5-cube
(Redirected from 5-hypercube)
5-cube | |
---|---|
Rank | 5 |
Type | Regular |
Notation | |
Bowers style acronym | Pent |
Coxeter diagram | x4o3o3o3o () |
Schläfli symbol | {4,3,3,3} |
Tapertopic notation | 11111 |
Toratopic notation | IIIII |
Bracket notation | [IIIII] |
Elements | |
Tera | 10 tesseracts |
Cells | 40 cubes |
Faces | 80 squares |
Edges | 80 |
Vertices | 32 |
Vertex figure | Pentachoron, edge length √2 |
Petrie polygons | 16 5D decagons |
Measures (edge length 1) | |
Circumradius | |
Edge radius | 1 |
Face radius | |
Cell radius | |
Inradius | |
Hypervolume | 1 |
Diteral angle | 90° |
Height | 1 |
Central density | 1 |
Number of external pieces | 10 |
Level of complexity | 1 |
Related polytopes | |
Army | Pent |
Regiment | Pent |
Dual | 5-orthoplex |
Conjugate | None |
Abstract & topological properties | |
Flag count | 3840 |
Euler characteristic | 2 |
Orientable | Yes |
Skeleton | Q 5 |
Properties | |
Symmetry | B5, order 3840 |
Flag orbits | 1 |
Convex | Yes |
Net count | 9694[1] |
Nature | Tame |
The 5-cube, also called the decateron, penteract, or pent, is a regular 5-polytope. It has 10 tesseracts as facets, joining 5 to a vertex. It is the 5-dimensional hypercube. As such, it is also a tesseractic prism and square-cube duoprism.
Like the hypercubes of every other dimension, the penteract can tile 5D Euclidean space in the penteractic pentacomb.
It can be alternated into a 5-demicube, which is uniform.
Vertex coordinates[edit | edit source]
The vertices of a 5-cube of edge length 1, centered at the origin, are given by:
- .
Representations[edit | edit source]
A 5-cube has the following Coxeter diagrams:
- x4o3o3o3o () (full symmetry)
- x x4o3o3o () (B4×A1 symmetry, tesseractic prism)
- x4o x4o3o () (B3×B2 symmetry, square-cubic duoprism)
- x x x4o3o () (B3×K2 symmetry, cubic prismatic prism)
- x x4o x4o () (B2×B2×A1 symmetry, square duoprismatic prism)
- x x x x4o () (B2×K3 symmetry, square prismatic prismatic prism)
- x x x x x () (K5 symmetry, all five dimensions separate)
- xx4oo3oo3oo&#x (B4 axial, tesseract atop tesseract)
- xx xx4oo3oo&#x (B3×A1 axial, cubic prism bases)
- xx4oo xx4oo&#x (B3×B2 axial, square duoprismatic bases)
- xx xx xx4oo&#x (B2×K2 axial, square prismatic prism bases)
- xx xx xx xx&#x (K4 symmetry, bases have four separate dimensions)
- oqo xxx4ooo3ooo&#xt (B3×A1 axial, cell-first)
- xxxx4oooo oqoo3ooqo&#xt (B2×A1 axial, face-first)
- xxxxx oqooo3ooqoo3oooqo&#xt (A3×A1 axial, edge-first)
- oqoooo3ooqooo3oooqoo3ooooqo&#xt (A4 axial, vertex-first)
- qo3oo3oq *b3oo3oo&#zx (D5 symmetry)
Gallery[edit | edit source]
-
A rotating penteract, with two opposite tesseracts highlighted.
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A penteract projected into a rhombic icosahedron envelope.
-
Stereographic projection of a penteract.
External links[edit | edit source]
- Bowers, Jonathan. "Category 1: Primary Polytera" (#2).
- Klitzing, Richard. "pent".
- Wikipedia contributors. "5-cube".
- Hi.gher.Space Wiki Contributors. "Geoteron".