5cell 
Runcinated 5cell 
Runcitruncated 5cell 
Omnitruncated 5cell (Runcicantitruncated 5cell) 
Orthogonal projections in A_{4} Coxeter plane 

In fourdimensional geometry, a runcinated 5cell is a convex uniform 4polytope, being a runcination (a 3rd order truncation, up to faceplaning) of the regular 5cell.
There are 3 unique degrees of runcinations of the 5cell, including with permutations, truncations, and cantellations.
Runcinated 5cell  
Schlegel diagram with half of the tetrahedral cells visible.  
Type  Uniform 4polytope  
Schläfli symbol  t_{0,3}{3,3,3}  
Coxeter diagram  or or  
Cells  30  10 (3.3.3) 20 (3.4.4) 
Faces  70  40 {3} 30 {4} 
Edges  60  
Vertices  20  
Vertex figure  (Elongated equilateraltriangular antiprism)  
Symmetry group  Aut(A_{4}), [[3,3,3]], order 240  
Properties  convex, isogonal isotoxal  
Uniform index  4 5 6 
The runcinated 5cell or small prismatodecachoron is constructed by expanding the cells of a 5cell radially and filling in the gaps with triangular prisms (which are the face prisms and edge figures) and tetrahedra (cells of the dual 5cell). It consists of 10 tetrahedra and 20 triangular prisms. The 10 tetrahedra correspond with the cells of a 5cell and its dual.
Topologically, under its highest symmetry, [[3,3,3]], there is only one geometrical form, containing 10 tetrahedra and 20 uniform triangular prisms. The rectangles are always squares because the two pairs of edges correspond to the edges of the two sets of 5 regular tetrahedra each in dual orientation, which are made equal under extended symmetry.
E. L. Elte identified it in 1912 as a semiregular polytope.
Two of the ten tetrahedral cells meet at each vertex. The triangular prisms lie between them, joined to them by their triangular faces and to each other by their square faces. Each triangular prism is joined to its neighbouring triangular prisms in anti orientation (i.e., if edges A and B in the shared square face are joined to the triangular faces of one prism, then it is the other two edges that are joined to the triangular faces of the other prism); thus each pair of adjacent prisms, if rotated into the same hyperplane, would form a gyrobifastigium.
The runcinated 5cell can be dissected by a central cuboctahedron into two tetrahedral cupola. This dissection is analogous to the 3D cuboctahedron being dissected by a central hexagon into two triangular cupola.
A_{k} Coxeter plane 
A_{4}  A_{3}  A_{2} 

Graph  
Dihedral symmetry  [[5]] = [10]  [4]  [[3]] = [6] 
View inside of a 3sphere projection Schlegel diagram with its 10 tetrahedral cells 
Net 
The Cartesian coordinates of the vertices of an origincentered runcinated 5cell with edge length 2 are:


An alternate simpler set of coordinates can be made in 5space, as 20 permutations of:
This construction exists as one of 32 orthant facets of the runcinated 5orthoplex.
A second construction in 5space, from the center of a rectified 5orthoplex is given by coordinate permutations of:
Its 20 vertices represent the root vectors of the simple Lie group A_{4}. It is also the vertex figure for the 5cell honeycomb in 4space.
The maximal crosssection of the runcinated 5cell with a 3dimensional hyperplane is a cuboctahedron. This crosssection divides the runcinated 5cell into two tetrahedral hypercupolae consisting of 5 tetrahedra and 10 triangular prisms each.
The tetrahedronfirst orthographic projection of the runcinated 5cell into 3dimensional space has a cuboctahedral envelope. The structure of this projection is as follows:
The regular skew polyhedron, {4,63}, exists in 4space with 6 squares around each vertex, in a zigzagging nonplanar vertex figure. These square faces can be seen on the runcinated 5cell, using all 60 edges and 20 vertices. The 40 triangular faces of the runcinated 5cell can be seen as removed. The dual regular skew polyhedron, {6,43}, is similarly related to the hexagonal faces of the bitruncated 5cell.
Runcitruncated 5cell  
Schlegel diagram with cuboctahedral cells shown  
Type  Uniform 4polytope  
Schläfli symbol  t_{0,1,3}{3,3,3}  
Coxeter diagram  
Cells  30  5 (3.6.6) 10 (4.4.6) 10 (3.4.4) 5 (3.4.3.4) 
Faces  120  40 {3} 60 {4} 20 {6} 
Edges  150  
Vertices  60  
Vertex figure  (Rectangular pyramid)  
Coxeter group  A_{4}, [3,3,3], order 120  
Properties  convex, isogonal  
Uniform index  7 8 9 
The runcitruncated 5cell or prismatorhombated pentachoron is composed of 60 vertices, 150 edges, 120 faces, and 30 cells. The cells are: 5 truncated tetrahedra, 10 hexagonal prisms, 10 triangular prisms, and 5 cuboctahedra. Each vertex is surrounded by five cells: one truncated tetrahedron, two hexagonal prisms, one triangular prism, and one cuboctahedron; the vertex figure is a rectangular pyramid.
A_{k} Coxeter plane 
A_{4}  A_{3}  A_{2} 

Graph  
Dihedral symmetry  [5]  [4]  [3] 
Schlegel diagram with its 40 blue triangular faces and its 60 green quad faces. 
Central part of Schlegel diagram. 
The Cartesian coordinates of an origincentered runcitruncated 5cell having edge length 2 are:
Coordinates  




The vertices can be more simply constructed on a hyperplane in 5space, as the permutations of:
This construction is from the positive orthant facet of the runcitruncated 5orthoplex.
Omnitruncated 5cell  
Schlegel diagram with half of the truncated octahedral cells shown.  
Type  Uniform 4polytope  
Schläfli symbol  t_{0,1,2,3}{3,3,3}  
Coxeter diagram  or or  
Cells  30  10 (4.6.6) 20 (4.4.6) 
Faces  150  90{4} 60{6} 
Edges  240  
Vertices  120  
Vertex figure  Phyllic disphenoid  
Coxeter group  Aut(A_{4}), [[3,3,3]], order 240  
Properties  convex, isogonal, zonotope  
Uniform index  8 9 10 
The omnitruncated 5cell or great prismatodecachoron is composed of 120 vertices, 240 edges, 150 faces (90 squares and 60 hexagons), and 30 cells. The cells are: 10 truncated octahedra, and 20 hexagonal prisms. Each vertex is surrounded by four cells: two truncated octahedra, and two hexagonal prisms, arranged in two chiral irregular tetrahedral vertex figures.
Coxeter calls this Hinton's polytope after C. H. Hinton, who described it in his book The Fourth Dimension in 1906. It forms a uniform honeycomb which Coxeter calls Hinton's honeycomb.^{[1]}
A_{k} Coxeter plane 
A_{4}  A_{3}  A_{2} 

Graph  
Dihedral symmetry  [[5]] = [10]  [4]  [[3]] = [6] 
Omnitruncated 5cell 
Dual to omnitruncated 5cell 
Perspective Schlegel diagram Centered on truncated octahedron 
Stereographic projection 
Just as the truncated octahedron is the permutohedron of order 4, the omnitruncated 5cell is the permutohedron of order 5.^{[2]} The omnitruncated 5cell is a zonotope, the Minkowski sum of five line segments parallel to the five lines through the origin and the five vertices of the 5cell.
The omnitruncated 5cell honeycomb can tessellate 4dimensional space by translational copies of this cell, each with 3 hypercells around each face. This honeycomb's Coxeter diagram is .^{[3]} Unlike the analogous honeycomb in three dimensions, the bitruncated cubic honeycomb which has three different Coxeter group Wythoff constructions, this honeycomb has only one such construction.^{[1]}
The omnitruncated 5cell has extended pentachoric symmetry, [[3,3,3]], order 240. The vertex figure of the omnitruncated 5cell represents the Goursat tetrahedron of the [3,3,3] Coxeter group. The extended symmetry comes from a 2fold rotation across the middle order3 branch, and is represented more explicitly as [2^{+}[3,3,3]].
The Cartesian coordinates of the vertices of an origincentered omnitruncated 5cell having edge length 2 are:



These vertices can be more simply obtained in 5space as the 120 permutations of (0,1,2,3,4). This construction is from the positive orthant facet of the runcicantitruncated 5orthoplex, t_{0,1,2,3}{3,3,3,4}, .
The full snub 5cell or omnisnub 5cell, defined as an alternation of the omnitruncated 5cell, can not be made uniform, but it can be given Coxeter diagram , and symmetry [[3,3,3]]^{+}, order 120, and constructed from 90 cells: 10 icosahedrons, 20 octahedrons, and 60 tetrahedrons filling the gaps at the deleted vertices. It has 300 faces (triangles), 270 edges, and 60 vertices.
Topologically, under its highest symmetry, [[3,3,3]]^{+}, the 10 icosahedra have T (chiral tetrahedral) symmetry, while the 20 octahedra have D_{3} symmetry and the 60 tetrahedra have C_{2} symmetry^{[4]}.
These polytopes are a part of a family 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 