5-simplex |
Rectified 5-simplex |
Birectified 5-simplex |
| Orthogonal projections in A5 Coxeter plane | ||
|---|---|---|
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.
Contents |
| Rectified 5-simplex Rectified hexateron (rix) |
||
|---|---|---|
A5 Coxeter plane projection [6] symmetry |
||
| Type | uniform polyteron | |
| Schläfli symbol | t1{34} | |
| Coxeter-Dynkin diagram | ||
| 4-faces | 12 | 6 {3,3,3} 6 t1{3,3,3} |
| Cells | 45 | 15 {3,3} 30 t1{3,3} |
| Faces | 80 | 80 {3} |
| Edges | 60 | |
| Vertices | 15 | |
| Vertex figure | {}x{3,3} |
|
| Coxeter group | A5, [34], order 720 | |
| 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 hypercells (6 5-cell and 6 rectified 5-cells).
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.
| Ak Coxeter plane |
A5 | A4 |
|---|---|---|
| Graph | ||
| Dihedral symmetry | [6] | [5] |
| Ak Coxeter plane |
A3 | A2 |
| Graph | ||
| Dihedral symmetry | [4] | [3] |
Stereographic projection of spherical form |
| Birectified 5-simplex Birectified hexateron (dot) |
||
|---|---|---|
A5 Coxeter plane projection [6] symmetry |
||
| Type | uniform polyteron | |
| Schläfli symbol | t2{34} | |
| Coxeter-Dynkin diagram | ||
| 4-faces | 12 | 12 t1{3,3,3} |
| Cells | 60 | 30 {3,3} 30 t1{3,3} |
| Faces | 120 | 120 {3} |
| Edges | 90 | |
| Vertices | 20 | |
| Vertex figure | {3}x{3} |
|
| Coxeter group | A5, [[34]], order 1440 | |
| 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).
The birectified hexateron is the intersection of two regular hexatera in dual configuration. As such, it is also the intersection of a hexeract with the hyperplane that bisects the hexeract's long diagonal orthogonally. In this sense it is the 5-dimensional analog of the regular hexagon, octahedron, and bitruncated pentachoron. This characterization yields simple coordinates for the vertices of a birectified hexateron 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 hexacross.
The birectified 5-simplex is the vertex figure for the 6 dimensional 122 polytope.
| Ak Coxeter plane |
A5 | A4 |
|---|---|---|
| Graph | ||
| Dihedral symmetry | [6] | [[5]]=[10] |
| Ak Coxeter plane |
A3 | A2 |
| Graph | ||
| Dihedral symmetry | [4] | [[3]]=[6] |
This polytope is the vertex figure of the 6-demicube, and the edge figure of the uniform 231 polytope.
It is also one of 19 uniform polytera based on the [3,3,3,3] Coxeter group, all shown here in A5 Coxeter plane orthographic projections. (Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple having progressively more vertices)
t0 |
t1 |
t2 |
t0,1 |
t0,2 |
t1,2 |
t0,3 |
t1,3 |
t0,4 |
t0,1,2 |
t0,1,3 |
t0,2,3 |
t1,2,3 |
t0,1,4 |
t0,2,4 |
t0,1,2,3 |
t0,1,2,4 |
t0,1,3,4 |
t0,1,2,3,4 |