Construction

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.

Structure

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]

More information A4, k-face ...

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

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Projections

The truncated tetrahedron-first Schlegel diagram 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.

Images

More information Ak Coxeter plane, A4 ...

orthographic projections

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

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• 
  net
• 
  stereographic projection
  (centered on truncated tetrahedron)

Alternate names

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

Coordinates

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.

Related polytopes

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

Symmetry

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.

Alternative names

• 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)

Images

Coordinates

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

More information , ...

Coordinates

${\displaystyle \pm \left({\sqrt {\frac {5}{2}}},\ {\frac {5}{\sqrt {6}}},\ {\frac {2}{\sqrt {3}}},\ 0\right)}$ ${\displaystyle \pm \left({\sqrt {\frac {5}{2}}},\ {\frac {5}{\sqrt {6}}},\ {\frac {-1}{\sqrt {3}}},\ \pm 1\right)}$ ${\displaystyle \pm \left({\sqrt {\frac {5}{2}}},\ {\frac {1}{\sqrt {6}}},\ {\frac {4}{\sqrt {3}}},\ 0\right)}$ ${\displaystyle \pm \left({\sqrt {\frac {5}{2}}},\ {\frac {1}{\sqrt {6}}},\ {\frac {-2}{\sqrt {3}}},\ \pm 2\right)}$

${\displaystyle \pm \left({\sqrt {\frac {5}{2}}},\ -{\sqrt {\frac {3}{2}}},\ \pm {\sqrt {3}},\ \pm 1\right)}$ ${\displaystyle \pm \left({\sqrt {\frac {5}{2}}},\ -{\sqrt {\frac {3}{2}}},\ 0,\ \pm 2\right)}$ ${\displaystyle \pm \left(0,\ 2{\sqrt {\frac {2}{3}}},\ {\frac {4}{\sqrt {3}}},\ 0\right)}$ ${\displaystyle \pm \left(0,\ 2{\sqrt {\frac {2}{3}}},\ {\frac {-2}{\sqrt {3}}},\ \pm 2\right)}$

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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).

Related polytopes

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

More information ...

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	( )∨( )	{ }×{ }	{ }∨{ }	{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	∩	∩	∩	∩	∩	∩	∩

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Configuration

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.^[2]

More information Element, fk ...

Element	fk	f0	f1		f2			f3
	f0	30	2	2	1	4	1	2	2
	f1	2	30	*	1	2	0	2	1
		2	*	30	0	2	1	1	2
	f2	3	3	0	10	*	*	2	0
		6	3	3	*	20	*	1	1
		3	0	3	*	*	10	0	2
	f3	12	12	6	4	4	0	5	*
		12	6	12	0	4	4	*	5

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Related regular skew polyhedron

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

More information 20 of length ...

Disphenoidal 30-cell
Type	perfect[3] polychoron
Symbol	f1,2A4[3]
Coxeter
Cells	30 congruent tetragonal disphenoids
Faces	60 congruent isosceles triangles   (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

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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).