5cell 
Truncated 5cell 
Bitruncated 5cell  
Schlegel diagrams centered on [3,3] (cells at opposite at [3,3]) 
In geometry, a truncated 5cell is a uniform 4polytope (4dimensional uniform polytope) formed as the truncation of the regular 5cell.
There are two degrees of truncations, including a bitruncation.
Truncated 5cell  

Schlegel diagram (tetrahedron cells visible)  
Type  Uniform 4polytope  
Schläfli symbol  t_{0,1}{3,3,3} t{3,3,3}  
Coxeter diagram  
Cells  10  5 (3.3.3) 5 (3.6.6) 
Faces  30  20 {3} 10 {6} 
Edges  40  
Vertices  20  
Vertex figure  Equilateraltriangular pyramid  
Symmetry group  A_{4}, [3,3,3], order 120  
Properties  convex, isogonal  
Uniform index  2 3 4 
The truncated 5cell, truncated pentachoron or truncated 4simplex is bounded by 10 cells: 5 tetrahedra, and 5 truncated tetrahedra. Each vertex is surrounded by 3 truncated tetrahedra and one tetrahedron; the vertex figure is an elongated tetrahedron.
The truncated 5cell may be constructed from the 5cell by truncating its vertices at 1/3 of its edge length. This transforms the 5 tetrahedral cells into truncated tetrahedra, and introduces 5 new tetrahedral cells positioned near the original vertices.
The truncated tetrahedra are joined to each other at their hexagonal faces, and to the tetrahedra at their triangular faces.
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.^{[1]}
A_{4}  kface  f_{k}  f_{0}  f_{1}  f_{2}  f_{3}  kfigure  Notes  

A_{2}  ( )  f_{0}  20  1  3  3  3  3  1  {3}v( )  A_{4}/A_{2} = 5!/3! = 20  
A_{2}A_{1}  { }  f_{1}  2  10  *  3  0  3  0  {3}  A_{4}/A_{2}A_{1} = 5!/3!/2 = 10  
A_{1}A_{1}  2  *  30  1  2  2  1  { }v( )  A_{4}/A_{1}A_{1} = 5!/2/2 = 30  
A_{2}A_{1}  t{3}  f_{2}  6  3  3  10  *  2  0  { }  A_{4}/A_{2}A_{1} = 5!/3!/2 = 10  
A_{2}  {3}  3  0  3  *  20  1  1  A_{4}/A_{2} = 5!/3! = 20  
A_{3}  t{3,3}  f_{3}  12  6  12  4  4  5  *  ( )  A_{4}/A_{3} = 5!/4! = 5  
{3,3}  4  0  6  0  4  *  5 
The truncated tetrahedronfirst Schlegel diagram projection of the truncated 5cell into 3dimensional space has the following structure:
This layout of cells in projection is analogous to the layout of faces in the facefirst projection of the truncated tetrahedron into 2dimensional space. The truncated 5cell is the 4dimensional analogue of the truncated tetrahedron.
A_{k} Coxeter plane 
A_{4}  A_{3}  A_{2} 

Graph  
Dihedral symmetry  [5]  [4]  [3] 
stereographic projection
(centered on truncated tetrahedron)
The Cartesian coordinates for the vertices of an origincentered truncated 5cell having edge length 2 are:


More simply, the vertices of the truncated 5cell can be constructed on a hyperplane in 5space as permutations of (0,0,0,1,2) or of (0,1,2,2,2). These coordinates come from positive orthant facets of the truncated pentacross and bitruncated penteract respectively.
The convex hull of the truncated 5cell and its dual (assuming that they are congruent) is a nonuniform polychoron composed of 60 cells: 10 tetrahedra, 20 octahedra (as triangular antiprisms), 30 tetrahedra (as tetragonal disphenoids), and 40 vertices. Its vertex figure is a hexakis triangular cupola.
Bitruncated 5cell  

Schlegel diagram with alternate cells hidden.  
Type  Uniform 4polytope  
Schläfli symbol  t_{1,2}{3,3,3} 2t{3,3,3}  
Coxeter diagram  or or  
Cells  10 (3.6.6)  
Faces  40  20 {3} 20 {6} 
Edges  60  
Vertices  30  
Vertex figure  ({ }v{ })  
dual polytope  Disphenoidal 30cell  
Symmetry group  Aut(A_{4}), [[3,3,3]], order 240  
Properties  convex, isogonal, isotoxal, isochoric  
Uniform index  5 6 7 
The bitruncated 5cell (also called a bitruncated pentachoron, decachoron and 10cell) is a 4dimensional polytope, or 4polytope, composed of 10 cells in the shape of truncated tetrahedra.
Topologically, under its highest symmetry, [[3,3,3]], there is only one geometrical form, containing 10 uniform truncated tetrahedra. The hexagons are always regular because of the polychoron's inversion symmetry, of which the regular hexagon is the only such case among ditrigons (an isogonal hexagon with 3fold symmetry).
E. L. Elte identified it in 1912 as a semiregular polytope.
Each hexagonal face of the truncated tetrahedra is joined in complementary orientation to the neighboring truncated tetrahedron. Each edge is shared by two hexagons and one triangle. Each vertex is surrounded by 4 truncated tetrahedral cells in a tetragonal disphenoid vertex figure.
The bitruncated 5cell is the intersection of two pentachora in dual configuration. As such, it is also the intersection of a penteract with the hyperplane that bisects the penteract's long diagonal orthogonally. In this sense it is a 4dimensional analog of the regular octahedron (intersection of regular tetrahedra in dual configuration / tesseract bisection on long diagonal) and the regular hexagon (equilateral triangles / cube). The 5dimensional analog is the birectified 5simplex, and the dimensional analog is the polytope whose Coxeter–Dynkin diagram is linear with rings on the middle one or two nodes.
The bitruncated 5cell is one of the two nonregular convex uniform 4polytopes which are celltransitive. The other is the bitruncated 24cell, which is composed of 48 truncated cubes.
This 4polytope has a higher extended pentachoric symmetry (2×A_{4}, [[3,3,3]]), doubled to order 240, because the element corresponding to any element of the underlying 5cell can be exchanged with one of those corresponding to an element of its dual.
A_{k} Coxeter plane 
A_{4}  A_{3}  A_{2} 

Graph  
Dihedral symmetry  [[5]] = [10]  [4]  [[3]] = [6] 
stereographic projection of spherical 4polytope (centred on a hexagon face) 
Net (polytope) 
The Cartesian coordinates of an origincentered bitruncated 5cell having edge length 2 are:
Coordinates  



More simply, the vertices of the bitruncated 5cell can be constructed on a hyperplane in 5space as permutations of (0,0,1,2,2). These represent positive orthant facets of the bitruncated pentacross. Another 5space construction, centered on the origin are all 20 permutations of (1,1,0,1,1).
The bitruncated 5cell can be seen as the intersection of two regular 5cells in dual positions. = ∩ .
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  ( )∨( )  { }×{ } 
{ }∨{ } 
{3}×{3} 
{3}∨{3} 
{3,3}×{3,3}  {3,3}∨{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 
∩ 
∩ 
∩ 
∩ 
∩  ∩  ∩ 
The regular skew polyhedron, {6,43}, exists in 4space with 4 hexagonal around each vertex, in a zigzagging nonplanar vertex figure. These hexagonal faces can be seen on the bitruncated 5cell, using all 60 edges and 30 vertices. The 20 triangular faces of the bitruncated 5cell can be seen as removed. The dual regular skew polyhedron, {4,63}, is similarly related to the square faces of the runcinated 5cell.
Disphenoidal 30cell  

Type  perfect^{[2]} polychoron  
Symbol  f_{1,2}A_{4}^{[2]}  
Coxeter  
Cells  30 congruent tetragonal disphenoids  
Faces  60 congruent isosceles triangles (2 short edges)  
Edges  40  20 of length 20 of length 
Vertices  10  
Vertex figure  (Triakis tetrahedron)  
Dual  Bitruncated 5cell  
Coxeter group  Aut(A_{4}), [[3,3,3]], order 240  
Orbit vector  (1, 2, 1, 1)  
Properties  convex, isochoric 
The disphenoidal 30cell is the dual of the bitruncated 5cell. It is a 4dimensional polytope (or polychoron) derived from the 5cell. It is the convex hull of two 5cells in opposite orientations.
Being the dual of a uniform polychoron, it is celltransitive, consisting of 30 congruent tetragonal disphenoids. In addition, it is vertextransitive under the group Aut(A_{4}).
These polytope are from a set of 9 uniform 4polytope 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 