5-cell

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 the 4-simplex (Coxeter's polytope[3]), 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.

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)
TypeConvex regular 4-polytope
Schläfli symbol{3,3,3}
Coxeter diagram
Cells5 {3,3}
Faces10 {3}
Edges10
Vertices5
Vertex figure
(tetrahedron)
Petrie polygonpentagon
Coxeter groupA4, [3,3,3]
DualSelf-dual
Propertiesconvex, isogonal, isotoxal, isohedral
Uniform index1
Vertex figure: tetrahedron
Net

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

The convex hull of the 5-cell and its dual (assuming that they are congruent) is the disphenoidal 30-cell, dual of the bitruncated 5-cell.

Alternative names
  • Pentachoron
  • 4-simplex
  • Pentatope
  • Pentahedroid (Henry Parker Manning)
  • Pen (Jonathan Bowers: for pentachoron)[4]
  • 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°.

As a configuration

This configuration matrix represents the 5-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 5-cell. The nondiagonal numbers say how many of the column's element occur in or at the row's element. This self-dual polytope's matrix is identical to its 180 degree rotation.[5]

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 a 4-dimensional pyramid with a tetrahedral base and four tetrahedral sides.)

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:

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:

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 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

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

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. = ∩ .

This compound can be seen as the 4D analogue of the 2D hexagram {​6⁄2} and the 3D compound of two tetrahedra.

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 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.

Citations
  1. N.W. Johnson: Geometries and Transformations, (2018) ISBN 978-1-107-10340-5 Chapter 11: Finite Symmetry Groups, 11.5 Spherical Coxeter groups, p.249
  2. Matila Ghyka, The geometry of Art and Life (1977), p.68
  3. Coxeter 1973, p. 120, §7.2. see illustration Fig 7.2A.
  4. Category 1: Regular Polychora
  5. Coxeter 1973, p. 12, §1.8. Configurations.
  6. Coxeter 1991, p. 30, §4.2. The Crystallographic regular polytopes.

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, H.S.M. (1973). Regular Polytopes (3rd ed.). New York: Dover.
      • p. 120, §7.2. see illustration Fig 7.2A
      • p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
    • Coxeter, H.S.M. (1991), Regular Complex Polytopes (2nd ed.), Cambridge: Cambridge University Press
    • 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
      • (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)
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform 4-polytope 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 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
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