# Rectified 5-cell

 Rectified 5-cell Schlegel diagram with the 5 tetrahedral cells shown. Type Uniform polychoron Schläfli symbol t1{3,3,3} Coxeter-Dynkin diagram Cells 10 5 {3,3} 5 3.3.3.3 Faces 30 {3} Edges 30 Vertices 10 Vertex figure Triangular prism Symmetry group A4, [3,3,3], order 120 Petrie Polygon Pentagon Properties convex, isogonal, isotoxal Uniform index 1 2 3

In four-dimensional geometry, the rectified 5-cell is a uniform polychoron 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.

It is one of three semiregular polychora made of two or more cells which are platonic solids, discovered by Thorold Gosset in his 1900 paper. He called it a Tetroctahedric for being made of tetrahedron and octahedron cells.

The vertex figure of the rectified 5-cell is a uniform triangular prism, formed by three octahedra around the sides, and two tetrahedra on the opposite ends.

## Images

orthographic projections
Ak
Coxeter plane
A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]
 stereographic projection (centered on octahedron) Net (polytope) Tetrahedron-centered perspective projection into 3D space, with nearest tetrahedron to the 4D viewpoint rendered in red, and the 4 surrounding octahedra in green. Cells lying on the far side of the polytope have been culled for clarity (although they can be discerned from the edge outlines). The rotation is only of the 3D projection image, in order to show its structure, not a rotation in 4D space.

## Coordinates

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

 $\left(\sqrt{\frac{2}{5}},\ \frac{2}{\sqrt{6}},\ \frac{2}{\sqrt{3}},\ 0 \right)$ $\left(\sqrt{\frac{2}{5}},\ \frac{2}{\sqrt{6}},\ \frac{-1}{\sqrt{3}},\ \pm1\right)$ $\left(\sqrt{\frac{2}{5}},\ \frac{-2}{\sqrt{6}},\ \frac{1}{\sqrt{3}},\ \pm1\right)$ $\left(\sqrt{\frac{2}{5}},\ \frac{-2}{\sqrt{6}},\ \frac{-2}{\sqrt{3}},\ 0 \right)$ $\left(\frac{-3}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\ \frac{1}{\sqrt{3}},\ \pm1\right)$ $\left(\frac{-3}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\ \frac{-2}{\sqrt{3}},\ 0 \right)$ $\left(\frac{-3}{\sqrt{10}},\ -\sqrt{\frac{3}{2}},\ 0,\ 0 \right)$

More simply, the vertices of the rectified 5-cell can be positioned on a hyperplane in 5-space as permutations of (0,0,0,1,1) or (0,0,1,1,1). These construction can be seen as positive orthant facets of the rectified pentacross or birectified penteract respectively.

## Related polychora

This polytope is the vertex figure of the 5-demicube, and the edge figure of the uniform 221 polytope.

It is also one of 9 uniform polychora 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} t0,1{3,3,3} t1{3,3,3} t0,2{3,3,3} t1,2{3,3,3} t0,1,2{3,3,3} t0,3{3,3,3} t0,1,3{3,3,3} t0,1,2,3{3,3,3}
Coxeter-Dynkin
diagram
Schlegel
diagram
A4
Coxeter plane
Graph
A3 Coxeter plane
Graph
A2 Coxeter plane
Graph

### Related polytopes and honeycombs

The rectified 5-cell is second in a dimensional series of semiregular polytopes. Each progressive uniform polytope is constructed as the vertex figure of the previous polytope. Thorold Gosset identified this series in 1900 as containing all regular polytope facets, containing all simplexes and orthoplexes (tetrahedrons and octahedrons in the case of the rectified 5-cell). In Coxeter's notation the rectified 5-cell is given the symbol 021.

k21 figures in n dimensional
Space Finite Euclidean Hyperbolic
En 3 4 5 6 7 8 9 10
Coxeter
group
E3=A2×A1 E4=A4 E5=D5 E6 E7 E8 E9 = ${\tilde{E}}_{8}$ = E8+ E10 = E8++
Coxeter
diagram
Symmetry
(order)
[3-1,2,1]
(12)
[30,2,1]
(120)
[31,2,1]
(192)
[32,2,1]
(51,840)
[33,2,1]
(2,903,040)
[34,2,1]
(696,729,600)
[35,2,1]
(∞)
[36,2,1]
(∞)
Graph
Name −121 021 121 221 321 421 521 621