5-cell (4-simplex) | |
---|---|

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 | A_{4}, [3,3,3] |

Dual | Self-dual |

Properties | convex, isogonal, isotoxal, isohedral |

Uniform index | 1 |

In geometry, the **5-cell** is the convex 4-polytope with Schläfli symbol {3,3,3}. It is a 5-vertex four-dimensional object bounded by five tetrahedral cells.^{ [lower-alpha 1] } It is also known as a **C _{5}**,

- Alternative names
- Geometry
- Structure
- As a configuration
- Coordinates
- Boerdijk–Coxeter helix
- Projections
- Irregular 5-cells
- Orthoschemes
- Isometries
- Compound
- Related polytopes and honeycombs
- Notes
- Citations
- References
- External links

The **regular 5-cell** is bounded by five regular tetrahedra, and is one of the six regular convex 4-polytopes (the four-dimensional analogues of the Platonic solids). A regular 5-cell can be constructed from a regular tetrahedron by adding a fifth vertex one edge length distant from all the vertices of the tetrahedron. This cannot be done in 3-dimensional space. The regular 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, and none of the triangles and match sticks intersect one another.* No solution exists in three dimensions.

- Pentachoron (5-point 4-polytope)
- Hypertetrahedron (4-dimensional analogue of the tetrahedron)
- 4-simplex (4-dimensional simplex)
- Tetrahedral pyramid (4-dimensional hyperpyramid with a tetrahedral base)
- Pentatope
- Pentahedroid (Henry Parker Manning)
- Pen (Jonathan Bowers: for pentachoron)
^{ [4] }

The 5-cell is the 4-dimensional simplex, the simplest possible 4-polytope. As such it is the first in the sequence of 6 convex regular 4-polytopes (in order of size and complexity).^{ [lower-alpha 2] }

Regular convex 4-polytopes | |||||||
---|---|---|---|---|---|---|---|

Symmetry group | A_{4} | B_{4} | F_{4} | H_{4} | |||

Name | 5-cell Hyper-tetrahedron | 16-cell Hyper-octahedron | 8-cell Hyper-cube | 24-cell
| 600-cell Hyper-icosahedron | 120-cell Hyper-dodecahedron | |

Schläfli symbol | {3, 3, 3} | {3, 3, 4} | {4, 3, 3} | {3, 4, 3} | {3, 3, 5} | {5, 3, 3} | |

Coxeter mirrors | |||||||

Mirror dihedrals | 𝝅/3𝝅/3𝝅/3𝝅/2𝝅/2𝝅/2 | 𝝅/3𝝅/3𝝅/4𝝅/2𝝅/2𝝅/2 | 𝝅/4𝝅/3𝝅/3𝝅/2𝝅/2𝝅/2 | 𝝅/3𝝅/4𝝅/3𝝅/2𝝅/2𝝅/2 | 𝝅/3𝝅/3𝝅/5𝝅/2𝝅/2𝝅/2 | 𝝅/5𝝅/3𝝅/3𝝅/2𝝅/2𝝅/2 | |

Graph | |||||||

Vertices | 5 tetrahedral | 8 octahedral | 16 tetrahedral | 24 cubical | 120 icosahedral | 600 tetrahedral | |

Edges | 10 triangular | 24 square | 32 triangular | 96 triangular | 720 pentagonal | 1200 triangular | |

Faces | 10 triangles | 32 triangles | 24 squares | 96 triangles | 1200 triangles | 720 pentagons | |

Cells | 5 tetrahedra | 16 tetrahedra | 8 cubes | 24 octahedra | 600 tetrahedra | 120 dodecahedra | |

Tori | 1 5-tetrahedron | 2 8-tetrahedron | 2 4-cube | 4 6-octahedron | 20 30-tetrahedron | 12 10-dodecahedron | |

Inscribed | 120 in 120-cell | 675 in 120-cell | 2 16-cells | 3 8-cells | 25 24-cells | 10 600-cells | |

Great polygons | 2 𝝅/2 squares x 3 | 4 𝝅/2 rectangles x 3 | 4 𝝅/3 hexagons x 4 | 12 𝝅/5 decagons x 6 | 50 𝝅/15 dodecagons x 4 | ||

Petrie polygons | 1 pentagon | 1 octagon | 2 octagons | 2 dodecagons | 4 30-gons | 20 30-gons | |

Isocline polygrams | 1 octagram _{3} √4 | 2 octagram _{3}√4 | 4 hexagram _{2} √3 | 4 30-gram _{2} √1 | 20 30-gram _{2}√1 | ||

Long radius | |||||||

Edge length | |||||||

Short radius | |||||||

Area | |||||||

Volume | |||||||

4-Content |

A 5-cell is formed by any five points which are not all in the same hyperplane (as a tetrahedron is formed by any four points which are not all in the same plane, and a triangle is formed by any three points which are not all in the same line). Therefore any arbitrarily chosen five vertices of *any* 4-polytope constitute a 5-cell, though not usually a regular 5-cell. The *regular* 5-cell is not found within any of the other regular convex 4-polytopes except one: the 600-vertex 120-cell is a compound of 120 regular 5-cells.^{ [lower-alpha 3] }

When a net of five tetrahedra is folded up in 4-dimensional space such that each tetrahedron is face bonded to the other four, the resulting 5-cell has a total of 5 vertices, 10 edges and 10 faces. Four edges meet at each vertex, and three tetrahedral cells meet at each edge.

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

The convex hull of two 5-cells in dual configuration is the disphenoidal 30-cell, dual of the bitruncated 5-cell.

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.^{ [7] }

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.^{ [8] }

The Cartesian coordinates of the vertices of an origin-centered regular 5-cell having edge length 2 and radius √1.6 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 and radius √3.2:

The vertices of a 4-simplex (with edge √2 and radius 1) 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.

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.

The A_{4} Coxeter plane projects the 5-cell into a regular pentagon and pentagram. The A_{3} Coxeter plane projection of the 5-cell is that of a square pyramid. The A_{2} Coxeter plane projection of the regular 5-cell is that of a triangular bipyramid (two tetrahedra joined face-to-face) with the two opposite vertices centered.

A_{k}Coxeter plane | A_{4} | A_{3} | A_{2} |
---|---|---|---|

Graph | |||

Dihedral symmetry | [5] | [4] | [3] |

Projections to 3 dimensions | |
---|---|

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. |

In the case of simplexes such as the 5-cell, certain irregular forms are in some sense more fundamental than the regular form. Although regular 5-cells cannot fill 4-space or the regular 4-polytopes, there are irregular 5-cells which do. These **characteristic 5-cells** are the fundamental domains of the different symmetry groups which give rise to the various 4-polytopes.

A **4-orthoscheme** is a 5-cell where all 10 faces are right triangles.^{ [lower-alpha 1] } An orthoscheme is an irregular simplex that is the convex hull of a tree in which all edges are mutually perpendicular. In a 4-dimensional orthoscheme, the tree consists of four perpendicular edges connecting all five vertices in a linear path that makes three right-angled turns. The elements of an orthoscheme are also orthoschemes (just as the elements of a regular simplex are also regular simplexes). Each tetrahedral cell of a 4-orthoscheme is a 3-orthoscheme, and each triangular face is a 2-orthoscheme (a right triangle).

Orthoschemes are the characteristic simplexes of the regular polytopes, because each regular polytope is generated by reflections in the bounding facets of its particular characteristic orthoscheme.^{ [9] } For example, the special case of the 4-orthoscheme with equal-length perpendicular edges is the characteristic orthoscheme of the 4-cube (also called the *tesseract* or *8-cell*), the 4-dimensional analogue of the 3-dimensional cube. If the three perpendicular edges of the 4-orthoscheme are of unit length, then all its edges are of length √1, √2, √3, or √4, precisely the chord lengths of the unit 4-cube (the lengths of the 4-cube's edges and its various diagonals). Therefore this 4-orthoscheme fits within the 4-cube, and the 4-cube (like every regular convex polytope) can be dissected into instances of its characteristic orthoscheme.

A 3-orthoscheme is easily illustrated, but a 4-orthoscheme is more difficult to visualize. A 4-orthoscheme is a tetrahedral pyramid with a 3-orthoscheme as its base. It has four more edges than the 3-orthoscheme, joining the four vertices of the base to its apex (the fifth vertex of the 5-cell). Pick out any one of the 3-orthoschemes of the six shown in the 3-cube illustration. Notice that it touches four of the cube's eight vertices, and those four vertices are linked by a 3-edge path that makes two right-angled turns. Imagine that this 3-orthoscheme is the base of a 4-orthoscheme, so that from each of those four vertices, an unseen 4-orthoscheme edge connects to a fifth apex vertex (which is outside the 3-cube and does not appear in the illustration at all). Although the four additional edges all reach the same apex vertex, they will all be of different lengths. The first of them, at one end of the 3-edge orthogonal path, extends that path with a fourth orthogonal √1 edge by making a third 90 degree turn and reaching perpendicularly into the fourth dimension to the apex. The second of the four additional edges is a √2 diagonal of a cube face (not of the illustrated 3-cube, but of another of the tesseract's eight 3-cubes).^{ [lower-alpha 4] } The third additional edge is a √3 diagonal of a 3-cube (again, not the original illustrated 3-cube). The fourth additional edge (at the other end of the orthogonal path) is a long diameter of the tesseract itself, of length √4. It reaches through the exact center of the tesseract to the antipodal vertex (a vertex of the opposing 3-cube), which is the apex. Thus the **characteristic 5-cell of the 4-cube** has four √1 edges, three √2 edges, two √3 edges, and one √4 edge.

The 4-cube can be dissected into 24 such 4-orthoschemes eight different ways, with six 4-orthoschemes surrounding each of four orthogonal √4 tesseract long diameters. The 4-cube can also be dissected into 384 *smaller* instances of this same characteristic 4-orthoscheme, just one way, by all of its symmetry hyperplanes at once, which divide it into 384 4-orthoschemes that all meet at the center of the 4-cube.^{ [lower-alpha 5] }

More generally, any regular polytope can be dissected into *g* instances of its characteristic orthoscheme that all meet at the regular polytope's center.^{ [10] } The number *g* is the *order* of the polytope, the number of reflected instances of its characteristic orthoscheme that comprise the polytope when a *single* mirror-surfaced orthoscheme instance is reflected in its own facets.^{ [lower-alpha 6] } More generally still, characteristic simplexes are able to fill uniform polytopes because they possess all the requisite elements of the polytope. They also possess all the requisite angles between elements (from 90 degrees on down). The characteristic simplexes are the genetic codes of polytopes: like a Swiss Army knife, they contain one of everything needed to construct the polytope by replication.

Every regular polytope, including the regular 5-cell, has its characteristic orthoscheme.^{ [lower-alpha 7] } There is a 4-orthoscheme which is the **characteristic 5-cell of the regular 5-cell**. It is a tetrahedral pyramid based on the characteristic tetrahedron of the regular tetrahedron. The regular 5-cell can be dissected into 120 instances of this characteristic 4-orthoscheme just one way, by all of its symmetry hyperplanes at once, which divide it into 120 4-orthoschemes that all meet at the center of the regular 5-cell.^{ [lower-alpha 8] }

Characteristics of the regular 5-cell^{ [14] } | |||||
---|---|---|---|---|---|

edge^{ [15] } | arc | dihedral^{ [16] } | |||

𝒍 | 104°30′40″ | 75°29′20″ | |||

𝟀 | 75°29′20″ | 60° | |||

𝝓 | 52°15′20″ | 60° | |||

𝟁 | 52°15′20″ | 60° | |||

75°29′20″ | 90° | ||||

52°15′20″ | 90° | ||||

52°15′20″ | 90° | ||||

37°44′40″ |

The characteristic 5-cell (4-orthoscheme) of the regular 5-cell has four more edges than its base characteristic tetrahedron (3-orthoscheme), which join the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 5-cell).^{ [lower-alpha 9] } If the regular 5-cell has unit radius and edge length 𝒍 = , its characteristic 5-cell's ten edges have lengths , , (the exterior right triangle face, the *characteristic triangle* 𝟀, 𝝓, 𝟁), plus , , (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the *characteristic radii* of the regular tetrahedron), plus , , , (edges which are the characteristic radii of the regular 5-cell). The 4-edge path along orthogonal edges of the orthoscheme is , , , , first from a regular 5-cell vertex to a regular 5-cell edge center, then turning 90° to a regular 5-cell face center, then turning 90° to a regular 5-cell tetrahedral cell center, then turning 90° to the regular 5-cell center.^{ [lower-alpha 10] }

There are many lower symmetry forms of the 5-cell, including these found as 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 | 3-2 fusil | Pentagonal hyperdisphenoid | |

Schläfli | {3,3,3} | {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 with Schläfli symbols ( )∨{3,3}.

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 | ||
---|---|---|---|---|---|---|

Schläfli | {3,3}∨( ) | {3}∨( )∨( ) | { }∨{ }∨( ) | |||

Schlegel diagram | ||||||

Name Coxeter | t_{12}α_{5} | t_{12}γ_{5} | t_{012}α_{5} | t_{012}γ_{5} | t_{123}α_{5} | t_{123}γ_{5} |

Symmetry | [2,1,1], order 2 | [2^{+},1,1], order 2 | [ ]^{+}, order 1 | ||
---|---|---|---|---|---|

Schläfli | { }∨( )∨( )∨( ) | ( )∨( )∨( )∨( )∨( ) | |||

Schlegel diagram | |||||

Name Coxeter | t_{0123}α_{5} | t_{0123}γ_{5} | t_{0123}β_{5} | t_{01234}α_{5} | t_{01234}γ_{5} |

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} | t_{0,3}{3,3,3} | t_{0,1,3}{3,3,3} | t_{0,1,2,3}{3,3,3} |
---|---|---|---|---|---|---|---|---|---|

Coxeter | |||||||||

Schlegel |

1_{k2} figures in n dimensions | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Space | Finite | Euclidean | Hyperbolic | ||||||||

n | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |||

Coxeter group | E_{3}=A_{2}A_{1} | E_{4}=A_{4} | E_{5}=D_{5} | E_{6} | E_{7} | E_{8} | E_{9} = = E_{8}^{+} | E_{10} = = E_{8}^{++} | |||

Coxeter diagram | |||||||||||

Symmetry (order) | [3^{−1,2,1}] | [3^{0,2,1}] | [3^{1,2,1}] | [[3^{2,2,1}]] | [3^{3,2,1}] | [3^{4,2,1}] | [3^{5,2,1}] | [3^{6,2,1}] | |||

Order | 12 | 120 | 1,920 | 103,680 | 2,903,040 | 696,729,600 | ∞ | ||||

Graph | - | - | |||||||||

Name | 1_{−1,2} | 1_{02} | 1_{12} | 1_{22} | 1_{32} | 1_{42} | 1_{52} | 1_{62} |

2_{k1} figures in n dimensions | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Space | Finite | Euclidean | Hyperbolic | ||||||||

n | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |||

Coxeter group | E_{3}=A_{2}A_{1} | E_{4}=A_{4} | E_{5}=D_{5} | E_{6} | E_{7} | E_{8} | E_{9} = = E_{8}^{+} | E_{10} = = E_{8}^{++} | |||

Coxeter diagram | |||||||||||

Symmetry | [3^{−1,2,1}] | [3^{0,2,1}] | [[3^{1,2,1}]] | [3^{2,2,1}] | [3^{3,2,1}] | [3^{4,2,1}] | [3^{5,2,1}] | [3^{6,2,1}] | |||

Order | 12 | 120 | 384 | 51,840 | 2,903,040 | 696,729,600 | ∞ | ||||

Graph | - | - | |||||||||

Name | 2_{−1,1} | 2_{01} | 2_{11} | 2_{21} | 2_{31} | 2_{41} | 2_{51} | 2_{61} |

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

{p,3,3} polytopes | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Space | S^{3} | H^{3} | |||||||||

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 {3,3,p} regular 4-polytopes with tetrahedral cells, along with the 16-cell {3,3,4} and 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 | S^{3} | H^{3} | |||||||||

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,∞} |

It is self-dual like the 24-cell {3,4,3}, having a palindromic {3,p,3} Schläfli symbol.

{3,p,3} polytopes | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Space | S^{3} | H^{3} | |||||||||

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 | S^{3} | Euclidean E^{3} | H^{3} | ||||||||

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,∞} |

- 1 2 A 5-cell's 5 vertices form 5 tetrahedral cells face-bonded to each other, with a total of 10 edges and 10 triangular faces.
- ↑ The convex regular 4-polytopes can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. Each greater polytope in the sequence is
*rounder*than its predecessor, enclosing more content^{ [5] }within the same radius. The 4-simplex (5-cell) is the limit smallest case, and the 120-cell is the largest. Complexity (as measured by comparing configuration matrices or simply the number of vertices) follows the same ordering. This provides an alternative numerical naming scheme for regular polytopes in which the 5-cell is the 5-point 4-polytope: first in the ascending sequence that runs to the 600-point 4-polytope. - ↑ The regular 120-cell has a curved 3-dimensional boundary surface consisting of 120 regular dodecahedron cells. It also has 120 disjoint regular 5-cells inscribed in it.
^{ [6] }These are not 3-dimensional cells but 4-dimensional objects which share the 120-cell's center point, and collectively cover all 600 of its vertices. - ↑ The 4-cube (tesseract) contains eight 3-cubes (so it is also called the 8-cell). Each 3-cube is face-bonded to six others (that entirely surround it), but entirely disjoint from the one other 3-cube which lies opposite and parallel to it on the other side of the 8-cell.
- ↑ The dissection of the 4-cube into 384 4-orthoschemes is 16 of the dissections into 24 4-orthoschemes. First, each 4-cube edge is divided into 2 smaller edges, so each square face is divided into 4 smaller squares, each cubical cell is divided into 8 smaller cubes, and the entire 4-cube is divided into 16 smaller 4-cubes. Then each smaller 4-cube is divided into 24 4-orthoschemes that meet at the center of the original 4-cube.
- ↑ For a regular
*k*-polytope, the Coxeter-Dynkin diagram of the characteristic*k-*orthoscheme is the*k*-polytope's diagram without the generating point ring. The regular*k-*polytope is subdivided by its symmetry (*k*-1)-elements into*g*instances of its characteristic*k*-orthoscheme that surround its center, where*g*is the*order*of the*k*-polytope's symmetry group.^{ [11] } - ↑ A regular polytope of dimension
*k*has a characteristic*k*-orthoscheme, and also a characteristic (*k*-1)-orthoscheme. A regular 4-polytope has a characteristic 5-cell (4-orthoscheme) into which it is subdivided by its (3-dimensional) hyperplanes of symmetry, and also a characteristic tetrahedron (3-orthoscheme) into which its surface is subdivided by its cells' (2-dimensional) planes of symmetry. After subdividing its (3-dimensional) surface into characteristic tetrahedra surrounding each cell center, its (4-dimensional) interior can be subdivided into characteristic 5-cells by adding radii joining the vertices of the surface characteristic tetrahedra to the 4-polytope's center.^{ [12] }The interior tetrahedra and triangles thus formed will also be orthoschemes. - ↑ The 120 congruent
^{ [13] }4-orthoschemes of the regular 5-cell occur in two mirror-image forms, 60 of each. Each 4-orthoscheme is cell-bonded to 4 others of the opposite chirality (by the 4 of its 5 tetrahedral cells that lie in the interior of the regular 5-cell). If the 60 left-handed 4-orthoschemes are colored red and the 60 right-handed 4-orthoschemes are colored black, each red 5-cell is surrounded by 4 black 5-cells and vice versa, in a pattern 4-dimensionally analogous to a checkerboard (if checkerboards had triangles instead of squares). - ↑ The four edges of each 4-orthoscheme which meet at the center of a regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.
- ↑ If the regular 5-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths , , (the exterior right triangle face, the
*characteristic triangle*), plus , , (the other three edges of the exterior 3-orthoscheme facet the*characteristic tetrahedron*), plus , , , (edges that are the characteristic radii of the regular 5-cell).^{ [14] }The 4-edge path along orthogonal edges of the orthoscheme is , , , .

- ↑ 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 - ↑ Matila Ghyka,
*The geometry of Art and Life*(1977), p.68 - ↑ Coxeter 1973, p. 120, §7.2. see illustration Fig 7.2A.
- ↑ Category 1: Regular Polychora
- ↑ Coxeter 1973, pp. 292–293, Table I(ii): The sixteen regular polytopes {
*p,q,r*} in four dimensions; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius. - ↑ Coxeter 1973, p. 305, Table VII: Regular Compounds in Four Dimensions.
- ↑ Coxeter 1973, p. 12, §1.8. Configurations.
- ↑ Coxeter 1991, p. 30, §4.2. The Crystallographic regular polytopes.
- ↑ Coxeter 1973, pp. 198–202, §11.7 Regular figures and their truncations.
- ↑ Kim & Rote 2016, pp. 17–20, §10 The Coxeter Classification of Four-Dimensional Point Groups.
- ↑ Coxeter 1973, pp. 130–133, §7.6 The symmetry group of the general regular polytope.
- ↑ Coxeter 1973, p. 130, §7.6; "simplicial subdivision".
- ↑ Coxeter 1973, §3.1 Congruent transformations.
- 1 2 Coxeter 1973, pp. 292–293, Table I(ii); "5-cell, 𝛼
_{4}". - ↑ Coxeter 1973, p. 139, §7.9 The characteristic simplex.
- ↑ Coxeter 1973, p. 290, Table I(ii); "dihedral angles".

A **cuboctahedron** is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is radially equilateral.

In geometry, a **cube** is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.

In geometry, a **regular icosahedron** is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most faces.

In geometry, an **octahedron** is a polyhedron with eight faces. The term is most commonly used to refer to the **regular** octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.

In geometry, a **Platonic solid** is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent regular polygons, and the same number of faces meet at each vertex. There are only five such polyhedra:

In geometry, a **tetrahedron**, also known as a **triangular pyramid**, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces.

In geometry, a **tesseract** is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eight cubical cells. The tesseract is one of the six convex regular 4-polytopes.

In geometry, the **24-cell** is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,4,3}. It is also called **C _{24}**, or the

In geometry, the **600-cell** is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,5}. It is also known as the **C _{600}**,

In geometry, the **16-cell** is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,4}. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. It is also called **C _{16}**,

In geometry, a **pyramid** is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a *lateral face*. It is a conic solid with polygonal base. A pyramid with an n-sided base has *n* + 1 vertices, *n* + 1 faces, and 2*n* edges. All pyramids are self-dual.

In geometry, the **snub 24-cell** or **snub disicositetrachoron** is a convex uniform 4-polytope composed of 120 regular tetrahedral and 24 icosahedral cells. Five tetrahedra and three icosahedra meet at each vertex. In total it has 480 triangular faces, 432 edges, and 96 vertices. One can build it from the 600-cell by diminishing a select subset of icosahedral pyramids and leaving only their icosahedral bases, thereby removing 480 tetrahedra and replacing them with 24 icosahedra.

The **tetrahedral-octahedral honeycomb**, **alternated cubic honeycomb** is a quasiregular space-filling tessellation in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2.

In geometry, a **truncated tesseract** is a uniform 4-polytope formed as the truncation of the regular tesseract.

In geometry, a **truncated 5-cell** is a uniform 4-polytope formed as the truncation of the regular 5-cell.

In geometry, a **quasiregular polyhedron** is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are vertex-transitive and edge-transitive, hence a step closer to regular polyhedra than the semiregular, which are merely vertex-transitive.

In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope. It has six vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 5-cell facets. It has a dihedral angle of cos^{−1}(1/5), or approximately 78.46°.

The **Boerdijk–Coxeter helix**, named after H. S. M. Coxeter and A. H. Boerdijk, is a linear stacking of regular tetrahedra, arranged so that the edges of the complex that belong to only one tetrahedron form three intertwined helices. There are two chiral forms, with either clockwise or counterclockwise windings. Unlike any other stacking of Platonic solids, the Boerdijk–Coxeter helix is not rotationally repetitive in 3-dimensional space. Even in an infinite string of stacked tetrahedra, no two tetrahedra will have the same orientation, because the helical pitch per cell is not a rational fraction of the circle. However, modified forms of this helix have been found which are rotationally repetitive, and in 4-dimensional space this helix repeats in rings of exactly 30 tetrahedral cells that tessellate the 3-sphere surface of the 600-cell, one of the six regular convex polychora.

In geometry, a **Schläfli orthoscheme** is a type of simplex. The orthoscheme is the generalization of the right triangle to simplex figures of any number of dimensions. Orthoschemes are defined by a sequence of edges that are mutually orthogonal. They were introduced by Ludwig Schläfli, who called them *orthoschemes* and studied their volume in Euclidean, hyperbolic, and spherical geometries. H. S. M. Coxeter later named them after Schläfli. As right triangles provide the basis for trigonometry, orthoschemes form the basis of a trigonometry of *n* dimensions, as developed by Schoute who called it **polygonometry**. J.-P. Sydler and Børge Jessen studied orthoschemes extensively in connection with Hilbert's third problem.

In geometry, the **Hessian polyhedron** is a regular complex polyhedron _{3}{3}_{3}{3}_{3}, , in . It has 27 vertices, 72 _{3}{} edges, and 27 _{3}{3}_{3} faces. It is self-dual.

- 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]

- (Paper 22) H.S.M. Coxeter,

- Coxeter, H.S.M. (1973).
- Kim, Heuna; Rote, G. (2016). "Congruence Testing of Point Sets in 4 Dimensions". arXiv: 1603.07269 [cs.CG].
- 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)

- N.W. Johnson:

- Weisstein, Eric W. "Pentatope".
*MathWorld*. - Klitzing, Richard. "4D uniform polytopes (polychora) x3o3o3o - pen".
- Der 5-Zeller (5-cell) Marco Möller's Regular polytopes in R
^{4}(German) - Jonathan Bowers, Regular polychora
- Java3D Applets
- pyrochoron

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