Truncated 5-cell

The truncated 5-cell may be constructed from the 5-cell 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 f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time.[1]

A4		k-face	fk	f0	f1		f2		f3		k-figure	Notes
A2		( )	f0	20	1	3	3	3	3	1	{3}v( )	A4/A2 = 5!/3! = 20
A2A1		{ }	f1	2	10	*	3	0	3	0	{3}	A4/A2A1 = 5!/3!/2 = 10
A1A1				2	*	30	1	2	2	1	{ }v( )	A4/A1A1 = 5!/2/2 = 30
A2A1		t{3}	f2	6	3	3	10	*	2	0	{ }	A4/A2A1 = 5!/3!/2 = 10
A2		{3}		3	0	3	*	20	1	1		A4/A2 = 5!/3! = 20
A3		t{3,3}	f3	12	6	12	4	4	5	*	( )	A4/A3 = 5!/4! = 5
		{3,3}		4	0	6	0	4	*	5

The tetrahedron-first parallel projection of the truncated 5-cell into 3-dimensional space has the following structure:

• The projection envelope is a truncated tetrahedron.
• One of the truncated tetrahedral cells project onto the entire envelope.
• One of the tetrahedral cells project onto a tetrahedron lying at the center of the envelope.
• Four flattened tetrahedra are joined to the triangular faces of the envelope, and connected to the central tetrahedron via 4 radial edges. These are the images of the remaining 4 tetrahedral cells.
• Between the central tetrahedron and the 4 hexagonal faces of the envelope are 4 irregular truncated tetrahedral volumes, which are the images of the 4 remaining truncated tetrahedral cells.

This layout of cells in projection is analogous to the layout of faces in the face-first projection of the truncated tetrahedron into 2-dimensional space. The truncated 5-cell is the 4-dimensional analogue of the truncated tetrahedron.

orthographic projections

Ak Coxeter plane	A4	A3	A2
Graph
Dihedral symmetry	[5]	[4]	[3]

• 
  net
• 
  stereographic projection
  (centered on truncated tetrahedron)

• Truncated pentatope
• Truncated 4-simplex
• Truncated pentachoron (Acronym: tip) (Jonathan Bowers)

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

More simply, the vertices of the truncated 5-cell can be constructed on a hyperplane in 5-space 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 5-cell 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.

Vertex figure

This 4-polytope has a higher extended pentachoric symmetry (2×A₄, [[3,3,3]]), doubled to order 240, because the element corresponding to any element of the underlying 5-cell can be exchanged with one of those corresponding to an element of its dual.

• Bitruncated 5-cell (Norman W. Johnson)
• 10-cell as a cell-transitive 4-polytope
• Bitruncated pentachoron
• Bitruncated pentatope
• Bitruncated 4-simplex
• Decachoron (Acronym: deca) (Jonathan Bowers)

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

More simply, the vertices of the bitruncated 5-cell can be constructed on a hyperplane in 5-space as permutations of (0,0,1,2,2). These represent positive orthant facets of the bitruncated pentacross. Another 5-space construction, centered on the origin are all 20 permutations of (-1,-1,0,1,1).

The bitruncated 5-cell can be seen as the intersection of two regular 5-cells in dual positions. = ∩ .

Isotopic uniform truncated simplices

Dim.	2	3	4	5	6	7	8
Name Coxeter	Hexagon = t{3} = {6}	Octahedron = r{3,3} = {31,1} = {3,4} ${\displaystyle \left\{{\begin{array}{l}3\\3\end{array}}\right\}}$	Decachoron 2t{33}	Dodecateron 2r{34} = {32,2} ${\displaystyle \left\{{\begin{array}{l}3,3\\3,3\end{array}}\right\}}$	Tetradecapeton 3t{35}	Hexadecaexon 3r{36} = {33,3} ${\displaystyle \left\{{\begin{array}{l}3,3,3\\3,3,3\end{array}}\right\}}$	Octadecazetton 4t{37}
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	∩	∩	∩	∩	∩	∩	∩

A 3D net for {6,4|3}, with pairs of yellow triangles folded together into 4D and removed

The regular skew polyhedron, {6,4|3}, exists in 4-space with 4 hexagonal around each vertex, in a zig-zagging nonplanar vertex figure. These hexagonal faces can be seen on the bitruncated 5-cell, using all 60 edges and 30 vertices. The 20 triangular faces of the bitruncated 5-cell can be seen as removed. The dual regular skew polyhedron, {4,6|3}, is similarly related to the square faces of the runcinated 5-cell.

Disphenoidal 30-cell
Type	perfect[2] polychoron
Symbol	f1,2A4[2]
Coxeter
Cells	30 congruent tetragonal disphenoids
Faces	60 congruent isosceles   (2 short edges)
Edges	40	20 of length ${\displaystyle \scriptstyle 1}$ 20 of length ${\displaystyle \scriptstyle {\sqrt {3/5}}}$
Vertices	10
Vertex figure	(Triakis tetrahedron)
Dual	Bitruncated 5-cell
Coxeter group	Aut(A4), [[3,3,3]], order 240
Orbit vector	(1, 2, 1, 1)
Properties	convex, isochoric

The disphenoidal 30-cell is the dual of the bitruncated 5-cell. It is a 4-dimensional polytope (or polychoron) derived from the 5-cell. It is the convex hull of two 5-cells in opposite orientations.

Being the dual of a uniform polychoron, it is cell-transitive, consisting of 30 congruent tetragonal disphenoids. In addition, it is vertex-transitive under the group Aut(A₄).