| 5-cube Penteract |
|
|---|---|
Orthogonal projection inside Petrie polygon Central orange vertex is doubled |
|
| Type | Regular 5-polytope |
| Family | hypercube |
| Schläfli symbols | {4,3,3,3} {4,3,3}x{} {4,3}x{4} {4,3}x{}x{} {4}x{4}x{} {4}x{}x{}x{} {}x{}x{}x{}x{} |
| Coxeter-Dynkin diagrams | |
| Hypercells | 10 tesseracts |
| Cells | 40 cubes |
| Faces | 80 squares |
| Edges | 80 |
| Vertices | 32 |
| Vertex figure | 5-cell |
| Petrie polygon | decagon |
| Coxeter group | BC5, [3,3,3,4] |
| Dual | 5-orthoplex |
| Properties | convex |
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 hypercells.
It is represented by Schläfli symbol {4,33}, constructed as 3 tesseracts, {4,3,3}, around each cubic ridge. It can be called a penteract, derived from combining the name tesseract (the 4-cube) with pente for five (dimensions) in Greek. It can also be called a regular deca-5-tope or decateron, being a 5 dimensional polytope constructed from 10 regular facets.
Contents |
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.
Cartesian coordinates for the vertices of a 5-cube centered at the origin and edge length 2 are
while the interior of the same consists of all points (x0, x1, x2, x3, x4) with -1 < xi < 1.
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.
| Coxeter plane | B5 | B4 / D5 | B3 / D4 / A2 |
|---|---|---|---|
| Graph | |||
| Dihedral symmetry | [10] | [8] | [6] |
| Coxeter plane | B2 | A3 | |
| Graph | |||
| Dihedral symmetry | [4] | [4] |
Wireframe skew direction |
B5 Coxeter plane |
This oriention shows columns of vertices positioned a vertex-edge-vertex distance from one vertex on the left to one vertex on the right, and edges attaching adjacent columns of vertices. The number of vertices in each column represents rows in Pascal's triangle, being 1:5:10:10:5:1. |
Vertex-edge graph. |
A perspective projection 3D to 2D of stereographic projection 4D to 3D of Schlegel diagram 5D to 4D. |
Animation of a 5D rotation of a 5-cube perspective projection to 3D. |
This polytope is one of 31 uniform polytera generated from the regular 5-cube or 5-orthoplex.