5-cell | Runcinated 5-cell |
Runcitruncated 5-cell | Omnitruncated 5-cell (Runcicantitruncated 5-cell) |
Orthogonal projections in A4 Coxeter plane |
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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 | ||
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.
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.
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]
fk | 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 | |
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 |
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.
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 | Net |
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:
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:
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.
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.
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 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 | ||
Schlegel diagram with cuboctahedral 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.
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]
fk | f0 | f1 | f2 | f3 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
f0 | 60 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | |
f1 | 2 | 30 | * | * | 2 | 2 | 0 | 0 | 0 | 1 | 2 | 1 | 0 | |
2 | * | 60 | * | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | ||
2 | * | * | 60 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | ||
f2 | 6 | 3 | 3 | 0 | 20 | * | * | * | * | 1 | 1 | 0 | 0 | |
4 | 2 | 0 | 2 | * | 30 | * | * | * | 0 | 1 | 1 | 0 | ||
3 | 0 | 3 | 0 | * | * | 20 | * | * | 1 | 0 | 0 | 1 | ||
4 | 0 | 2 | 2 | * | * | * | 30 | * | 0 | 1 | 0 | 1 | ||
3 | 0 | 0 | 3 | * | * | * | * | 20 | 0 | 0 | 1 | 1 | ||
f3 | 12 | 6 | 12 | 0 | 4 | 0 | 4 | 0 | 0 | 5 | * | * | * | |
12 | 6 | 6 | 6 | 2 | 3 | 0 | 3 | 0 | * | 10 | * | * | ||
6 | 3 | 0 | 6 | 0 | 3 | 0 | 0 | 2 | * | * | 10 | * | ||
12 | 0 | 12 | 12 | 0 | 0 | 4 | 6 | 4 | * | * | * | 5 |
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. |
The Cartesian coordinates of an origin-centered runcitruncated 5-cell having edge length 2 are:
Coordinates | ||
---|---|---|
The vertices can be more simply constructed on a hyperplane in 5-space, as the permutations of:
This construction is from the positive orthant facet of the runcitruncated 5-orthoplex.
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. [3]
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]
fk | f0 | f1 | f2 | f3 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
f0 | 120 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |
f1 | 2 | 60 | * | * | * | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | |
2 | * | 60 | * | * | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | ||
2 | * | * | 60 | * | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | ||
2 | * | * | * | 60 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | ||
f2 | 6 | 3 | 3 | 0 | 0 | 20 | * | * | * | * | * | 1 | 1 | 0 | 0 | |
4 | 2 | 0 | 2 | 0 | * | 30 | * | * | * | * | 1 | 0 | 1 | 0 | ||
4 | 2 | 0 | 0 | 2 | * | * | 30 | * | * | * | 0 | 1 | 1 | 0 | ||
6 | 0 | 3 | 3 | 0 | * | * | * | 20 | * | * | 1 | 0 | 0 | 1 | ||
4 | 0 | 2 | 0 | 2 | * | * | * | * | 30 | * | 0 | 1 | 0 | 1 | ||
6 | 0 | 0 | 3 | 3 | * | * | * | * | * | 20 | 0 | 0 | 1 | 1 | ||
f3 | 24 | 12 | 12 | 12 | 0 | 4 | 6 | 0 | 4 | 0 | 0 | 5 | * | * | * | |
12 | 6 | 6 | 0 | 6 | 2 | 0 | 3 | 0 | 3 | 0 | * | 10 | * | * | ||
12 | 6 | 0 | 6 | 6 | 0 | 3 | 3 | 0 | 0 | 2 | * | * | 10 | * | ||
24 | 0 | 12 | 12 | 12 | 0 | 0 | 0 | 4 | 6 | 4 | * | * | * | 5 |
Ak Coxeter plane | A4 | A3 | A2 |
---|---|---|---|
Graph | |||
Dihedral symmetry | [[5]] = [10] | [4] | [[3]] = [6] |
Omnitruncated 5-cell | Dual to omnitruncated 5-cell |
Perspective Schlegel diagram Centered on truncated octahedron | Stereographic projection |
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.
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 . [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]
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]].
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}, .
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.
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. [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 | |||||||||
Schlegel diagram | |||||||||
A4 Coxeter plane Graph | |||||||||
A3 Coxeter plane Graph | |||||||||
A2 Coxeter plane Graph |
A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is radially equilateral.
In four-dimensional geometry, a runcinated tesseract is a convex uniform 4-polytope, being a runcination of the regular tesseract.
In four-dimensional geometry, the rectified 5-cell is a uniform 4-polytope composed of 5 regular tetrahedral and 5 regular octahedral cells. Each edge has one tetrahedron and two octahedra. Each vertex has two tetrahedra and three octahedra. In total it has 30 triangle faces, 30 edges, and 10 vertices. Each vertex is surrounded by 3 octahedra and 2 tetrahedra; the vertex figure is a triangular prism.
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 called 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.
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.
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 cantellated 24-cell is a convex uniform 4-polytope, being a cantellation of the regular 24-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 five-dimensional geometry, a stericated 5-simplex is a convex uniform 5-polytope with fourth-order truncations (sterication) of the regular 5-simplex.
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.
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.