5-cell

For the sequence of fifth element numbers of Pascal's triangle, see Pentatope number.

Regular 5-cell (pentachoron) (4-simplex)
Schlegel diagram (vertices and edges)
Type	Convex regular 4-polytope
Schläfli symbol	{3,3,3}
Coxeter-Dynkin diagram
Cells	5 {3,3}
Faces	10 {3}
Edges	10
Vertices	5
Vertex figure	(tetrahedron)
Petrie polygon	pentagon
Coxeter group	A4, [3,3,3]
Dual	Self-dual
Properties	convex, isogonal, isotoxal, isohedral
Uniform index	1

Vertex figure: tetrahedron

In geometry, the 5-cell is a four-dimensional object bounded by 5 tetrahedral cells. It is also known as the pentachoron, pentatope, or tetrahedral hyperpyramid. It is a 4-simplex, the simplest possible convex regular 4-polytope (four-dimensional analogue of a polyhedron), and is analogous to the tetrahedron in three dimensions and the triangle in two dimensions.

The regular 5-cell is bounded by regular tetrahedra, and is one of the six regular convex polychora, represented by Schläfli symbol {3,3,3}.

Geometry

The 5-cell is self-dual, and its vertex figure is a tetrahedron. Its maximal intersection with 3-dimensional space is the triangular prism. Its dihedral angle is cos⁻¹(1/4), or approximately 75.52°.

Construction

The 5-cell can be constructed from a tetrahedron by adding a 5th vertex such that it is equidistant from all the other vertices of the tetrahedron. (The 5-cell is essentially a 4-dimensional pyramid with a tetrahedral base.)

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

$\left( \frac{1}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\ \frac{1}{\sqrt{3}},\ \pm1\right)$

$\left( \frac{1}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\ \frac{-2}{\sqrt{3}},\ 0 \right)$

$\left( \frac{1}{\sqrt{10}},\ -\sqrt{\frac{3}{2}},\ 0,\ 0 \right)$

$\left( -2\sqrt{\frac{2}{5}},\ 0,\ 0,\ 0 \right)$

Another set of origin-centered coordinates in 4-space can be seen as a hyperpyramid with a regular tetrahedral base in 3-space, with edge length 2√2:

(1,1,1,-1), (1,-1,-1,-1), (-1,1,-1,-1), (-1,-1,1,-1), (0,0,0,4)

The vertices of a 4-simplex (with edge √2) can be more simply constructed on a hyperplane in 5-space, as permutations of (0,0,0,0,1) or (0,1,1,1,1); in these positions it is a facet of, respectively, the 5-orthoplex or the rectified penteract.

Projections

orthographic projections

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

Projections to 3 dimensions
Stereographic projection wireframe (edge projected onto a 3-sphere)	A 3D projection of a 5-cell performing a simple rotation
The vertex-first projection of the pentachoron into 3 dimensions has a tetrahedral projection envelope. The closest vertex of the pentachoron projects to the center of the tetrahedron, as shown here in red. The farthest cell projects onto the tetrahedral envelope itself, while the other 4 cells project onto the 4 flattened tetrahedral regions surrounding the central vertex.	The edge-first projection of the pentachoron into 3 dimensions has a triangular dipyramidal envelope. The closest edge (shown here in red) projects to the axis of the dipyramid, with the three cells surrounding it projecting to 3 tetrahedral volumes arranged around this axis at 120 degrees to each other. The remaining 2 cells project to the two halves of the dipyramid and are on the far side of the pentatope.
The face-first projection of the pentachoron into 3 dimensions also has a triangular dipyramidal envelope. The nearest face is shown here in red. The two cells that meet at this face projects to the two halves of the dipyramid. The remaining three cells are on the far side of the pentatope from the 4D viewpoint, and are culled from the image for clarity. They are arranged around the central axis of the dipyramid, just as in the edge-first projection.	The cell-first projection of the pentachoron into 3 dimensions has a tetrahedral envelope. The nearest cell projects onto the entire envelope, and, from the 4D viewpoint, obscures the other 4 cells; hence, they are not rendered here.

Alternative names

• Pentachoron
• 4-simplex
• Pentatope
• Pentahedroid (Henry Parker Manning)
• Pen (Jonathan Bowers: for pentachoron)
• Hyperpyramid

Related polytopes and honeycomb

The pentachoron (5-cell) is the simplest of 9 uniform polychora constructed from the [3,3,3] Coxeter group.

Name	5-cell	truncated 5-cell	rectified 5-cell	cantellated 5-cell	bitruncated 5-cell	cantitruncated 5-cell	runcinated 5-cell	runcitruncated 5-cell	omnitruncated 5-cell
Schläfli symbol	{3,3,3}	t0,1{3,3,3}	t1{3,3,3}	t0,2{3,3,3}	t1,2{3,3,3}	t0,1,2{3,3,3}	t0,3{3,3,3}	t0,1,3{3,3,3}	t0,1,2,3{3,3,3}
Coxeter-Dynkin diagram
Schlegel diagram
A4 Coxeter plane Graph
A3 Coxeter plane Graph
A2 Coxeter plane Graph

1_k2 figures in n dimensions

n	4	4	5	6	7	8	9	10
Coxeter group	E3=A2×A1	E4=A4	E5=D5	E6	E7	E8	E9 = ${\tilde{E}}_{8}$ = E8+	E10 = E8++
Coxeter diagram
Symmetry (order)	[3-1,2,1] (12)	[30,2,1] (120)	[31,2,1] (192)	[[32,2,1]] (103,680)	[33,2,1] (2,903,040)	[34,2,1] (696,729,600)	[35,2,1] (∞)	[36,2,1] (∞)
Graph							∞	∞
Name	1-1,2	102	112	122	132	142	152	162

2_k1 figures in n dimensions

n	3	4	5	6	7	8	9	10
Coxeter group	E3=A2×A1	E4=A4	E5=D5	E6	E7	E8	E9 = ${\tilde{E}}_{8}$ = E8+	E10 = E8++
Coxeter diagram
Symmetry (order)	[3-1,2,1] (12)	[30,2,1] (120)	[[31,2,1]] (384)	[32,2,1] (51,840)	[33,2,1] (2,903,040)	[34,2,1] (696,729,600)	[35,2,1] (∞)	[36,2,1] (∞)
Graph							∞	∞
Name	2-1,1	201	211	221	231	241	251	261

Other forms

The 5-cell can also be considered a tetrahedral pyramid, constructed as a tetrahedron base in a 3-space hyperplane, and an apex point above the hyperplane. The four sides of the pyramid are made of tetrahedron cells.

References

• T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
• H.S.M. Coxeter:
  • Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, p.296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
  • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p.296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
    • (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]
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1_n1)
• Norman Johnson Uniform Polytopes, Manuscript (1991)
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)

External links

• Weisstein, Eric W., "Pentatope", MathWorld.
• Olshevsky, George, Pentachoron at Glossary for Hyperspace.
  • 1. Convex uniform polychora based on the pentachoron - Model 1, George Olshevsky.
• Richard Klitzing, 4D uniform polytopes (polychora), x3o3o3o - pen
• Der 5-Zeller (5-cell) Marco Möller's Regular polytopes in R⁴ (German)
• Jonathan Bowers, Regular polychora
• Java3D Applets