# Runcinated 5-cell

Last updated
 Orthogonal projections in A4 Coxeter plane 5-cell Runcinated 5-cell Runcitruncated 5-cell Omnitruncated 5-cell(Runcicantitruncated 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.

## Contents

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 t0,3{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.

### 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.

### 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
Ak
Coxeter plane
A4A3A2
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:

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

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 withcuboctahedral cells shown Type Uniform 4-polytope Schläfli symbol t0,1,3{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

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)

### Images

orthographic projections
Ak
Coxeter plane
A4A3A2
Graph
Dihedral symmetry [5][4][3]
 Schlegel diagram with its 40 blue triangular faces and its 60 green quad faces. Central part of Schlegel diagram.

### Coordinates

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

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 Schlegel diagram with half of the truncated octahedral cells shown. Type Uniform 4-polytope Schläfli symbol t0,1,2,3{3,3,3} Coxeter diagram Cells 30 10 (4.6.6) 20 (4.4.6) Faces 150 90{4}60{6} Edges 240 Vertices 120 Vertex figure 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. [1]

### Images

orthographic projections
Ak
Coxeter plane
A4A3A2
Graph
Dihedral symmetry [[5]] = [10][4][[3]] = [6]
 Omnitruncated 5-cell Dual to omnitruncated 5-cell

### Perspective projections

 Perspective Schlegel diagram Centered on truncated octahedron Stereographic projection

### Permutohedron

Just as the truncated octahedron is the permutohedron of order 4, the omnitruncated 5-cell is the permutohedron of order 5. [2] 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.

### 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 . [3] 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. [1]

### 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]].

### Coordinates

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

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

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, t0,1,2,3{3,3,3,4}, .

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 D2d symmetry and 60 with C2v symmetry), and 240 vertices. Its vertex figure is an irregular triangular bipyramid.

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.

Vertex figure

### Full snub 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 , 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. [4]

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}
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

Schlegel
diagram
A4
Coxeter plane
Graph
A3 Coxeter plane
Graph
A2 Coxeter plane
Graph

## Notes

1. The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN   99-35678, ISBN   0-486-40919-8 (The classification of Zonohededra, page 73)
2. George Olshevsky, Uniform Panoploid Tetracombs, manuscript (2006): Lists the tessellation as [140 of 143] Great-prismatodecachoric tetracomb (Omnitruncated pentachoric 4d honeycomb)

## Related Research Articles

In four-dimensional geometry, a runcinated tesseract is a convex uniform 4-polytope, being a runcination of the regular tesseract.

In four-dimensional geometry, a cantellated tesseract is a convex uniform 4-polytope, being a cantellation of the regular tesseract.

The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation in Euclidean 3-space, made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron. It is a self-dual tessellation with Schläfli symbol {4,3,4}. John Horton Conway calls this honeycomb a cubille.

The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2.

The order-5 cubic honeycomb is one of four compact regular space-filling tessellations in hyperbolic 3-space. With Schläfli symbol {4,3,5}, it has five cubes {4,3} around each edge, and 20 cubes around each vertex. It is dual with the order-4 dodecahedral honeycomb.

The icosahedral honeycomb is one of four compact regular space-filling tessellations in hyperbolic 3-space. With Schläfli symbol {3,5,3}, there are three icosahedra around each edge, and 12 icosahedra around each vertex, in a regular dodecahedral vertex figure.

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

In four-dimensional geometry, a cantellated 5-cell is a convex uniform 4-polytope, being a cantellation of the regular 5-cell.

In four-dimensional geometry, a runcinated 24-cell is a convex uniform 4-polytope, being a runcination of the regular 24-cell.

In four-dimensional geometry, a runcinated 120-cell is a convex uniform 4-polytope, being a runcination of the regular 120-cell.

In the field of hyperbolic geometry, the hexagonal tiling honeycomb is one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is paracompact because it has cells composed of an infinite number of faces. Each cell is a hexagonal tiling whose vertices lie on a horosphere, a surface in hyperbolic space that approaches a single ideal point at infinity.

In hyperbolic 3-space, the order-6 tetrahedral honeycomb is a paracompact regular space-filling tessellation. It is paracompact because it has vertex figures composed of an infinite number of faces, and has all vertices as ideal points at infinity. With Schläfli symbol {3,3,6}, the order-6 tetrahedral honeycomb has six ideal tetrahedra around each edge. All vertices are ideal, with infinitely many tetrahedra existing around each vertex in a triangular tiling vertex figure.

In the field of hyperbolic geometry, the order-4 hexagonal tiling honeycomb arises as one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is paracompact because it has cells composed of an infinite number of faces. Each cell is a hexagonal tiling whose vertices lie on a horosphere: a flat plane in hyperbolic space that approaches a single ideal point at infinity.

The order-6 cubic honeycomb is a paracompact regular space-filling tessellation in hyperbolic 3-space. It is paracompact because it has vertex figures composed of an infinite number of facets, with all vertices as ideal points at infinity. With Schläfli symbol {4,3,6}, the honeycomb has six ideal cubes meeting along each edge. Its vertex figure is an infinite triangular tiling. Its dual is the order-4 hexagonal tiling honeycomb.

The order-6 dodecahedral honeycomb is one of 11 paracompact regular honeycombs in hyperbolic 3-space. It is paracompact because it has vertex figures composed of an infinite number of faces, with all vertices as ideal points at infinity. It has Schläfli symbol {5,3,6}, with six ideal dodecahedral cells surrounding each edge of the honeycomb. Each vertex is ideal, and surrounded by infinitely many dodecahedra. The honeycomb has a triangular tiling vertex figure.

The triangular tiling honeycomb is one of 11 paracompact regular space-filling tessellations in hyperbolic 3-space. It is called paracompact because it has infinite cells and vertex figures, with all vertices as ideal points at infinity. It has Schläfli symbol {3,6,3}, being composed of triangular tiling cells. Each edge of the honeycomb is surrounded by three cells, and each vertex is ideal with infinitely many cells meeting there. Its vertex figure is a hexagonal tiling.

In the geometry of hyperbolic 3-space, the square tiling honeycomb is one of 11 paracompact regular honeycombs. It is called paracompact because it has infinite cells, whose vertices exist on horospheres and converge to a single ideal point at infinity. Given by Schläfli symbol {4,4,3}, it has three square tilings, {4,4}, around each edge, and six square tilings around each vertex, in a cubic {4,3} vertex figure.

The order-4 octahedral honeycomb is a regular paracompact honeycomb in hyperbolic 3-space. It is paracompact because it has infinite vertex figures, with all vertices as ideal points at infinity. Given by Schläfli symbol {3,4,4}, it has four ideal octahedra around each edge, and infinite octahedra around each vertex in a square tiling vertex figure.

## References

Family An Bn I2(p) / Dn E6 / E7 / E8 / / 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