Rectified 5cell  
Schlegel diagram with the 5 tetrahedral cells shown.  
Type  Uniform 4polytope  
Schläfli symbol  t_{1}{3,3,3} or r{3,3,3} {3^{2,1}} =  
CoxeterDynkin diagram  
Cells  10  5 {3,3} 5 3.3.3.3 
Faces  30 {3}  
Edges  30  
Vertices  10  
Vertex figure  Triangular prism  
Symmetry group  A_{4}, [3,3,3], order 120  
Petrie Polygon  Pentagon  
Properties  convex, isogonal, isotoxal  
Uniform index  1 2 3 
In fourdimensional geometry, the rectified 5cell is a uniform 4polytope 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.
Topologically, under its highest symmetry, [3,3,3], there is only one geometrical form, containing 5 regular tetrahedra and 5 rectified tetrahedra (which is geometrically the same as a regular octahedron). It is also topologically identical to a tetrahedronoctahedron segmentochoron.^{[clarification needed]}
The vertex figure of the rectified 5cell is a uniform triangular prism, formed by three octahedra around the sides, and two tetrahedra on the opposite ends.^{[1]}
Despite having the same number of vertices as cells (10) and the same number of edges as faces (30), the rectified 5cell is not selfdual because the vertex figure (a uniform triangular prism) is not a dual of the polychoron's cells.
Seen in a configuration matrix, all incidence counts between elements are shown. The diagonal fvector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time.^{[2]}
A_{4}  kface  f_{k}  f_{0}  f_{1}  f_{2}  f_{3}  kfigure  Notes  

A_{2}A_{1}  ( )  f_{0}  10  6  3  6  3  2  {3}x{ }  A_{4}/A_{2}A_{1} = 5!/3!/2 = 10  
A_{1}A_{1}  { }  f_{1}  2  30  1  2  2  1  { }v( )  A_{4}/A_{1}A_{1} = 5!/2/2 = 30  
A_{2}A_{1}  {3}  f_{2}  3  3  10  *  2  0  { }  A_{4}/A_{2}A_{1} = 5!/3!/2 = 10  
A_{2}  3  3  *  20  1  1  A_{4}/A_{2} = 5!/3! = 20  
A_{3}  r{3,3}  f_{3}  6  12  4  4  5  *  ( )  A_{4}/A_{3} = 5!/4! = 5  
A_{3}  {3,3}  4  6  0  4  *  5 
Together with the simplex and 24cell, this shape and its dual (a polytope with ten vertices and ten triangular bipyramid facets) was one of the first 2simple 2simplicial 4polytopes known. This means that all of its twodimensional faces, and all of the twodimensional faces of its dual, are triangles. In 1997, Tom Braden found another dual pair of examples, by gluing two rectified 5cells together; since then, infinitely many 2simple 2simplicial polytopes have been constructed.^{[3]}^{[4]}
It is one of three semiregular 4polytope 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.^{[5]}
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as tC_{5}.
A_{k} Coxeter plane 
A_{4}  A_{3}  A_{2} 

Graph  
Dihedral symmetry  [5]  [4]  [3] 
stereographic projection (centered on octahedron) 
Net (polytope) 
Tetrahedroncentered 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. 
The Cartesian coordinates of the vertices of an origincentered rectified 5cell having edge length 2 are:
Coordinates  



More simply, the vertices of the rectified 5cell can be positioned on a hyperplane in 5space 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.
The convex hull of the rectified 5cell and its dual (assuming that they are congruent) is a nonuniform polychoron composed of 30 cells: 10 tetrahedra, 20 octahedra (as triangular antiprisms), and 20 vertices. Its vertex figure is a triangular bifrustum.
This polytope is the vertex figure of the 5demicube, and the edge figure of the uniform 2_{21} polytope.
It is also one of 9 Uniform 4polytopes constructed from the [3,3,3] Coxeter group.
Name  5cell  truncated 5cell  rectified 5cell  cantellated 5cell  bitruncated 5cell  cantitruncated 5cell  runcinated 5cell  runcitruncated 5cell  omnitruncated 5cell 

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} 
t_{0,3}{3,3,3}  t_{0,1,3}{3,3,3} t_{0,2,3}{3,3,3} 
t_{0,1,2,3}{3,3,3} 
Coxeter diagram 

Schlegel diagram 

A_{4} Coxeter plane Graph 

A_{3} Coxeter plane Graph 

A_{2} Coxeter plane Graph 
The rectified 5cell 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 5cell). The Coxeter symbol for the rectified 5cell is 0_{21}.
k_{21} figures in n dimensional  

Space  Finite  Euclidean  Hyperbolic  
E_{n}  3  4  5  6  7  8  9  10  
Coxeter group 
E_{3}=A_{2}A_{1}  E_{4}=A_{4}  E_{5}=D_{5}  E_{6}  E_{7}  E_{8}  E_{9} = = E_{8}^{+}  E_{10} = = E_{8}^{++}  
Coxeter diagram 

Symmetry  [3^{−1,2,1}]  [3^{0,2,1}]  [3^{1,2,1}]  [3^{2,2,1}]  [3^{3,2,1}]  [3^{4,2,1}]  [3^{5,2,1}]  [3^{6,2,1}]  
Order  12  120  1,920  51,840  2,903,040  696,729,600  ∞  
Graph      
Name  −1_{21}  0_{21}  1_{21}  2_{21}  3_{21}  4_{21}  5_{21}  6_{21} 
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 
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