5-simplex |
Rectified 5-simplex |
Birectified 5-simplex |
Orthogonal projections in A_{5} Coxeter plane |
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In five-dimensional geometry, a rectified 5-simplex is a convex uniform 5-polytope, being a rectification of the regular 5-simplex.
There are three unique degrees of rectifications, including the zeroth, the 5-simplex itself. Vertices of the rectified 5-simplex are located at the edge-centers of the 5-simplex. Vertices of the birectified 5-simplex are located in the triangular face centers of the 5-simplex.
Rectified 5-simplex Rectified hexateron (rix) | ||
---|---|---|
Type | uniform 5-polytope | |
Schläfli symbol | r{3^{4}} or | |
Coxeter diagram | or | |
4-faces | 12 | 6 {3,3,3} 6 r{3,3,3} |
Cells | 45 | 15 {3,3} 30 r{3,3} |
Faces | 80 | 80 {3} |
Edges | 60 | |
Vertices | 15 | |
Vertex figure | {}x{3,3} | |
Coxeter group | A_{5}, [3^{4}], order 720 | |
Dual | ||
Base point | (0,0,0,0,1,1) | |
Circumradius | 0.645497 | |
Properties | convex, isogonal isotoxal |
In five-dimensional geometry, a rectified 5-simplex is a uniform 5-polytope with 15 vertices, 60 edges, 80 triangular faces, 45 cells (15 tetrahedral, and 30 octahedral), and 12 4-faces (6 5-cell and 6 rectified 5-cells). It is also called 0_{3,1} for its branching Coxeter-Dynkin diagram, shown as .
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S^{1}
_{5}.
The vertices of the rectified 5-simplex can be more simply positioned on a hyperplane in 6-space as permutations of (0,0,0,0,1,1) or (0,0,1,1,1,1). These construction can be seen as facets of the rectified 6-orthoplex or birectified 6-cube respectively.
This configuration matrix represents the rectified 5-simplex. The rows and columns correspond to vertices, edges, faces, cells and 4-faces. The diagonal numbers say how many of each element occur in the whole rectified 5-simplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element.^{[1]}^{[2]}
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.^{[3]}
A_{5} | k-face | f_{k} | f_{0} | f_{1} | f_{2} | f_{3} | f_{4} | k-figure | notes | ||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
A_{3}A_{1} | ( ) | f_{0} | 15 | 8 | 4 | 12 | 6 | 8 | 4 | 2 | {3,3}x{ } | A_{5}/A_{3}A_{1} = 6!/4!/2 = 15 | |
A_{2}A_{1} | { } | f_{1} | 2 | 60 | 1 | 3 | 3 | 3 | 3 | 1 | {3}V( ) | A_{5}/A_{2}A_{1} = 6!/3!/2 = 60 | |
A_{2}A_{2} | r{3} | f_{2} | 3 | 3 | 20 | * | 3 | 0 | 3 | 0 | {3} | A_{5}/A_{2}A_{2} = 6!/3!/3! =20 | |
A_{2}A_{1} | {3} | 3 | 3 | * | 60 | 1 | 2 | 2 | 1 | { }x( ) | A_{5}/A_{2}A_{1} = 6!/3!/2 = 60 | ||
A_{3}A_{1} | r{3,3} | f_{3} | 6 | 12 | 4 | 4 | 15 | * | 2 | 0 | { } | A_{5}/A_{3}A_{1} = 6!/4!/2 = 15 | |
A_{3} | {3,3} | 4 | 6 | 0 | 4 | * | 30 | 1 | 1 | A_{5}/A_{3} = 6!/4! = 30 | |||
A_{4} | r{3,3,3} | f_{4} | 10 | 30 | 10 | 20 | 5 | 5 | 6 | * | ( ) | A_{5}/A_{4} = 6!/5! = 6 | |
A_{4} | {3,3,3} | 5 | 10 | 0 | 10 | 0 | 5 | * | 6 | A_{5}/A_{4} = 6!/5! = 6 |
Stereographic projection of spherical form |
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] |
The rectified 5-simplex, 0_{31}, is second in a dimensional series of uniform polytopes, expressed by Coxeter as 1_{3k} series. The fifth figure is a Euclidean honeycomb, 3_{31}, and the final is a noncompact hyperbolic honeycomb, 4_{31}. Each progressive uniform polytope is constructed from the previous as its vertex figure.
n | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|
Coxeter group |
A_{3}A_{1} | A_{5} | D_{6} | E_{7} | = E_{7}^{+} | =E_{7}^{++} |
Coxeter diagram |
||||||
Symmetry | [3^{−1,3,1}] | [3^{0,3,1}] | [3^{1,3,1}] | [3^{2,3,1}] | [3^{3,3,1}] | [3^{4,3,1}] |
Order | 48 | 720 | 23,040 | 2,903,040 | ∞ | |
Graph | - | - | ||||
Name | −1_{31} | 0_{31} | 1_{31} | 2_{31} | 3_{31} | 4_{31} |
Birectified 5-simplex Birectified hexateron (dot) | ||
---|---|---|
Type | uniform 5-polytope | |
Schläfli symbol | 2r{3^{4}} = {3^{2,2}} or | |
Coxeter diagram | or | |
4-faces | 12 | 12 r{3,3,3} |
Cells | 60 | 30 {3,3} 30 r{3,3} |
Faces | 120 | 120 {3} |
Edges | 90 | |
Vertices | 20 | |
Vertex figure | {3}x{3} | |
Coxeter group | A_{5}×2, [[3^{4}]], order 1440 | |
Dual | ||
Base point | (0,0,0,1,1,1) | |
Circumradius | 0.866025 | |
Properties | convex, isogonal isotoxal |
The birectified 5-simplex is isotopic, with all 12 of its facets as rectified 5-cells. It has 20 vertices, 90 edges, 120 triangular faces, 60 cells (30 tetrahedral, and 30 octahedral).
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S^{2}
_{5}.
It is also called 0_{2,2} for its branching Coxeter-Dynkin diagram, shown as . It is seen in the vertex figure of the 6-dimensional 1_{22}, .
The elements of the regular polytopes can be expressed in a configuration matrix. Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts (f-vectors). The nondiagonal elements represent the number of row elements are incident to the column element.^{[4]}^{[5]}
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.^{[6]}
A_{5} | k-face | f_{k} | f_{0} | f_{1} | f_{2} | f_{3} | f_{4} | k-figure | notes | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
A_{2}A_{2} | ( ) | f_{0} | 20 | 9 | 9 | 9 | 3 | 9 | 3 | 3 | 3 | {3}x{3} | A_{5}/A_{2}A_{2} = 6!/3!/3! = 20 | |
A_{1}A_{1}A_{1} | { } | f_{1} | 2 | 90 | 2 | 2 | 1 | 4 | 1 | 2 | 2 | { }∨{ } | A_{5}/A_{1}A_{1}A_{1} = 6!/2/2/2 = 90 | |
A_{2}A_{1} | {3} | f_{2} | 3 | 3 | 60 | * | 1 | 2 | 0 | 2 | 1 | { }v( ) | A_{5}/A_{2}A_{1} = 6!/3!/2 = 60 | |
A_{2}A_{1} | 3 | 3 | * | 60 | 0 | 2 | 1 | 1 | 2 | |||||
A_{3}A_{1} | {3,3} | f_{3} | 4 | 6 | 4 | 0 | 15 | * | * | 2 | 0 | { } | A_{5}/A_{3}A_{1} = 6!/4!/2 = 15 | |
A_{3} | r{3,3} | 6 | 12 | 4 | 4 | * | 30 | * | 1 | 1 | A_{5}/A_{3} = 6!/4! = 30 | |||
A_{3}A_{1} | {3,3} | 4 | 6 | 0 | 4 | * | * | 15 | 0 | 2 | A_{5}/A_{3}A_{1} = 6!/4!/2 = 15 | |||
A_{4} | r{3,3,3} | f_{4} | 10 | 30 | 20 | 10 | 5 | 5 | 0 | 6 | * | ( ) | A_{5}/A_{4} = 6!/5! = 6 | |
A_{4} | 10 | 30 | 10 | 20 | 0 | 5 | 5 | * | 6 |
The A5 projection has an identical appearance to Metatron's Cube.^{[7]}
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 birectified 5-simplex is the intersection of two regular 5-simplexes in dual configuration. The vertices of a birectification exist at the center of the faces of the original polytope(s). This intersection is analogous to the 3D stellated octahedron, seen as a compound of two regular tetrahedra and intersected in a central octahedron, while that is a first rectification where vertices are at the center of the original edges.
Dual 5-simplexes (red and blue), and their birectified 5-simplex intersection in green, viewed in A5 and A4 Coxeter planes. The simplexes overlap in the A5 projection and are drawn in magenta. |
It is also the intersection of a 6-cube with the hyperplane that bisects the 6-cube's long diagonal orthogonally. In this sense it is the 5-dimensional analog of the regular hexagon, octahedron, and bitruncated 5-cell. This characterization yields simple coordinates for the vertices of a birectified 5-simplex in 6-space: the 20 distinct permutations of (1,1,1,−1,−1,−1).
The vertices of the birectified 5-simplex can also be positioned on a hyperplane in 6-space as permutations of (0,0,0,1,1,1). This construction can be seen as facets of the birectified 6-orthoplex.
The birectified 5-simplex, 0_{22}, is second in a dimensional series of uniform polytopes, expressed by Coxeter as k_{22} series. The birectified 5-simplex is the vertex figure for the third, the 1_{22}. The fourth figure is a Euclidean honeycomb, 2_{22}, and the final is a noncompact hyperbolic honeycomb, 3_{22}. Each progressive uniform polytope is constructed from the previous as its vertex figure.
Space | Finite | Euclidean | Hyperbolic | ||
---|---|---|---|---|---|
n | 4 | 5 | 6 | 7 | 8 |
Coxeter group |
A_{2}A_{2} | E_{6} | =E_{6}^{+} | =E_{6}^{++} | |
Coxeter diagram |
|||||
Symmetry | [[3^{2,2,-1}]] | [[3^{2,2,0}]] | [[3^{2,2,1}]] | [[3^{2,2,2}]] | [[3^{2,2,3}]] |
Order | 72 | 1440 | 103,680 | ∞ | |
Graph | ∞ | ∞ | |||
Name | −1_{22} | 0_{22} | 1_{22} | 2_{22} | 3_{22} |
Dim. | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|
Name Coxeter |
Hexagon = t{3} = {6} |
Octahedron = r{3,3} = {3^{1,1}} = {3,4} |
Decachoron 2t{3^{3}} |
Dodecateron 2r{3^{4}} = {3^{2,2}} |
Tetradecapeton 3t{3^{5}} |
Hexadecaexon 3r{3^{6}} = {3^{3,3}} |
Octadecazetton 4t{3^{7}} |
Images | |||||||
Vertex figure | ( )v( ) | { }×{ } |
{ }v{ } |
{3}×{3} |
{3}v{3} |
{3,3}x{3,3} | {3,3}v{3,3} |
Facets | {3} | t{3,3} | r{3,3,3} | 2t{3,3,3,3} | 2r{3,3,3,3,3} | 3t{3,3,3,3,3,3} | |
As intersecting dual simplexes |
∩ |
∩ |
∩ |
∩ |
∩ | ∩ | ∩ |
This polytope is the vertex figure of the 6-demicube, and the edge figure of the uniform 2_{31} polytope.
It is also one of 19 uniform polytera 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 | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
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} |