Truncated 5-cell explained

In geometry, a truncated 5-cell is a uniform 4-polytope (4-dimensional uniform polytope) formed as the truncation of the regular 5-cell.

There are two degrees of truncations, including a bitruncation.

Truncated 5-cell

bgcolor=#e7dcc3 colspan=3	Truncated 5-cell
bgcolor=#ffffff align=center colspan=3	Schlegel diagram (tetrahedron cells visible)
Type	Uniform 4-polytope
Schläfli symbol	t0,1 t
Coxeter diagram
Cells	10
Faces	30	20 10
Edges	40
Vertices	20
Vertex figure	Equilateral-triangular pyramid
Symmetry group	A4, [3,3,3], order 120
Properties	convex, isogonal
Uniform index	2 3 4

The truncated 5-cell, truncated pentachoron or truncated 4-simplex 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.

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.

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

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.

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:

\left( (3)/(\sqrt{ 10) },\ \sqrt,\ \pm\sqrt,\ \pm1\right)\left( (3)/(\sqrt{ 10) },\ \sqrt,\ 0,\ \pm2\right)\left( (3)/(\sqrt{ 10) },\ \frac,\ \frac,\ \pm2\right)\left( (3)/(\sqrt{ 10) },\ \frac,\ \frac,\ 0 \right)\left( (3)/(\sqrt{ 10) },\ \frac,\ \frac,\ \pm1\right)\left( (3)/(\sqrt{ 10) },\ \frac,\ \frac,\ 0 \right)

\left( - \sqrt{ 2 \over 5 }, \sqrt{ 2 \over 3 }, (2)/(\sqrt{ 3) },\ \pm2\right)\left( - \sqrt{ 2 \over 5 }, \sqrt{ 2 \over 3 }, (-4)/(\sqrt{ 3) },\ 0 \right)\left( - \sqrt{ 2 \over 5 }, - \sqrt{ 6 }, 0 , 0 \right) \left( (-7)/(\sqrt{ 10) },\ \frac,\ \frac,\ \pm1\right)\left( (-7)/(\sqrt{ 10) },\ \frac,\ \frac,\ 0 \right)\left( (-7)/(\sqrt{ 10) },\ -\sqrt,\ 0,\ 0 \right)

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

Bitruncated 5-cell

bgcolor=#e7dcc3 colspan=3	Bitruncated 5-cell
bgcolor=#ffffff align=center colspan=3	Schlegel diagram with alternate cells hidden.
Type	Uniform 4-polytope
Schläfli symbol	t1,2 2t
Coxeter diagram	or or
Cells	10 (3.6.6)
Faces	40	20 20
Edges	60
Vertices	30
bgcolor=#e7dcc3 valign=center	Vertex figure	(v)
dual polytope	Disphenoidal 30-cell
Symmetry group	Aut(A4), [[3,3,3]], order 240|-|bgcolor=#e7dcc3|Properties|colspan=2|convex, isogonal, isotoxal, isochoric|-|bgcolor=#e7dcc3|Uniform index|colspan=2|5 6 7|}The bitruncated 5-cell (also called a bitruncated pentachoron, decachoron and 10-cell) is a 4-dimensional polytope, or 4-polytope, 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 3-fold 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 5-cell 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 4-dimensional 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 5-dimensional analog is the birectified 5-simplex, and the n -dimensional analog is the polytope whose Coxeter–Dynkin diagram is linear with rings on the middle one or two nodes. The bitruncated 5-cell is one of the two non-regular convex uniform 4-polytopes which are cell-transitive. The other is the bitruncated 24-cell, which is composed of 48 truncated cubes. 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) Coordinates 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). Related polytopes The bitruncated 5-cell can be seen as the intersection of two regular 5-cells in dual positions. = ∩ . 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. Element fk f0 f1 f2 f3 align=left bgcolor=#ffffe0 f0 30 2 2 1 4 1 2 2 align=left bgcolor=#ffffe0 f1 2 30 1 2 0 2 1 align=left bgcolor=#ffffe0 2 30 0 2 1 1 2 align=left bgcolor=#ffffe0 f2 3 3 0 10 2 0 align=left bgcolor=#ffffe0 6 3 3 20 1 1 align=left bgcolor=#ffffe0 3 0 3 10 0 2 align=left bgcolor=#ffffe0 f3 12 12 6 4 4 0 5 align=left bgcolor=#ffffe0 12 6 12 0 4 4 5 Related regular skew polyhedron The regular skew polyhedron,, 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,, is similarly related to the square faces of the runcinated 5-cell. Disphenoidal 30-cell bgcolor=#e7dcc3 colspan=3 Disphenoidal 30-cell Type perfect^[1] polychoron Symbol f1,2A4 Coxeter Cells 30 congruent tetragonal disphenoids Faces 60 congruent isosceles triangles   (2 short edges) Edges 40 20 of length \scriptstyle 1 20 of length \scriptstyle \sqrt{ 3 / 5 } Vertices 10 Vertex figure (Triakis tetrahedron) Dual Bitruncated 5-cell Coxeter group Aut(A4), [[3,3,3]], order 240|-|bgcolor=#e7dcc3|Orbit vector|colspan=2| (1, 2, 1, 1)|-|bgcolor=#e7dcc3|Properties|colspan=2|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). Related polytopes These polytope are from a set of 9 uniform 4-polytope constructed from the [3,3,3] Coxeter group. References H.S.M. Coxeter: H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591] (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, p. 88 (Chapter 5: Regular Skew Polyhedra in three and four dimensions and their topological analogues, Proceedings of the London Mathematics Society, Ser. 2, Vol 43, 1937.) Coxeter, H. S. M. Regular Skew Polyhedra in Three and Four Dimensions. Proc. London Math. Soc. 43, 33-62, 1937. Norman Johnson Uniform Polytopes, Manuscript (1991) N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966) x3x3o3o - tip, o3x3x3o - deca Specific ]

Notes and References

1. http://www.emis.de/journals/BAG/vol.43/no.1/b43h1gev.pdf On Perfect 4-Polytopes Gabor Gévay