Runcinated 5-cell


5-cell

Runcinated 5-cell

Runcitruncated 5-cell

Omnitruncated 5-cell
(Runcicantitruncated 5-cell)
Orthogonal projections in A4 Coxeter plane

In four-dimensional geometry, a runcinated 5-cell is a convex uniform polychoron, being a runcination (a 3rd order truncation) of the regular 5-cell.

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

Contents


Runcinated 5-cell

Runcinated 5-cell

Schlegel diagram with half of the tetrahedral cells visible.
Type Uniform polychoron
Schläfli symbol t0,3{3,3,3}
Coxeter-Dynkin 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 [[3,3,3]], order 240
Properties convex, isogonal isotoxal
Uniform index 4 5 6

The runcinated 5-cell 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 pentachoron). It consists of 10 tetrahedra and 20 triangular prisms. The 10 tetrahedra correspond with the cells of a 5-cell and its dual.

Alternative names

Root vectors

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

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.

Images

orthographic projections
Ak
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

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}},\  \pm1\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}},\  \pm1\right)
\pm\left(0,\                  2\sqrt{\frac{2}{3}},\ \frac{-2}{\sqrt{3}},\ 0\right)
   \left(0,\                  0,\                   \pm\sqrt{3},\         \pm1\right)
   \left(0,\                  0,\                   0,\                   \pm2\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.

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 hypercupolas 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:

  • 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'.

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 polychoron
Schläfli symbol t0,1,3{3,3,3}
Coxeter-Dynkin 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 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

Images

orthographic projections
Ak
Coxeter plane
A4 A3 A2
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:

\left(\frac{7}{\sqrt{10}},\ \sqrt{\frac{3}{2}},\    \pm\sqrt{3},\         \pm1\right)
\left(\frac{7}{\sqrt{10}},\ \sqrt{\frac{3}{2}},\    0,\                   \pm2\right)
\left(\frac{7}{\sqrt{10}},\ \frac{-1}{\sqrt{6}},\   \frac{2}{\sqrt{3}},\  \pm2\right)
\left(\frac{7}{\sqrt{10}},\ \frac{-1}{\sqrt{6}},\   \frac{-4}{\sqrt{3}},\ 0\right)
\left(\frac{7}{\sqrt{10}},\ \frac{-5}{\sqrt{6}},\   \frac{1}{\sqrt{3}},\  \pm1\right)
\left(\frac{7}{\sqrt{10}},\ \frac{-5}{\sqrt{6}},\   \frac{-2}{\sqrt{3}},\ 0\right)
\left(\sqrt{\frac{2}{5}},\  \pm\sqrt{6},\           \pm\sqrt{3},\         \pm1\right)
\left(\sqrt{\frac{2}{5}},\  \pm\sqrt{6},\           0,\                   \pm2\right)
\left(\sqrt{\frac{2}{5}},\  \sqrt{\frac{2}{3}},\    \frac{5}{\sqrt{3}},\  \pm1\right)
\left(\sqrt{\frac{2}{5}},\  \sqrt{\frac{2}{3}},\    \frac{-1}{\sqrt{3}},\ \pm3\right)
\left(\sqrt{\frac{2}{5}},\  \sqrt{\frac{2}{3}},\    \frac{-4}{\sqrt{3}},\ \pm2\right)
\left(\sqrt{\frac{2}{5}},\  -\sqrt{\frac{2}{3}},\   \frac{4}{\sqrt{3}},\  \pm2\right)
\left(\sqrt{\frac{2}{5}},\  -\sqrt{\frac{2}{3}},\   \frac{1}{\sqrt{3}},\  \pm3\right)
\left(\sqrt{\frac{2}{5}},\  -\sqrt{\frac{2}{3}},\   \frac{-5}{\sqrt{3}},\ \pm1\right)
\left(\frac{-3}{\sqrt{10}},\ \frac{5}{\sqrt{6}},\   \frac{2}{\sqrt{3}},\  \pm2\right)
\left(\frac{-3}{\sqrt{10}},\ \frac{5}{\sqrt{6}},\   \frac{-4}{\sqrt{3}},\ 0\right)
\left(\frac{-3}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\   \frac{4}{\sqrt{3}},\  \pm2\right)
\left(\frac{-3}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\   \frac{1}{\sqrt{3}},\  \pm3\right)
\left(\frac{-3}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\   \frac{-5}{\sqrt{3}},\ \pm1\right)
\left(\frac{-3}{\sqrt{10}},\ \frac{-7}{\sqrt{6}},\  \frac{2}{\sqrt{3}},\  0\right)
\left(\frac{-3}{\sqrt{10}},\ \frac{-7}{\sqrt{6}},\  \frac{-1}{\sqrt{3}},\ \pm1\right)
\left(-4\sqrt{\frac{2}{5}},\ 2\sqrt{\frac{2}{3}},\  \frac{1}{\sqrt{3}},\  \pm1\right)
\left(-4\sqrt{\frac{2}{5}},\ 2\sqrt{\frac{2}{3}},\  \frac{-2}{\sqrt{3}},\ 0\right)
\left(-4\sqrt{\frac{2}{5}},\ 0,\                    \pm\sqrt{3},\         \pm1\right)
\left(-4\sqrt{\frac{2}{5}},\ 0,\                    0,\                   \pm2\right)
\left(-4\sqrt{\frac{2}{5}},\ -2\sqrt{\frac{2}{3}},\ \frac{2}{\sqrt{3}},\  0\right)
\left(-4\sqrt{\frac{2}{5}},\ -2\sqrt{\frac{2}{3}},\ \frac{-1}{\sqrt{3}},\ 1\right)

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 polychoron
Schläfli symbol t0,1,2,3{3,3,3}
Coxeter-Dynkin diagram
Cells 30 10 (4.6.6)
20 (4.4.6)
Faces 150 90{4}
60{6}
Edges 240
Vertices 120
Vertex figure
(Chiral irregular tetrahedron)
Coxeter group A4, [[3,3,3]], order 240
Properties convex, isogonal, zonotope
Uniform index 8 9 10

The omnitruncated 5-cell 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 chiral irregular tetrahedral 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]

Alternative names

Images

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

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.


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-Dynkin 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]

Coordinates

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

\left(\pm\sqrt{10},\ \pm\sqrt{6},\ \pm\sqrt{3},\ \pm1\right)
\left(\pm\sqrt{10},\ \pm\sqrt{6},\ 0,\ \pm2\right)
\pm\left(\pm\sqrt{10},\ \sqrt{\frac{2}{3}},\ \frac{5}{\sqrt{3}},\ \pm1\right)
\pm\left(\pm\sqrt{10},\ \sqrt{\frac{2}{3}},\ \frac{-1}{\sqrt{3}},\ \pm3\right)
\pm\left(\pm\sqrt{10},\ \sqrt{\frac{2}{3}},\ \frac{-4}{\sqrt{3}},\ \pm2\right)
\left(\pm\sqrt{\frac{5}{2}},\ 3\sqrt{\frac{3}{2}},\ \pm\sqrt{3},\ \pm1\right)
\left(\pm\sqrt{\frac{5}{2}},\ 3\sqrt{\frac{3}{2}},\ 0,\ \pm2\right)
\pm\left(\sqrt{\frac{5}{2}},\ \frac{1}{\sqrt{6}},\ \frac{7}{\sqrt{3}},\ \pm1\right)
\pm\left(\sqrt{\frac{5}{2}},\ \frac{1}{\sqrt{6}},\ \frac{-2}{\sqrt{3}},\ \pm4\right)
\pm\left(\sqrt{\frac{5}{2}},\ \frac{1}{\sqrt{6}},\ \frac{-5}{\sqrt{3}},\ \pm3\right)
\pm\left(\sqrt{\frac{5}{2}},\ -\sqrt{\frac{3}{2}},\ \pm2\sqrt{3},\ \pm2\right)
\pm\left(\sqrt{\frac{5}{2}},\ -\sqrt{\frac{3}{2}},\ 0,\ \pm4\right)
\pm\left(\sqrt{\frac{5}{2}},\ \frac{-7}{\sqrt{6}},\ \frac{5}{\sqrt{3}},\ \pm1\right)
\pm\left(\sqrt{\frac{5}{2}},\ \frac{-7}{\sqrt{6}},\ \frac{-1}{\sqrt{3}},\ \pm3\right)
\pm\left(\sqrt{\frac{5}{2}},\ \frac{-7}{\sqrt{6}},\ \frac{-4}{\sqrt{3}},\ \pm2\right)
\pm\left(0,\ 4\sqrt{\frac{2}{3}},\ \frac{5}{\sqrt{3}},\ \pm1\right)
\pm\left(0,\ 4\sqrt{\frac{2}{3}},\ \frac{-1}{\sqrt{3}},\ \pm3\right)
\pm\left(0,\ 4\sqrt{\frac{2}{3}},\ \frac{-4}{\sqrt{3}},\ \pm2\right)
\pm\left(0,\ 2\sqrt{\frac{2}{3}},\ \frac{7}{\sqrt{3}},\ \pm1\right)
\pm\left(0,\ 2\sqrt{\frac{2}{3}},\ \frac{-2}{\sqrt{3}},\ \pm4\right)
\pm\left(0,\ 2\sqrt{\frac{2}{3}},\ \frac{-5}{\sqrt{3}},\ \pm3\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}, .

Related polytopes

These polytopes are a part of a family 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
Schlegel
diagram
A4
Coxeter plane
Graph
A3 Coxeter plane
Graph
A2 Coxeter plane
Graph

Notes

  1. ^ a b The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 (The classification of Zonohededra, page 73
  2. ^ The permutahedron of order 5
  3. ^ George Olshevsky, Uniform Panoploid Tetracombs, manuscript (2006): Lists the tessellation as [140 of 143] Great-prismatodecachoric tetracomb (Omnitruncated pentachoric 4d honeycomb)

References