# 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 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 a C5, pentachoron, pentatope, pentahedroid,[1] 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}.

## Alternative names

• Pentachoron
• 4-simplex
• Pentatope
• Pentahedroid (Henry Parker Manning)
• Pen (Jonathan Bowers: for pentachoron)[2]
• Hyperpyramid, tetrahedral pyramid

## 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 simplest set of coordinates is: (2,0,0,0), (0,2,0,0), (0,0,2,0), (0,0,0,2), (τ,τ,τ,τ), with edge length 2√2, where τ is the golden ratio.[3]

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:

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

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

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

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 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 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

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 symbol {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:

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
diagram
t12α5
t12γ5
t012α5
t012γ5
t123α5
t123γ5
Symmetry [2,1,1], order 2 [2+,1,1], order 2 [ ]+, order 1
Schlegel
diagram
Name
Coxeter
diagram
t0123α5
t0123γ5
t0123β5
t01234α5
t01234γ5

## Compound

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 birectified 5-cell. = .

## Related polytopes and honeycomb

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

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.

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 cell.

## References

1. ^ Matila Ghyka, The geometry of Art and Life (1977), p.68
2. ^ Category 1: Regular Polychora
3. ^ Coxeter, Regular Complex Polytopes, 1991, p. 30. 4.2 The Crystalographic regular polytopes
• 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: 1n1)
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)