Runcinated 5-cell

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

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 half-solid runcinated 5-cell.png
Schlegel diagram with half of the tetrahedral cells visible.
Type Uniform 4-polytope
Schläfli symbol t0,3{3,3,3}
Coxeter diagram CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Cells3010 (3.3.3) Tetrahedron.png
20 (3.4.4) Triangular prism.png
Faces7040 {3}
30 {4}
Edges60
Vertices20
Vertex figure Runcinated 5-cell verf.png
(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

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]

CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngfkf0f1f2f3
CDel node x.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node x.pngf020333631331
CDel node 1.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node x.pngf1230*2201210
CDel node x.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node 1.png2*300220121
CDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node x.pngf233020**1100
CDel node 1.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node 1.png422*30*0110
CDel node x.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node 1.png303**200011
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node x.pngf34604005***
CDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node 1.png663230*10**
CDel node 1.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node 1.png636032**10*
CDel node x.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png406004***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.

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

Images

orthographic projections
Ak
Coxeter plane
A4A3A2
Graph 4-simplex t03.svg 4-simplex t03 A3.svg 4-simplex t03 A2.svg
Dihedral symmetry [[5]] = [10][4][[3]] = [6]
Runcinated pentatope.png
View inside of a 3-sphere projection Schlegel diagram with its 10 tetrahedral cells
Small prismatodecachoron net.png
Net

Coordinates

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

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:

  • 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 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 half-solid runcitruncated 5-cell.png
Schlegel diagram with
cuboctahedral cells shown
Type Uniform 4-polytope
Schläfli symbol t0,1,3{3,3,3}
Coxeter diagram CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Cells305 Truncated tetrahedron.png (3.6.6)
10 Hexagonal prism.png (4.4.6)
10 Triangular prism.png (3.4.4)
5 Cuboctahedron.png (3.4.3.4)
Faces12040 {3}
60 {4}
20 {6}
Edges150
Vertices60
Vertex figure Runcitruncated 5-cell verf.png
(Rectangular pyramid)
Coxeter group A4, [3,3,3], order 120
Properties convex, isogonal
Uniform index 7 8 9
Net 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

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]

CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngfkf0f1f2f3
CDel node x.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node x.pngf060122221211211
CDel node 1.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node x.pngf1230**220001210
CDel node x.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node x.png2*60*101101101
CDel node x.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node 1.png2**60010110111
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node x.pngf2633020****1100
CDel node 1.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node 1.png4202*30***0110
CDel node x.pngCDel 2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node x.png3030**20**1001
CDel node x.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node 1.png4022***30*0101
CDel node x.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node 1.png3003****200011
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node x.pngf3126120404005***
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node 1.png1266623030*10**
CDel node 1.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node 1.png630603002**10*
CDel node x.pngCDel 2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png120121200464***5

Images

orthographic projections
Ak
Coxeter plane
A4A3A2
Graph 4-simplex t013.svg 4-simplex t013 A3.svg 4-simplex t013 A2.svg
Dihedral symmetry [5][4][3]
Runcitruncated 5cell.png
Schlegel diagram with its 40 blue triangular faces and its 60 green quad faces.
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:

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 half-solid omnitruncated 5-cell.png
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 CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Cells3010 Truncated octahedron.png (4.6.6)
20 Hexagonal prism.png (4.4.6)
Faces15090{4}
60{6}
Edges240
Vertices120
Vertex figure Omnitruncated 5-cell vertex figure.png 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

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]

CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngfkf0f1f2f3
CDel node x.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node x.pngf012011111111111111
CDel node 1.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node x.pngf1260***1110001110
CDel node x.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node x.png2*60**1001101101
CDel node x.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node x.png2**60*0101011011
CDel node x.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node 1.png2***600010110111
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node x.pngf26330020*****1100
CDel node 1.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node x.png42020*30****1010
CDel node 1.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node 1.png42002**30***0110
CDel node x.pngCDel 2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node x.png60330***20**1001
CDel node x.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node 1.png40202****30*0101
CDel node x.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node 1.pngCDel 3.pngCDel node 1.png60033*****200011
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node x.pngf32412121204604005***
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node 1.png126606203030*10**
CDel node 1.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node 1.pngCDel 3.pngCDel node 1.png126066033002**10*
CDel node x.pngCDel 2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png240121212000464***5

Images

orthographic projections
Ak
Coxeter plane
A4A3A2
Graph 4-simplex t0123.svg 4-simplex t0123 A3.svg 4-simplex t0123 A2.svg
Dihedral symmetry [[5]] = [10][4][[3]] = [6]
Net
Great prismatodecachoron net.png
Omnitruncated 5-cell
Dual gippid net.png
Dual to omnitruncated 5-cell

Perspective projections

Omnitruncated 5-cell.png
Perspective Schlegel diagram
Centered on truncated octahedron
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.

Orthogonal projection as a permutohedron 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 CDel branch 11.pngCDel 3ab.pngCDel nodes 11.pngCDel split2.pngCDel 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]].

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:

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}, CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png.

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.

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.

Alternated biomnitruncatodecachoron vertex figure.png
Vertex figure

Full snub 5-cell

Vertex figure for the omnisnub 5-cell 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 CDel branch hh.pngCDel 3ab.pngCDel 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]

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
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.pngCDel 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.pngCDel 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

Notes

  1. Klitzing, Richard. "x3o3o3x - spid".
  2. Klitzing, Richard. "x3x3o3x - prip".
  3. 1 2 The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN   99-35678, ISBN   0-486-40919-8 (The classification of Zonohededra, page 73)
  4. Klitzing, Richard. "x3x3x3x - gippid".
  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. "S3s3s3s".

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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.

<span class="mw-page-title-main">Order-5 cubic honeycomb</span> Regular tiling of hyperbolic 3-space

In hyperbolic geometry, 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.

<span class="mw-page-title-main">Truncated 5-cell</span>

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

<span class="mw-page-title-main">Cantellated 5-cell</span>

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

<span class="mw-page-title-main">Cantellated 24-cells</span>

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

<span class="mw-page-title-main">Runcinated 24-cells</span>

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

<span class="mw-page-title-main">Runcinated 120-cells</span>

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

<span class="mw-page-title-main">Stericated 5-simplexes</span>

In five-dimensional geometry, a stericated 5-simplex is a convex uniform 5-polytope with fourth-order truncations (sterication) of the regular 5-simplex.

<span class="mw-page-title-main">Hexagonal tiling honeycomb</span>

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.

<span class="mw-page-title-main">Order-6 tetrahedral honeycomb</span>

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.

<span class="mw-page-title-main">Order-4 hexagonal tiling honeycomb</span>

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.

<span class="mw-page-title-main">Order-6 cubic honeycomb</span>

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.

<span class="mw-page-title-main">Square tiling honeycomb</span>

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.

<span class="mw-page-title-main">Order-4 octahedral honeycomb</span>

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 / 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