5-cell

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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:

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

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

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

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

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,√5 - 1)

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

The A₄ Coxeter plane projects the 5-cell into a regular pentagon and pentagram.

orthographic projections
A_k 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}	t_0,1{3,3,3}	t₁{3,3,3}	t_0,2{3,3,3}	t_1,2{3,3,3}	t_0,1,2{3,3,3}	t_0,3{3,3,3}	t_0,1,3{3,3,3}	t_0,1,2,3{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)	[3^0,2,1] (120)	[3^1,2,1] (192)	[[3^2,2,1]] (103,680)	[3^3,2,1] (2,903,040)	[3^4,2,1] (696,729,600)	[3^5,2,1] (∞)	[3^6,2,1] (∞)
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)	[3^0,2,1] (120)	[[3^1,2,1]] (384)	[3^2,2,1] (51,840)	[3^3,2,1] (2,903,040)	[3^4,2,1] (696,729,600)	[3^5,2,1] (∞)	[3^6,2,1] (∞)
Graph							∞	∞
Name	2_-1,1	2₀₁	2₁₁	2₂₁	2₃₁	2₄₁	2₅₁	2₆₁

It is in the sequence of regular polychora: the tesseract {4,3,3}, 120-cell {5,3,3}, of Euclidean 4-space, and hexagonal tiling honeycomb {6,3,3} of hyperbolic space. All of these have a tetrahedral vertex figure.

{p,3,3}
Space	S³			H³
Name	{3,3,3}	{4,3,3}	{5,3,3}	{6,3,3}
Image
Cells	{3,3}	{4,3}	{5,3}	{6,3}
Vertex figure

It is similar to three regular polychora: the tesseract {4,3,3}, 600-cell {3,3,5} of Euclidean 4-space, and the order-6 tetrahedral honeycomb {3,3,6} of hyperbolic space. All of these have a tetrahedral cells.

{3,3,p}
Space	S³			H³
Name	{3,3,3}	{3,3,4}	{3,3,5}	{3,3,6}	{3,3,7}	{3,3,8}	... {3,3,∞}
Image
Vertex figure	{3,3}	{3,4}	{3,5}	{3,6}	{3,7}	{3,8}	{3,∞}

Irregular 5-cell

Uniform 5-polytope vertex figures

The tetrahedral pyramid is a special case of a 5-cell, a polyhedral pyramid, constructed as a regular 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.

Some uniform 5-polytopes have tetrahedral pyramid vertex figures:

Symmetry , [3,3], (*332)
Name Coxeter diagram	{ }×{3,3,3}	{ }×{4,3,3}	{ }×{5,3,3}	t{3,3,3,3}	t{4,3,3,3}	t{3,4,3,3}
Schlegel diagram

Other uniform 5-polytopes have irregular 5-cell vertex figures. The symmetry of a vertex figure of a uniform polytope is represented by removing the ringed nodes of the Coxeter diagram.

Symmetry	, [2,3], (*223)		, [3], (*33)		[2⁺,4], (2*2)	, [2], (*22)
Schlegel diagram
Name Coxeter diagram	t₁₂α₅	t₁₂γ₅	t₀₁₂α₅	t₀₁₂γ₅	t₁₂₃α₅	t₁₂₃γ₅

Symmetry	, [ ], (*)			[2]⁺, (22)	[ ]⁺, (1)
Schlegel diagram
Name Coxeter diagram	t₀₁₂₃α₅	t₀₁₂₃γ₅	t₀₁₂₃β₅	t₀₁₂₃₄α₅	t₀₁₂₃₄γ₅

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
  - (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

Convex regular polychora

5-cell	8-cell	16-cell	24-cell	120-cell	600-cell

{3,3,3} pentachoron	{4,3,3} tesseract	{3,3,4} hexadecachoron	{3,4,3} icositetrachoron	{5,3,3} hecatonicosachoron	{3,3,5} hexacosichoron

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	BC_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	5-cell	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	1₂₂ • 2₂₁
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	1₃₂ • 2₃₁ • 3₂₁
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	1₄₂ • 2₄₁ • 4₂₁
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes

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