5-cube penteract (pent) | ||
---|---|---|

Type | uniform 5-polytope | |

Schläfli symbol | {4,3,3,3} | |

Coxeter diagram | ||

4-faces | 10 | tesseracts |

Cells | 40 | cubes |

Faces | 80 | squares |

Edges | 80 | |

Vertices | 32 | |

Vertex figure | 5-cell | |

Coxeter group | B_{5}, [4,3^{3}], order 3840 | |

Dual | 5-orthoplex | |

Base point | (1,1,1,1,1,1) | |

Circumradius | sqrt(5)/2 = 1.118034 | |

Properties | convex, isogonal regular |

In five-dimensional geometry, a **5-cube** is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces.

- Related polytopes
- As a configuration
- Cartesian coordinates
- Images
- Projection
- Symmetry
- Prisms
- Related polytopes 2
- References
- External links

It is represented by Schläfli symbol {4,3,3,3} or {4,3^{3}}, constructed as 3 tesseracts, {4,3,3}, around each cubic ridge. It can be called a **penteract**, a portmanteau of the Greek word * pénte *, for 'five' (dimensions), and the word * tesseract * (the 4-cube). It can also be called a regular **deca-5-tope** or **decateron**, being a 5-dimensional polytope constructed from 10 regular facets.

It is a part of an infinite hypercube family. The dual of a 5-cube is the 5-orthoplex, of the infinite family of orthoplexes.

Applying an * alternation * operation, deleting alternating vertices of the 5-cube, creates another uniform 5-polytope, called a 5-demicube, which is also part of an infinite family called the demihypercubes.

The 5-cube can be seen as an *order-3 tesseractic honeycomb* on a 4-sphere. It is related to the Euclidean 4-space (order-4) tesseractic honeycomb and paracompact hyperbolic honeycomb order-5 tesseractic honeycomb.

This configuration matrix represents the 5-cube. The rows and columns correspond to vertices, edges, faces, cells, and 4-faces. The diagonal numbers say how many of each element occur in the whole 5-cube. The nondiagonal numbers say how many of the column's element occur in or at the row's element.^{ [1] }^{ [2] }

The Cartesian coordinates of the vertices of a 5-cube centered at the origin and having edge length 2 are

- (±1,±1,±1,±1,±1),

while this 5-cube's interior consists of all points (*x*_{0}, *x*_{1}, *x*_{2}, *x*_{3}, *x*_{4}) with -1 < *x*_{i} < 1 for all *i*.

*n*-cube Coxeter plane projections in the B_{k} Coxeter groups project into k-cube graphs, with power of two vertices overlapping in the projective graphs.

Coxeter plane | B_{5} | B_{4} / D_{5} | B_{3} / D_{4} / A_{2} |
---|---|---|---|

Graph | |||

Dihedral symmetry | [10] | [8] | [6] |

Coxeter plane | Other | B_{2} | A_{3} |

Graph | |||

Dihedral symmetry | [2] | [4] | [4] |

Wireframe skew direction | B5 Coxeter plane |

Vertex-edge graph. |

A perspective projection 3D to 2D of stereographic projection 4D to 3D of Schlegel diagram 5D to 4D. |

4D net of the 5-cube, perspective projected into 3D. |

The 5-cube can be projected down to 3 dimensions with a rhombic icosahedron envelope. There are 22 exterior vertices, and 10 interior vertices. The 10 interior vertices have the convex hull of a pentagonal antiprism. The 80 edges project into 40 external edges and 40 internal ones. The 40 cubes project into golden rhombohedra which can be used to dissect the rhombic icosahedron. The projection vectors are u = {1, φ, 0, -1, φ}, v = {φ, 0, 1, φ, 0}, w = {0, 1, φ, 0, -1}, where φ is the golden ratio, .

rhombic icosahedron | 5-cube | |
---|---|---|

Perspective | orthogonal | |

The *5-cube* has Coxeter group symmetry B_{5}, abstract structure , order 3840, containing 25 hyperplanes of reflection. The Schläfli symbol for the 5-cube, {4,3,3,3}, matches the Coxeter notation symmetry [4,3,3,3].

All hypercube have lower symmetry forms constructed as prisms. The 5-cube has 7 prismatic forms from the lowest 5-orthotope, { }^{5}, and upwards as orthogonal edges are constrained to be of equal length. The vertices in a prism are equal to the product of the vertices in the elements. The edges of a prism can be partitioned into the number of edges in an element times the number of vertices in all the other elements.

Description | Schläfli symbol | Coxeter-Dynkin diagram | Vertices | Edges | Coxeter notation Symmetry | Order |
---|---|---|---|---|---|---|

5-cube | {4,3,3,3} | 32 | 80 | [4,3,3,3] | 3840 | |

tesseractic prism | {4,3,3}×{ } | 16×2 = 32 | 64 + 16 = 80 | [4,3,3,2] | 768 | |

cube-square duoprism | {4,3}×{4} | 8×4 = 32 | 48 + 32 = 80 | [4,3,2,4] | 384 | |

cube-rectangle duoprism | {4,3}×{ }^{2} | 8×2^{2} = 32 | 48 + 2×16 = 80 | [4,3,2,2] | 192 | |

square-square duoprism prism | {4}^{2}×{ } | 4^{2}×2 = 32 | 2×32 + 16 = 80 | [4,2,4,2] | 128 | |

square-rectangular parallelepiped duoprism | {4}×{ }^{3} | 4×2^{3} = 32 | 32 + 3×16 = 80 | [4,2,2,2] | 64 | |

5-orthotope | { }^{5} | 2^{5} = 32 | 5×16 = 80 | [2,2,2,2] | 32 |

The *5-cube* is 5th in a series of hypercube:

Line segment | Square | Cube | 4-cube | 5-cube | 6-cube | 7-cube | 8-cube |

The regular skew polyhedron {4,5| 4} can be realized within the 5-cube, with its 32 vertices, 80 edges, and 40 square faces, and the other 40 square faces of the 5-cube become square *holes*.

This polytope is one of 31 uniform 5-polytopes generated from the regular 5-cube or 5-orthoplex.

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 **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. It is also known as a **C _{5}**,

In four-dimensional geometry, a **cantellated tesseract** is a convex uniform 4-polytope, being a cantellation of the regular tesseract.

In geometry, the **rectified tesseract**, **rectified 8-cell** is a uniform 4-polytope bounded by 24 cells: 8 cuboctahedra, and 16 tetrahedra. It has half the vertices of a runcinated tesseract, with its construction, called a **runcic tesseract**.

In four-dimensional geometry, a **cantellated 5-cell** is a convex uniform 4-polytope, being a cantellation of the regular 5-cell.

In five-dimensional geometry, a **5-orthoplex**, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron cells, 32 5-cell 4-faces.

In geometry, a **6-cube** is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube 5-faces.

In geometry, a **6-orthoplex**, or 6-cross polytope, is a regular 6-polytope with 12 vertices, 60 edges, 160 triangle faces, 240 tetrahedron cells, 192 5-cell *4-faces*, and 64 *5-faces*.

In geometry, a **7-cube** is a seven-dimensional hypercube with 128 vertices, 448 edges, 672 square faces, 560 cubic cells, 280 tesseract 4-faces, 84 penteract 5-faces, and 14 hexeract 6-faces.

In geometry, an **8-cube** is an eight-dimensional hypercube. It has 256 vertices, 1024 edges, 1792 square faces, 1792 cubic cells, 1120 tesseract 4-faces, 448 5-cube 5-faces, 112 6-cube 6-faces, and 16 7-cube 7-faces.

In geometry, a **9-cube** is a nine-dimensional hypercube with 512 vertices, 2304 edges, 4608 square faces, 5376 cubic cells, 4032 tesseract 4-faces, 2016 5-cube 5-faces, 672 6-cube 6-faces, 144 7-cube 7-faces, and 18 8-cube 8-faces.

In geometry, a **7-orthoplex**, or 7-cross polytope, is a regular 7-polytope with 14 vertices, 84 edges, 280 triangle faces, 560 tetrahedron cells, 672 5-cells *4-faces*, 448 *5-faces*, and 128 *6-faces*.

In geometry, an **8-orthoplex** or 8-cross polytope is a regular 8-polytope with 16 vertices, 112 edges, 448 triangle faces, 1120 tetrahedron cells, 1792 5-cells *4-faces*, 1792 *5-faces*, 1024 *6-faces*, and 256 *7-faces*.

In geometry, a **uniform 5-polytope** is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope facets.

In geometry, a **10-cube** is a ten-dimensional hypercube. It has 1024 vertices, 5120 edges, 11520 square faces, 15360 cubic cells, 13440 tesseract 4-faces, 8064 5-cube 5-faces, 3360 6-cube 6-faces, 960 7-cube 7-faces, 180 8-cube 8-faces, and 20 9-cube 9-faces.

In five-dimensional geometry, a **rectified 5-orthoplex** is a convex uniform 5-polytope, being a rectification of the regular 5-orthoplex.

In six-dimensional geometry, a **rectified 6-cube** is a convex uniform 6-polytope, being a rectification of the regular 6-cube.

In five-dimensional geometry, a **truncated 5-cube** is a convex uniform 5-polytope, being a truncation of the regular 5-cube.

In five-dimensional geometry, a **truncated 5-orthoplex** is a convex uniform 5-polytope, being a truncation of the regular 5-orthoplex.

- 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) **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,
- Norman Johnson
*Uniform Polytopes*, Manuscript (1991)- N.W. Johnson:
*The Theory of Uniform Polytopes and Honeycombs*, Ph.D. (1966)

- N.W. Johnson:
- Klitzing, Richard. "5D uniform polytopes (polytera) o3o3o3o4x - pent".

- Weisstein, Eric W. "Hypercube".
*MathWorld*. - Olshevsky, George. "Measure polytope".
*Glossary for Hyperspace*. Archived from the original on 4 February 2007. - Multi-dimensional Glossary: hypercube Garrett Jones

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