5-simplex |
Stericated 5-simplex | ||
Steritruncated 5-simplex |
Stericantellated 5-simplex | ||
Stericantitruncated 5-simplex |
Steriruncitruncated 5-simplex | ||
Steriruncicantitruncated 5-simplex (Omnitruncated 5-simplex) | |||
Orthogonal projections in A_{5} and A_{4} Coxeter planes |
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In five-dimensional geometry, a stericated 5-simplex is a convex uniform 5-polytope with fourth-order truncations (sterication) of the regular 5-simplex.
There are six unique sterications of the 5-simplex, including permutations of truncations, cantellations, and runcinations. The simplest stericated 5-simplex is also called an expanded 5-simplex, with the first and last nodes ringed, for being constructible by an expansion operation applied to the regular 5-simplex. The highest form, the steriruncicantitruncated 5-simplex is more simply called an omnitruncated 5-simplex with all of the nodes ringed.
Stericated 5-simplex | ||
Type | Uniform 5-polytope | |
Schläfli symbol | 2r2r{3,3,3,3} 2r{3^{2,2}} = | |
Coxeter-Dynkin diagram | or | |
4-faces | 62 | 6+6 {3,3,3} 15+15 {}×{3,3} 20 {3}×{3} |
Cells | 180 | 60 {3,3} 120 {}×{3} |
Faces | 210 | 120 {3} 90 {4} |
Edges | 120 | |
Vertices | 30 | |
Vertex figure | Tetrahedral antiprism | |
Coxeter group | A_{5}×2, [[3,3,3,3]], order 1440 | |
Properties | convex, isogonal, isotoxal |
A stericated 5-simplex can be constructed by an expansion operation applied to the regular 5-simplex, and thus is also sometimes called an expanded 5-simplex. It has 30 vertices, 120 edges, 210 faces (120 triangles and 90 squares), 180 cells (60 tetrahedra and 120 triangular prisms) and 62 4-faces (12 5-cells, 30 tetrahedral prisms and 20 3-3 duoprisms).
The maximal cross-section of the stericated hexateron with a 4-dimensional hyperplane is a runcinated 5-cell. This cross-section divides the stericated hexateron into two pentachoral hypercupolas consisting of 6 5-cells, 15 tetrahedral prisms and 10 3-3 duoprisms each.
The vertices of the stericated 5-simplex can be constructed on a hyperplane in 6-space as permutations of (0,1,1,1,1,2). This represents the positive orthant facet of the stericated 6-orthoplex.
A second construction in 6-space, from the center of a rectified 6-orthoplex is given by coordinate permutations of:
The Cartesian coordinates in 5-space for the normalized vertices of an origin-centered stericated hexateron are:
Its 30 vertices represent the root vectors of the simple Lie group A_{5}. It is also the vertex figure of the 5-simplex honeycomb.
A_{k} Coxeter plane |
A_{5} | A_{4} |
---|---|---|
Graph | ||
Dihedral symmetry | [6] | [[5]]=[10] |
A_{k} Coxeter plane |
A_{3} | A_{2} |
Graph | ||
Dihedral symmetry | [4] | [[3]]=[6] |
orthogonal projection with [6] symmetry |
Steritruncated 5-simplex | ||
Type | Uniform 5-polytope | |
Schläfli symbol | t_{0,1,4}{3,3,3,3} | |
Coxeter-Dynkin diagram | ||
4-faces | 62 | 6 t{3,3,3} 15 {}×t{3,3} 20 {3}×{6} 15 {}×{3,3} 6 t_{0,3}{3,3,3} |
Cells | 330 | |
Faces | 570 | |
Edges | 420 | |
Vertices | 120 | |
Vertex figure | ||
Coxeter group | A_{5} [3,3,3,3], order 720 | |
Properties | convex, isogonal |
The coordinates can be made in 6-space, as 180 permutations of:
This construction exists as one of 64 orthant facets of the steritruncated 6-orthoplex.
A_{k} Coxeter plane |
A_{5} | A_{4} |
---|---|---|
Graph | ||
Dihedral symmetry | [6] | [5] |
A_{k} Coxeter plane |
A_{3} | A_{2} |
Graph | ||
Dihedral symmetry | [4] | [3] |
Stericantellated 5-simplex | ||
Type | Uniform 5-polytope | |
Schläfli symbol | t_{0,2,4}{3,3,3,3} | |
Coxeter-Dynkin diagram | or | |
4-faces | 62 | 12 rr{3,3,3} 30 rr{3,3}x{} 20 {3}×{3} |
Cells | 420 | 60 rr{3,3} 240 {}×{3} 90 {}×{}×{} 30 r{3,3} |
Faces | 900 | 360 {3} 540 {4} |
Edges | 720 | |
Vertices | 180 | |
Vertex figure | ||
Coxeter group | A_{5}×2, [[3,3,3,3]], order 1440 | |
Properties | convex, isogonal |
The coordinates can be made in 6-space, as permutations of:
This construction exists as one of 64 orthant facets of the stericantellated 6-orthoplex.
A_{k} Coxeter plane |
A_{5} | A_{4} |
---|---|---|
Graph | ||
Dihedral symmetry | [6] | [[5]]=[10] |
A_{k} Coxeter plane |
A_{3} | A_{2} |
Graph | ||
Dihedral symmetry | [4] | [[3]]=[6] |
Stericantitruncated 5-simplex | ||
Type | Uniform 5-polytope | |
Schläfli symbol | t_{0,1,2,4}{3,3,3,3} | |
Coxeter-Dynkin diagram | ||
4-faces | 62 | |
Cells | 480 | |
Faces | 1140 | |
Edges | 1080 | |
Vertices | 360 | |
Vertex figure | ||
Coxeter group | A_{5} [3,3,3,3], order 720 | |
Properties | convex, isogonal |
The coordinates can be made in 6-space, as 360 permutations of:
This construction exists as one of 64 orthant facets of the stericantitruncated 6-orthoplex.
A_{k} Coxeter plane |
A_{5} | A_{4} |
---|---|---|
Graph | ||
Dihedral symmetry | [6] | [5] |
A_{k} Coxeter plane |
A_{3} | A_{2} |
Graph | ||
Dihedral symmetry | [4] | [3] |
Steriruncitruncated 5-simplex | ||
Type | Uniform 5-polytope | |
Schläfli symbol | t_{0,1,3,4}{3,3,3,3} 2t{3^{2,2}} | |
Coxeter-Dynkin diagram | or | |
4-faces | 62 | 12 t_{0,1,3}{3,3,3} 30 {}×t{3,3} 20 {6}×{6} |
Cells | 450 | |
Faces | 1110 | |
Edges | 1080 | |
Vertices | 360 | |
Vertex figure | ||
Coxeter group | A_{5}×2, [[3,3,3,3]], order 1440 | |
Properties | convex, isogonal |
The coordinates can be made in 6-space, as 360 permutations of:
This construction exists as one of 64 orthant facets of the steriruncitruncated 6-orthoplex.
A_{k} Coxeter plane |
A_{5} | A_{4} |
---|---|---|
Graph | ||
Dihedral symmetry | [6] | [[5]]=[10] |
A_{k} Coxeter plane |
A_{3} | A_{2} |
Graph | ||
Dihedral symmetry | [4] | [[3]]=[6] |
Omnitruncated 5-simplex | ||
Type | Uniform 5-polytope | |
Schläfli symbol | t_{0,1,2,3,4}{3,3,3,3} 2tr{3^{2,2}} | |
Coxeter-Dynkin diagram |
or | |
4-faces | 62 | 12 t_{0,1,2,3}{3,3,3} 30 {}×tr{3,3} 20 {6}×{6} |
Cells | 540 | 360 t{3,4} 90 {4,3} 90 {}×{6} |
Faces | 1560 | 480 {6} 1080 {4} |
Edges | 1800 | |
Vertices | 720 | |
Vertex figure | Irregular 5-cell | |
Coxeter group | A_{5}×2, [[3,3,3,3]], order 1440 | |
Properties | convex, isogonal, zonotope |
The omnitruncated 5-simplex has 720 vertices, 1800 edges, 1560 faces (480 hexagons and 1080 squares), 540 cells (360 truncated octahedra, 90 cubes, and 90 hexagonal prisms), and 62 4-faces (12 omnitruncated 5-cells, 30 truncated octahedral prisms, and 20 6-6 duoprisms).
The vertices of the truncated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,1,2,3,4,5). These coordinates come from the positive orthant facet of the steriruncicantitruncated 6-orthoplex, t_{0,1,2,3,4}{3^{4},4}, .
A_{k} Coxeter plane |
A_{5} | A_{4} |
---|---|---|
Graph | ||
Dihedral symmetry | [6] | [[5]]=[10] |
A_{k} Coxeter plane |
A_{3} | A_{2} |
Graph | ||
Dihedral symmetry | [4] | [[3]]=[6] |
The omnitruncated 5-simplex is the permutohedron of order 6. It is also a zonotope, the Minkowski sum of six line segments parallel to the six lines through the origin and the six vertices of the 5-simplex.
Orthogonal projection, vertices labeled as a permutohedron. |
The omnitruncated 5-simplex honeycomb is constructed by omnitruncated 5-simplex facets with 3 facets around each ridge. It has Coxeter-Dynkin diagram of .
Coxeter group | |||||
---|---|---|---|---|---|
Coxeter-Dynkin | |||||
Picture | |||||
Name | Apeirogon | Hextille | Omnitruncated 3-simplex honeycomb |
Omnitruncated 4-simplex honeycomb |
Omnitruncated 5-simplex honeycomb |
Facets |
These polytopes are a part of 19 uniform 5-polytopes based on the [3,3,3,3] Coxeter group, all shown here in A_{5} Coxeter plane orthographic projections. (Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple having progressively more vertices)
A5 polytopes | |||||||||||
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t_{0} |
t_{1} |
t_{2} |
t_{0,1} |
t_{0,2} |
t_{1,2} |
t_{0,3} | |||||
t_{1,3} |
t_{0,4} |
t_{0,1,2} |
t_{0,1,3} |
t_{0,2,3} |
t_{1,2,3} |
t_{0,1,4} | |||||
t_{0,2,4} |
t_{0,1,2,3} |
t_{0,1,2,4} |
t_{0,1,3,4} |
t_{0,1,2,3,4} |