5-cell

Regular 5-cell (pentachoron) (4-simplex)
Schlegel diagram (vertices and edges)
Type	Convex regular 4-polytope
Schläfli symbol	{3,3,3}
Coxeter 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

Net

In geometry, the 5-cell is a four-dimensional object bounded by 5 tetrahedral cells. It is also known as a C₅, pentachoron^[1], pentatope, pentahedroid,^[2] or tetrahedral pyramid. It is a 4-simplex, the simplest possible convex regular 4-polytope (four-dimensional analogue of a Platonic solid), and is analogous to the tetrahedron in three dimensions and the triangle in two dimensions. The pentachoron is a four dimensional pyramid with a tetrahedral base.

The regular 5-cell is bounded by regular tetrahedra, and is one of the six regular convex 4-polytopes, represented by Schläfli symbol {3,3,3}.

The 5-cell is a solution to the problem: Make 10 equilateral triangles, all of the same size, using 10 matchsticks, where each side of every triangle is exactly one matchstick. No solution exists in three dimensions.

Alternative names[edit]

• Pentachoron
• 4-simplex
• Pentatope
• Pentahedroid (Henry Parker Manning)
• Pen (Jonathan Bowers: for pentachoron)^[3]
• Hyperpyramid, tetrahedral pyramid

Geometry[edit]

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°.

As a configuration[edit]

The elements of a regular polytopes can be expressed in a configuration matrix. Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts (f-vectors). The nondiagonal elements represent the number of row elements are incident to the column element. The configurations for dual polytopes can be seen by rotating the matrix elements by 180 degrees. The self-dual simplex is identical by this rotation.^[4][5]

${\displaystyle {\begin{bmatrix}{\begin{matrix}5&4&6&4\\2&10&3&3\\3&3&10&2\\4&6&4&5\end{matrix}}\end{bmatrix}}}$

Construction[edit]

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 simplest set of coordinates is: (2,0,0,0), (0,2,0,0), (0,0,2,0), (0,0,0,2), (φ,φ,φ,φ), with edge length 2√2, where φ is the golden ratio.^[6]

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

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

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

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

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

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:

${\displaystyle \left(1,1,1,{\frac {-1}{\sqrt {5}}}\right)}$

${\displaystyle \left(1,-1,-1,{\frac {-1}{\sqrt {5}}}\right)}$

${\displaystyle \left(-1,1,-1,{\frac {-1}{\sqrt {5}}}\right)}$

${\displaystyle \left(-1,-1,1,{\frac {-1}{\sqrt {5}}}\right)}$

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

The vertices of a 4-simplex (with edge √2) can be more simply constructed on a hyperplane in 5-space, as (distinct) 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.

Boerdijk–Coxeter helix[edit]

A 5-cell can be constructed as a Boerdijk–Coxeter helix of five chained tetrahedra, folded into a 4-dimensional ring. The 10 triangle faces can be seen in a 2D net within a triangular tiling, with 6 triangles around every vertex, although folding into 4-dimensions causes edges to coincide. The purple edges represent the Petrie polygon of the 5-cell.

Projections[edit]

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

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 5-cell into 3 dimensions has a tetrahedral projection envelope. The closest vertex of the 5-cell 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 5-cell 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 5-cell 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 project 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 5-cell 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.

Irregular 5-cell[edit]

There are many lower symmetry forms, including these found in uniform polytope vertex figures:

Symmetry	[3,3,3] Order 120	[3,3,1] Order 24	[3,2,1] Order 12	[3,1,1] Order 6	[5,2]+ Order 10
Name	Regular 5-cell	Tetrahedral pyramid	Triangular-pyramidal pyramid		Pentagonal hyperdisphenoid
Schläfli	{3,3,3}	{3,3} ∨ ( )	{3} ∨ { }
Example Vertex figure	5-simplex	Truncated 5-simplex	Bitruncated 5-simplex	Cantitruncated 5-simplex	Omnitruncated 4-simplex honeycomb

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.

Many uniform 5-polytopes have tetrahedral pyramid vertex figures:

Symmetry [3,3,1], order 24

Schlegel diagram
Name Coxeter	{ }×{3,3,3}	{ }×{4,3,3}	{ }×{5,3,3}	t{3,3,3,3}	t{4,3,3,3}	t{3,4,3,3}

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	[3,2,1], order 12		[3,1,1], order 6		[2+,4,1], order 8	[2,1,1], order 4
Schlegel diagram
Name Coxeter	t12α5	t12γ5	t012α5	t012γ5	t123α5	t123γ5

Symmetry	[2,1,1], order 2			[2+,1,1], order 2	[ ]+, order 1
Schlegel diagram
Name Coxeter	t0123α5	t0123γ5	t0123β5	t01234α5	t01234γ5

Compound[edit]

The compound of two 5-cells in dual configurations can be seen in this A5 Coxeter plane projection, with a red and blue 5-cell vertices and edges. This compound has [[3,3,3]] symmetry, order 240. The intersection of these two 5-cells is a uniform bitruncated 5-cell. = ∩ .

Related polytopes and honeycomb[edit]

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

Schläfli	{3,3,3}	t{3,3,3}	r{3,3,3}	rr{3,3,3}	2t{3,3,3}	tr{3,3,3}	t0,3{3,3,3}	t0,1,3{3,3,3}	t0,1,2,3{3,3,3}
Coxeter
Schlegel

1k2 figures in n dimensions
Space	Finite						Euclidean	Hyperbolic
n	3	4	5	6	7	8	9	10
Coxeter group	E3=A2A1	E4=A4	E5=D5	E6	E7	E8	E9 = ${\displaystyle {\tilde {E}}_{8}}$ = E8+	E10 = ${\displaystyle {\bar {T}}_{8}}$ = E8++
Coxeter diagram
Symmetry (order)	[3−1,2,1]	[30,2,1]	[31,2,1]	[[32,2,1]]	[33,2,1]	[34,2,1]	[35,2,1]	[36,2,1]
Order	12	120	192	103,680	2,903,040	696,729,600	∞
Graph							-	-
Name	1−1,2	102	112	122	132	142	152	162

2k1 figures in n dimensions
Space	Finite						Euclidean	Hyperbolic
n	3	4	5	6	7	8	9	10
Coxeter group	E3=A2A1	E4=A4	E5=D5	E6	E7	E8	E9 = ${\displaystyle {\tilde {E}}_{8}}$ = E8+	E10 = ${\displaystyle {\bar {T}}_{8}}$ = E8++
Coxeter diagram
Symmetry	[3−1,2,1]	[30,2,1]	[[31,2,1]]	[32,2,1]	[33,2,1]	[34,2,1]	[35,2,1]	[36,2,1]
Order	12	120	384	51,840	2,903,040	696,729,600	∞
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} polytopes
Space		S3			H3
Form		Finite			Paracompact	Noncompact
Name		{3,3,3}	{4,3,3}	{5,3,3}	{6,3,3}	{7,3,3}	{8,3,3}	...{∞,3,3}
Image
Cells {p,3}		{3,3}	{4,3}	{5,3}	{6,3}	{7,3}	{8,3}	{∞,3}

It is one of three regular 4-polytopes with tetrahedral cells, along with the 16-cell {3,3,5}, 600-cell {3,3,5}. The order-6 tetrahedral honeycomb {3,3,6} of hyperbolic space also has tetrahedral cells.

{3,3,p} polytopes
Space	S3			H3
Form	Finite			Paracompact	Noncompact
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,∞}

{3,p,3} polytopes
Space	S3		H3
Form	Finite		Compact	Paracompact	Noncompact
{3,p,3}	{3,3,3}	{3,4,3}	{3,5,3}	{3,6,3}	{3,7,3}	{3,8,3}	... {3,∞,3}
Image
Cells	{3,3}	{3,4}	{3,5}	{3,6}	{3,7}	{3,8}	{3,∞}
Vertex figure	{3,3}	{4,3}	{5,3}	{6,3}	{7,3}	{8,3}	{∞,3}

{p,3,p} regular honeycombs
Space	S3	Euclidean E3	H3
Form	Finite	Affine	Compact	Paracompact	Noncompact
Name	{3,3,3}	{4,3,4}	{5,3,5}	{6,3,6}	{7,3,7}	{8,3,8}	...{∞,3,∞}
Image
Cells	{3,3}	{4,3}	{5,3}	{6,3}	{7,3}	{8,3}	{∞,3}
Vertex figure	{3,3}	{3,4}	{3,5}	{3,6}	{3,7}	{3,8}	{3,∞}

References[edit]

• 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, edited 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[edit]

• Weisstein, Eric W. "Pentatope". MathWorld. 
• Olshevsky, George. "Pentachoron". Glossary for Hyperspace. Archived from the original on 4 February 2007. 
  • 1. Convex uniform polychora based on the pentachoron - Model 1, George Olshevsky.
• Klitzing, Richard. "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
• pyrochoron