5-cube explained

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.

It is represented by Schläfli symbol or, constructed as 3 tesseracts,, around each cubic ridge.

Related polytopes

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]

\begin{bmatrix}\begin{matrix} 32&5&10&10&5\ 2&80&4&6&4\ 4&4&80&3&3\ 8&12&6&40&2\ 16&32&24&8&10\end{matrix}\end{bmatrix}

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 planeB5B4 / D5B3 / D4 / A2
Graph
Dihedral symmetry[10][8][6]
Coxeter planeOtherB2A3
Graph
Dihedral symmetry[2][4][4]

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 =, v =, w =, where φ is the golden ratio,

1+\sqrt{5
}.It is also possible to project penteracts into three-dimensional space, similarly to projecting a cube into two-dimensional space.

Symmetry

The 5-cube has Coxeter group symmetry B5, abstract structure

C2\wrS5

, order 3840, containing 25 hyperplanes of reflection. The Schläfli symbol for the 5-cube,, matches the Coxeter notation symmetry [4,3,3,3].

Prisms

All hypercubes 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.

DescriptionSchläfli symbolCoxeter-Dynkin diagramVerticesEdgesCoxeter notation
Symmetry
Order
5-cube3280[4,3,3,3]3840
tesseractic prism×16×2 = 3264 + 16 = 80 [4,3,3,2]768
cube-square duoprism×8×4 = 3248 + 32 = 80 [4,3,2,4]384
cube-rectangle duoprism×28×22 = 3248 + 2×16 = 80 [4,3,2,2]192
square-square duoprism prism2×42×2 = 322×32 + 16 = 80 [4,2,4,2]128
square-rectangular parallelepiped duoprism×34×23 = 3232 + 3×16 = 80 [4,2,2,2]64
5-orthotope525 = 325×16 = 80[2,2,2,2]32

Related polytopes

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

The regular skew polyhedron 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.

References

  1. Coxeter, Regular Polytopes, sec 1.8 Configurations
  2. Coxeter, Complex Regular Polytopes, p.117

External links