangular prism cells.
The other 4 triangular faces are joined to the central tetrahedron via 4 triangular prisms (distorted by projection). These are the images of another 4 of the triangular prism cells.
This accounts for half of the runcinated 5-cell (5 tetrahedra and 10 triangular prisms), which may be thought of as the 'northern hemisphere'.

The other half, the 'southern hemisphere', corresponds to an isomorphic division of the cuboctahedron in dual orientation, in which the central tetrahedron is dual to the one in the first half. The triangular faces of the cuboctahedron join the triangular prisms in one hemisphere to the flattened tetrahedra in the other hemisphere, and vice versa. Thus, the southern hemisphere contains another 5 tetrahedra and another 10 triangular prisms, making the total of 10 tetrahedra and 20 triangular prisms.

Related skew polyhedron

The regular skew polyhedron, {4,6|3}, exists in 4-space with 6 squares around each vertex, in a zig-zagging nonplanar vertex figure. These square faces can be seen on the runcinated 5-cell, using all 60 edges and 20 vertices. The 40 triangular faces of the runcinated 5-cell can be seen as removed. The dual regular skew polyhedron, {6,4|3}, is similarly related to the hexagonal faces of the bitruncated 5-cell.

Runcitruncated 5-cell

Runcitruncated 5-cell
File:Schlegel half-solid runcitruncated 5-cell.png Schlegel diagram with cuboctahedral cells shown
Type	Uniform 4-polytope
Schläfli symbol	t_0,1,3{3,3,3}
Coxeter diagram	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png
Cells	30	5 File:Truncated tetrahedron.png(3.6.6) 10 File:Hexagonal prism.png(4.4.6) 10 File:Triangular prism.png(3.4.4) 5 File:Cuboctahedron.png(3.4.3.4)
Faces	120	40 {3} 60 {4} 20 {6}
Edges	150
Vertices	60
Vertex figure	File:Runcitruncated 5-cell verf.png (Rectangular pyramid)
Coxeter group	A₄, [3,3,3], order 120
Properties	convex, isogonal
Uniform index	7 8 9

File:Prismatorhombated pentachoron net.png

Net

The runcitruncated 5-cell or prismatorhombated pentachoron is composed of 60 vertices, 150 edges, 120 faces, and 30 cells. The cells are: 5 truncated tetrahedra, 10 hexagonal prisms, 10 triangular prisms, and 5 cuboctahedra. Each vertex is surrounded by five cells: one truncated tetrahedron, two hexagonal prisms, one triangular prism, and one cuboctahedron; the vertex figure is a rectangular pyramid.

Alternative names

Runcitruncated pentachoron
Runcitruncated 4-simplex
Diprismatodispentachoron
Prismatorhombated pentachoron (Acronym: prip) (Jonathan Bowers)

Configuration

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.^[2]

File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png	f_k	f₀	f₁			f₂					f₃
File:CDel node x.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node x.png	f₀	60	1	2	2	2	2	1	2	1	1	2	1	1
File:CDel node 1.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node x.png	f₁	2	30	*	*	2	2	0	0	0	1	2	1	0
File:CDel node x.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node x.png		2	*	60	*	1	0	1	1	0	1	1	0	1
File:CDel node x.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node 1.png		2	*	*	60	0	1	0	1	1	0	1	1	1
File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node x.png	f₂	6	3	3	0	20	*	*	*	*	1	1	0	0
File:CDel node 1.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node 1.png		4	2	0	2	*	30	*	*	*	0	1	1	0
File:CDel node x.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node x.png		3	0	3	0	*	*	20	*	*	1	0	0	1
File:CDel node x.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node 1.png		4	0	2	2	*	*	*	30	*	0	1	0	1
File:CDel node x.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png		3	0	0	3	*	*	*	*	20	0	0	1	1
File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node x.png	f₃	12	6	12	0	4	0	4	0	0	5	*	*	*
File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node 1.png		12	6	6	6	2	3	0	3	0	*	10	*	*
File:CDel node 1.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png		6	3	0	6	0	3	0	0	2	*	*	10	*
File:CDel node x.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png		12	0	12	12	0	0	4	6	4	*	*	*	5

Images

orthographic projections
A_k Coxeter plane	A₄	A₃	A₂
Graph	File:4-simplex t013.svg	File:4-simplex t013 A3.svg	File:4-simplex t013 A2.svg
Dihedral symmetry	[5]	[4]	[3]

File:Runcitruncated 5cell.png Schlegel diagram with its 40 blue triangular faces and its 60 green quad faces.	File:Runcitruncated 5cell part.png Central part of Schlegel diagram.

Coordinates

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

Coordinates
$(\frac{7}{\sqrt{10}}, \sqrt{\frac{3}{2}}, \pm \sqrt{3}, \pm 1)$ $(\frac{7}{\sqrt{10}}, \sqrt{\frac{3}{2}}, 0, \pm 2)$ $(\frac{7}{\sqrt{10}}, \frac{- 1}{\sqrt{6}}, \frac{2}{\sqrt{3}}, \pm 2)$ $(\frac{7}{\sqrt{10}}, \frac{- 1}{\sqrt{6}}, \frac{- 4}{\sqrt{3}}, 0)$ $(\frac{7}{\sqrt{10}}, \frac{- 5}{\sqrt{6}}, \frac{1}{\sqrt{3}}, \pm 1)$ $(\frac{7}{\sqrt{10}}, \frac{- 5}{\sqrt{6}}, \frac{- 2}{\sqrt{3}}, 0)$ $(\sqrt{\frac{2}{5}}, \pm \sqrt{6}, \pm \sqrt{3}, \pm 1)$ $(\sqrt{\frac{2}{5}}, \pm \sqrt{6}, 0, \pm 2)$ $(\sqrt{\frac{2}{5}}, \sqrt{\frac{2}{3}}, \frac{5}{\sqrt{3}}, \pm 1)$	$(\sqrt{\frac{2}{5}}, \sqrt{\frac{2}{3}}, \frac{- 1}{\sqrt{3}}, \pm 3)$ $(\sqrt{\frac{2}{5}}, \sqrt{\frac{2}{3}}, \frac{- 4}{\sqrt{3}}, \pm 2)$ $(\sqrt{\frac{2}{5}}, - \sqrt{\frac{2}{3}}, \frac{4}{\sqrt{3}}, \pm 2)$ $(\sqrt{\frac{2}{5}}, - \sqrt{\frac{2}{3}}, \frac{1}{\sqrt{3}}, \pm 3)$ $(\sqrt{\frac{2}{5}}, - \sqrt{\frac{2}{3}}, \frac{- 5}{\sqrt{3}}, \pm 1)$ $(\frac{- 3}{\sqrt{10}}, \frac{5}{\sqrt{6}}, \frac{2}{\sqrt{3}}, \pm 2)$ $(\frac{- 3}{\sqrt{10}}, \frac{5}{\sqrt{6}}, \frac{- 4}{\sqrt{3}}, 0)$ $(\frac{- 3}{\sqrt{10}}, \frac{1}{\sqrt{6}}, \frac{4}{\sqrt{3}}, \pm 2)$ $(\frac{- 3}{\sqrt{10}}, \frac{1}{\sqrt{6}}, \frac{1}{\sqrt{3}}, \pm 3)$	$(\frac{- 3}{\sqrt{10}}, \frac{1}{\sqrt{6}}, \frac{- 5}{\sqrt{3}}, \pm 1)$ $(\frac{- 3}{\sqrt{10}}, \frac{- 7}{\sqrt{6}}, \frac{2}{\sqrt{3}}, 0)$ $(\frac{- 3}{\sqrt{10}}, \frac{- 7}{\sqrt{6}}, \frac{- 1}{\sqrt{3}}, \pm 1)$ $(- 4 \sqrt{\frac{2}{5}}, 2 \sqrt{\frac{2}{3}}, \frac{1}{\sqrt{3}}, \pm 1)$ $(- 4 \sqrt{\frac{2}{5}}, 2 \sqrt{\frac{2}{3}}, \frac{- 2}{\sqrt{3}}, 0)$ $(- 4 \sqrt{\frac{2}{5}}, 0, \pm \sqrt{3}, \pm 1)$ $(- 4 \sqrt{\frac{2}{5}}, 0, 0, \pm 2)$ $(- 4 \sqrt{\frac{2}{5}}, - 2 \sqrt{\frac{2}{3}}, \frac{2}{\sqrt{3}}, 0)$ $(- 4 \sqrt{\frac{2}{5}}, - 2 \sqrt{\frac{2}{3}}, \frac{- 1}{\sqrt{3}}, 1)$

The vertices can be more simply constructed on a hyperplane in 5-space, as the permutations of:

(0,1,1,2,3)

This construction is from the positive orthant facet of the runcitruncated 5-orthoplex.

Omnitruncated 5-cell

Omnitruncated 5-cell
File:Schlegel half-solid omnitruncated 5-cell.png Schlegel diagram with half of the truncated octahedral cells shown.
Type	Uniform 4-polytope
Schläfli symbol	t_0,1,2,3{3,3,3}
Coxeter diagram	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png
Cells	30	10 File:Truncated octahedron.png(4.6.6) 20 File:Hexagonal prism.png(4.4.6)
Faces	150	90{4} 60{6}
Edges	240
Vertices	120
Vertex figure	File:Omnitruncated 5-cell vertex figure.png File:Omnitruncated 5-cell verf.png Phyllic disphenoid
Coxeter group	Aut(A₄), [[3,3,3]], order 240
Properties	convex, isogonal, zonotope
Uniform index	8 9 10

The omnitruncated 5-cell or great prismatodecachoron is composed of 120 vertices, 240 edges, 150 faces (90 squares and 60 hexagons), and 30 cells. The cells are: 10 truncated octahedra, and 20 hexagonal prisms. Each vertex is surrounded by four cells: two truncated octahedra, and two hexagonal prisms, arranged in two phyllic disphenoidal vertex figures.

Coxeter calls this Hinton's polytope after C. H. Hinton, who described it in his book The Fourth Dimension in 1906. It forms a uniform honeycomb which Coxeter calls Hinton's honeycomb.^[3]

Alternative names

Omnitruncated 5-cell
Omnitruncated pentachoron
Omnitruncated 4-simplex
Great prismatodecachoron (Acronym: gippid) (Jonathan Bowers)
Hinton's polytope (Coxeter)

Configuration

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.^[4]

File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png	f_k	f₀	f₁				f₂						f₃
File:CDel node x.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node x.png	f₀	120	1	1	1	1	1	1	1	1	1	1	1	1	1	1
File:CDel node 1.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node x.png	f₁	2	60	*	*	*	1	1	1	0	0	0	1	1	1	0
File:CDel node x.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node x.png		2	*	60	*	*	1	0	0	1	1	0	1	1	0	1
File:CDel node x.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel 2.pngFile:CDel node x.png		2	*	*	60	*	0	1	0	1	0	1	1	0	1	1
File:CDel node x.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node 1.png		2	*	*	*	60	0	0	1	0	1	1	0	1	1	1
File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node x.png	f₂	6	3	3	0	0	20	*	*	*	*	*	1	1	0	0
File:CDel node 1.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel 2.pngFile:CDel node x.png		4	2	0	2	0	*	30	*	*	*	*	1	0	1	0
File:CDel node 1.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node 1.png		4	2	0	0	2	*	*	30	*	*	*	0	1	1	0
File:CDel node x.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 2.pngFile:CDel node x.png		6	0	3	3	0	*	*	*	20	*	*	1	0	0	1
File:CDel node x.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node 1.png		4	0	2	0	2	*	*	*	*	30	*	0	1	0	1
File:CDel node x.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png		6	0	0	3	3	*	*	*	*	*	20	0	0	1	1
File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 2.pngFile:CDel node x.png	f₃	24	12	12	12	0	4	6	0	4	0	0	5	*	*	*
File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node 1.png		12	6	6	0	6	2	0	3	0	3	0	*	10	*	*
File:CDel node 1.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png		12	6	0	6	6	0	3	3	0	0	2	*	*	10	*
File:CDel node x.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png		24	0	12	12	12	0	0	0	4	6	4	*	*	*	5

Images

orthographic projections
A_k Coxeter plane	A₄	A₃	A₂
Graph	File:4-simplex t0123.svg	File:4-simplex t0123 A3.svg	File:4-simplex t0123 A2.svg
Dihedral symmetry	[[5]] = [10]	[4]	[[3]] = [6]

Net
File:Great prismatodecachoron net.png Omnitruncated 5-cell	File:Dual gippid net.png Dual to omnitruncated 5-cell

Perspective projections


File:Omnitruncated 5-cell.png Perspective Schlegel diagram Centered on truncated octahedron	File:Omnitruncated simplex stereographic.png Stereographic projection

Permutohedron

Just as the truncated octahedron is the permutohedron of order 4, the omnitruncated 5-cell is the permutohedron of order 5.^[5] The omnitruncated 5-cell is a zonotope, the Minkowski sum of five line segments parallel to the five lines through the origin and the five vertices of the 5-cell.

File:Omnitruncated 5Cell as Permutohedron.svg

Orthogonal projection as a permutohedron

Tessellations

The omnitruncated 5-cell honeycomb can tessellate 4-dimensional space by translational copies of this cell, each with 3 hypercells around each face. This honeycomb's Coxeter diagram is File:CDel branch 11.pngFile:CDel 3ab.pngFile:CDel nodes 11.pngFile:CDel split2.pngFile:CDel node 1.png.^[6] Unlike the analogous honeycomb in three dimensions, the bitruncated cubic honeycomb which has three different Coxeter group Wythoff constructions, this honeycomb has only one such construction.^[3]

Symmetry

The omnitruncated 5-cell has extended pentachoric symmetry, [[3,3,3]], order 240. The vertex figure of the omnitruncated 5-cell represents the Goursat tetrahedron of the [3,3,3] Coxeter group. The extended symmetry comes from a 2-fold rotation across the middle order-3 branch, and is represented more explicitly as [2⁺[3,3,3]].

File:Omnitruncated 5-cell vertex figure.png

Coordinates

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

$(\pm \sqrt{10}, \pm \sqrt{6}, \pm \sqrt{3}, \pm 1)$ $(\pm \sqrt{10}, \pm \sqrt{6}, 0, \pm 2)$ $\pm (\pm \sqrt{10}, \sqrt{\frac{2}{3}}, \frac{5}{\sqrt{3}}, \pm 1)$ $\pm (\pm \sqrt{10}, \sqrt{\frac{2}{3}}, \frac{- 1}{\sqrt{3}}, \pm 3)$ $\pm (\pm \sqrt{10}, \sqrt{\frac{2}{3}}, \frac{- 4}{\sqrt{3}}, \pm 2)$ $(\sqrt{\frac{5}{2}}, 3 \sqrt{\frac{3}{2}}, \pm \sqrt{3}, \pm 1)$ $(- \sqrt{\frac{5}{2}}, - 3 \sqrt{\frac{3}{2}}, \pm \sqrt{3}, \pm 1)$ $(\sqrt{\frac{5}{2}}, 3 \sqrt{\frac{3}{2}}, 0, \pm 2)$ $(- \sqrt{\frac{5}{2}}, - 3 \sqrt{\frac{3}{2}}, 0, \pm 2)$	$\pm (\sqrt{\frac{5}{2}}, \frac{1}{\sqrt{6}}, \frac{7}{\sqrt{3}}, \pm 1)$ $\pm (\sqrt{\frac{5}{2}}, \frac{1}{\sqrt{6}}, \frac{- 2}{\sqrt{3}}, \pm 4)$ $\pm (\sqrt{\frac{5}{2}}, \frac{1}{\sqrt{6}}, \frac{- 5}{\sqrt{3}}, \pm 3)$ $\pm (\sqrt{\frac{5}{2}}, - \sqrt{\frac{3}{2}}, \pm 2 \sqrt{3}, \pm 2)$ $\pm (\sqrt{\frac{5}{2}}, - \sqrt{\frac{3}{2}}, 0, \pm 4)$ $\pm (\sqrt{\frac{5}{2}}, \frac{- 7}{\sqrt{6}}, \frac{5}{\sqrt{3}}, \pm 1)$ $\pm (\sqrt{\frac{5}{2}}, \frac{- 7}{\sqrt{6}}, \frac{- 1}{\sqrt{3}}, \pm 3)$	$\pm (\sqrt{\frac{5}{2}}, \frac{- 7}{\sqrt{6}}, \frac{- 4}{\sqrt{3}}, \pm 2)$ $\pm (0, 4 \sqrt{\frac{2}{3}}, \frac{5}{\sqrt{3}}, \pm 1)$ $\pm (0, 4 \sqrt{\frac{2}{3}}, \frac{- 1}{\sqrt{3}}, \pm 3)$ $\pm (0, 4 \sqrt{\frac{2}{3}}, \frac{- 4}{\sqrt{3}}, \pm 2)$ $\pm (0, 2 \sqrt{\frac{2}{3}}, \frac{7}{\sqrt{3}}, \pm 1)$ $\pm (0, 2 \sqrt{\frac{2}{3}}, \frac{- 2}{\sqrt{3}}, \pm 4)$ $\pm (0, 2 \sqrt{\frac{2}{3}}, \frac{- 5}{\sqrt{3}}, \pm 3)$

These vertices can be more simply obtained in 5-space as the 120 permutations of (0,1,2,3,4). This construction is from the positive orthant facet of the runcicantitruncated 5-orthoplex, t_0,1,2,3{3,3,3,4}, File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png.

Related polytopes

Nonuniform variants with [3,3,3] symmetry and two types of truncated octahedra can be doubled by placing the two types of truncated octahedra on each other to produce a nonuniform polychoron with 10 truncated octahedra, two types of 40 hexagonal prisms (20 ditrigonal prisms and 20 ditrigonal trapezoprisms), two kinds of 90 rectangular trapezoprisms (30 with D_2d symmetry and 60 with C_2v symmetry), and 240 vertices. Its vertex figure is an irregular triangular bipyramid.

File:Biomnitruncatodecachoron vertex figure.png
Vertex figure

This polychoron can then be alternated to produce another nonuniform polychoron with 10 icosahedra, two types of 40 octahedra (20 with S₆ symmetry and 20 with D₃ symmetry), three kinds of 210 tetrahedra (30 tetragonal disphenoids, 60 phyllic disphenoids, and 120 irregular tetrahedra), and 120 vertices. It has a symmetry of [[3,3,3]⁺], order 120.

File:Alternated biomnitruncatodecachoron vertex figure.png
Vertex figure

Full snub 5-cell

File:Snub 5-cell verf.png

Vertex figure for the omnisnub 5-cell

The full snub 5-cell or omnisnub 5-cell, defined as an alternation of the omnitruncated 5-cell, cannot be made uniform, but it can be given Coxeter diagram File:CDel branch hh.pngFile:CDel 3ab.pngFile:CDel nodes hh.png, and symmetry [[3,3,3]]⁺, order 120, and constructed from 90 cells: 10 icosahedrons, 20 octahedrons, and 60 tetrahedrons filling the gaps at the deleted vertices. It has 300 faces (triangles), 270 edges, and 60 vertices.

Topologically, under its highest symmetry, [[3,3,3]]⁺, the 10 icosahedra have T (chiral tetrahedral) symmetry, while the 20 octahedra have D₃ symmetry and the 60 tetrahedra have C₂ symmetry.^[7]

Related polytopes

These polytopes are a part of a family 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} 3t{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}	t_0,3{3,3,3}	t_0,1,3{3,3,3} t_0,2,3{3,3,3}	t_0,1,2,3{3,3,3}
Coxeter diagram	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png	File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png	File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png
Schlegel diagram	File:Schlegel wireframe 5-cell.png	File:Schlegel half-solid truncated pentachoron.png	File:Schlegel half-solid rectified 5-cell.png	File:Schlegel half-solid cantellated 5-cell.png	File:Schlegel half-solid bitruncated 5-cell.png	File:Schlegel half-solid cantitruncated 5-cell.png	File:Schlegel half-solid runcinated 5-cell.png	File:Schlegel half-solid runcitruncated 5-cell.png	File:Schlegel half-solid omnitruncated 5-cell.png
A₄ Coxeter plane Graph	File:4-simplex t0.svg	File:4-simplex t01.svg	File:4-simplex t1.svg	File:4-simplex t02.svg	File:4-simplex t12.svg	File:4-simplex t012.svg	File:4-simplex t03.svg	File:4-simplex t013.svg	File:4-simplex t0123.svg
A₃ Coxeter plane Graph	File:4-simplex t0 A3.svg	File:4-simplex t01 A3.svg	File:4-simplex t1 A3.svg	File:4-simplex t02 A3.svg	File:4-simplex t12 A3.svg	File:4-simplex t012 A3.svg	File:4-simplex t03 A3.svg	File:4-simplex t013 A3.svg	File:4-simplex t0123 A3.svg
A₂ Coxeter plane Graph	File:4-simplex t0 A2.svg	File:4-simplex t01 A2.svg	File:4-simplex t1 A2.svg	File:4-simplex t02 A2.svg	File:4-simplex t12 A2.svg	File:4-simplex t012 A2.svg	File:4-simplex t03 A2.svg	File:4-simplex t013 A2.svg	File:4-simplex t0123 A2.svg

Notes

^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
^ ^a ^b The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678, Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). (The classification of Zonohededra, page 73)
^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
^ The permutahedron of order 5
^ George Olshevsky, Uniform Panoploid Tetracombs, manuscript (2006): Lists the tessellation as [140 of 143] Great-prismatodecachoric tetracomb (Omnitruncated pentachoric 4d honeycomb)
^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).

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, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, wiley.com, Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
  - (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.
1. Convex uniform polychora based on the pentachoron – Model 5, 8, and 9, George Olshevsky.
Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). o3x3x3o - spid, x3x3o3x - prip, x3x3x3x - gippid

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	$A n$	$B n$	$I 2 (p)$ / $D n$	$E 6$ / $E 7$ / $E 8$ / $F 4$ / $G 2$	$H n$
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	1₂₂ • 2₂₁
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	1₃₂ • 2₃₁ • 3₂₁
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	1₄₂ • 2₄₁ • 4₂₁
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds • Polytope operations

[1] Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).

[2] Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).

[cox-3] The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678, Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). (The classification of Zonohededra, page 73)

[4] Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).

[5] The permutahedron of order 5

[6] George Olshevsky, Uniform Panoploid Tetracombs, manuscript (2006): Lists the tessellation as [140 of 143] Great-prismatodecachoric tetracomb (Omnitruncated pentachoric 4d honeycomb)

[7] Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).

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Runcinated 5-cell

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