5-cube

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5-cube
penteract (pent)
Type uniform 5-polytope
Schläfli symbol {4,3,3,3}
Coxeter diagram CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
4-faces10 tesseracts
Cells40 cubes
Faces80 squares
Edges80
Vertices32
Vertex figure 5-cube verf.png
5-cell
Coxeter group B5, [4,33], 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.

Contents

It is represented by Schläfli symbol {4,3,3,3} or {4,33}, 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.

As a configuration

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]

Cartesian coordinates

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 (x0, x1, x2, x3, x4) with -1 < xi < 1 for all i.

Images

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

Orthographic projections
Coxeter plane B5B4 / D5B3 / D4 / A2
Graph 5-cube t0.svg 4-cube t0.svg 5-cube t0 B3.svg
Dihedral symmetry [10][8][6]
Coxeter planeOtherB2A3
Graph 5-cube column graph.svg 5-cube t0 B2.svg 5-cube t0 A3.svg
Dihedral symmetry[2][4][4]
More orthographic projections
2d of 5d 3.svg
Wireframe skew direction
5-cubePetrie.svg
B5 Coxeter plane
Graph
Penteract graph.svg
Vertex-edge graph.
Perspective projections
Penteract projected.png
A perspective projection 3D to 2D of stereographic projection 4D to 3D of Schlegel diagram 5D to 4D.
Net
The Net of 5-cube.png
4D net of the 5-cube, perspective projected into 3D.

Projection

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 icosahedron5-cube
Perspectiveorthogonal
Rhombic icosahedron.png Dual dodecahedron t1 H3.png 5-cube t0.svg

Symmetry

The 5-cube has Coxeter group symmetry B5, 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].

Prisms

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 VerticesEdges Coxeter notation
Symmetry
Order
5-cube{4,3,3,3}CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png3280[4,3,3,3]3840
tesseractic prism{4,3,3}×{ }CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png16×2 = 3264 + 16 = 80[4,3,3,2]768
cube-square duoprism {4,3}×{4}CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.png8×4 = 3248 + 32 = 80[4,3,2,4]384
cube-rectangle duoprism{4,3}×{ }2CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png8×22 = 3248 + 2×16 = 80[4,3,2,2]192
square-square duoprism prism{4}2×{ }CDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.png42×2 = 322×32 + 16 = 80[4,2,4,2]128
square-rectangular parallelepiped duoprism{4}×{ }3CDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png4×23 = 3232 + 3×16 = 80[4,2,2,2]64
5-orthotope { }5CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png25 = 325×16 = 80[2,2,2,2]32

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

Petrie polygon orthographic projections
1-simplex t0.svg 2-cube.svg 3-cube graph.svg 4-cube graph.svg 5-cube graph.svg 6-cube graph.svg 7-cube graph.svg 8-cube.svg
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.

B5 polytopes
5-cube t4.svg
β5
5-cube t3.svg
t1β5
5-cube t2.svg
t2γ5
5-cube t1.svg
t1γ5
5-cube t0.svg
γ5
5-cube t34.svg
t0,1β5
5-cube t24.svg
t0,2β5
5-cube t23.svg
t1,2β5
5-cube t14.svg
t0,3β5
5-cube t13.svg
t1,3γ5
5-cube t12.svg
t1,2γ5
5-cube t04.svg
t0,4γ5
5-cube t03.svg
t0,3γ5
5-cube t02.svg
t0,2γ5
5-cube t01.svg
t0,1γ5
5-cube t234.svg
t0,1,2β5
5-cube t134.svg
t0,1,3β5
5-cube t124.svg
t0,2,3β5
5-cube t123.svg
t1,2,3γ5
5-cube t034.svg
t0,1,4β5
5-cube t024.svg
t0,2,4γ5
5-cube t023.svg
t0,2,3γ5
5-cube t014.svg
t0,1,4γ5
5-cube t013.svg
t0,1,3γ5
5-cube t012.svg
t0,1,2γ5
5-cube t1234.svg
t0,1,2,3β5
5-cube t0234.svg
t0,1,2,4β5
5-cube t0134.svg
t0,1,3,4γ5
5-cube t0124.svg
t0,1,2,4γ5
5-cube t0123.svg
t0,1,2,3γ5
5-cube t01234.svg
t0,1,2,3,4γ5

Related Research Articles

<span class="mw-page-title-main">Tesseract</span> Four-dimensional analogue of the cube

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.

<span class="mw-page-title-main">5-cell</span> Four-dimensional analogue of the tetrahedron

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 C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It is the 4-simplex (Coxeter's polytope), the simplest possible convex 4-polytope, and is analogous to the tetrahedron in three dimensions and the triangle in two dimensions. The 5-cell is a 4-dimensional pyramid with a tetrahedral base and four tetrahedral sides.

<span class="mw-page-title-main">Cantellated tesseract</span>

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

<span class="mw-page-title-main">Rectified tesseract</span>

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.

<span class="mw-page-title-main">Cantellated 5-cell</span>

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<span class="mw-page-title-main">5-orthoplex</span>

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.

<span class="mw-page-title-main">6-cube</span> 6-dimensional hypercube

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<span class="mw-page-title-main">6-orthoplex</span>

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.

<span class="mw-page-title-main">7-cube</span> 7-dimensional hypercube

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.

<span class="mw-page-title-main">8-cube</span> 8-dimensional hypercube

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.

<span class="mw-page-title-main">9-cube</span> 9-dimensional hypercube

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.

<span class="mw-page-title-main">7-orthoplex</span>

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.

<span class="mw-page-title-main">8-orthoplex</span>

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.

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<span class="mw-page-title-main">10-cube</span> 10-dimensional hypercube

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.

<span class="mw-page-title-main">Rectified 5-orthoplexes</span>

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

<span class="mw-page-title-main">Rectified 6-cubes</span>

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

<span class="mw-page-title-main">Truncated 5-cubes</span>

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

<span class="mw-page-title-main">Truncated 5-orthoplexes</span>

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

References

  1. Coxeter, Regular Polytopes, sec 1.8 Configurations
  2. Coxeter, Complex Regular Polytopes, p.117
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds