Runcinated 5-cell

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.

Orthogonal projections in A₄ Coxeter plane
5-cell	Runcinated 5-cell
Runcitruncated 5-cell	Omnitruncated 5-cell (Runcicantitruncated 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
Schlegel diagram with half of the tetrahedral cells visible.
Type	Uniform 4-polytope
Schläfli symbol	t_0,3{3,3,3}
Coxeter diagram
Cells	30	10 (3.3.3) 20 (3.4.4)
Faces	70	40 {3} 30 {4}
Edges	60
Vertices	20
Vertex figure	(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]

f_k	f₀	f₁		f₂			f₃
f₀	20	3	3	3	6	3	1	3	3	1
f₁	2	30	*	2	2	0	1	2	1	0
f₁	2	*	30	0	2	2	0	1	2	1
f₂	3	3	0	20	*	*	1	1	0	0
	4	2	2	*	30	*	0	1	1	0
	3	0	3	*	*	20	0	0	1	1
f₃	4	6	0	4	0	0	5	*	*	*
	6	6	3	2	3	0	*	10	*	*
	6	3	6	0	3	2	*	*	10	*
	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.

Images

orthographic projections
A_k Coxeter plane	A₄	A₃	A₂
Graph
Dihedral symmetry	[[5]] = [10]	[4]	[[3]] = [6]

View inside of a 3-sphere projection Schlegel diagram with its 10 tetrahedral cells	Net

Coordinates

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

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

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
Schlegel diagram with cuboctahedral cells shown
Type	Uniform 4-polytope
Schläfli symbol	t_0,1,3{3,3,3}
Coxeter diagram
Cells	30	5 (3.6.6) 10 (4.4.6) 10 (3.4.4) 5 (3.4.3.4)
Faces	120	40 {3} 60 {4} 20 {6}
Edges	150
Vertices	60
Vertex figure	(Rectangular pyramid)
Coxeter group	A₄, [3,3,3], order 120
Properties	convex, isogonal
Uniform index	7 8 9

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.