Penteract

Penteract
(OFF file)
Rank	5
Type	Regular
Space	Spherical
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
Measures (edge length 1)
Circumradius	$\frac{\sqrt5}{2} \approx 1.11803$
Edge radius	1
Face radius	$\frac{\sqrt3}{2} \approx 0.86603$
Cell radius	$\frac{\sqrt2}{2} \approx 0.70111$
Inradius	$\frac12 = 0.5$
Hypervolume	1
Diteral angle	90°
Height	1
Central density	1
Number of pieces	10
Level of complexity	1
Related polytopes
Army	Pent
Regiment	Pent
Dual	Triacontaditeron
Conjugate	None
Abstract properties
Flag count	3840
Net count	9694
Euler characteristic	2
Topological properties
Orientable	Yes
Properties
Symmetry	B5, order 3840
Convex	Yes
Nature	Tame

The penteract, or pent, also called the 5-cube or decateron, is one of the 3 regular polytera. 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 demipenteract, which is uniform.

Vertex coordinates

The vertices of a penteract of edge length 1, centered at the origin, are given by:

• $\left(±\frac12,\,±\frac12,\,±\frac12,\,±\frac12,\,±\frac12\right).$

Representations

A penteract has the following Coxeter diagrams:

• x4o3o3o3o (full ysmmetry)
• x x4o3o3o (BC4×A1 symmetry, tesseractic prism)
• x4o x4o3o (BC3×BC2 symmetry, square-cubic duoprism)
• x x x4o3o (BC3×A1×A1 symmetry, cubic prismatic prism)
• x x4o x4o (BC2×BC2×A1 symmetry, square duoprismatic prism)
• x x x x4o (BC2×A1×A1×A1 symmetry, square prismatic prismatic prism)
• x x x x x (A1×A1×A1×A1×A1 symmetry, all five dimensions separate)
• xx4oo3oo3oo&#x (BC4 axial, tesseract atop tesseract)
• xx xx4oo3oo&#x (BC3×A1 axial, cubic prism bases)
• xx4oo xx4oo&#x (BC2×BC2 axial, square duoprismatic bases)
• xx xx xx4oo&#x (BC2×A1×A1 axial, square prismatic prism bases)
• xx xx xx xx&#x (A1×A1×A1×A1 symetry, bases have four separate dimensions)
• oqo xxx4ooo3ooo&#xt (BC3×A1 axial, cell-first)
• xxxx4oooo oqoo3ooqo&#xt (BC2×A1 axial, face-first)
• xxxxx oqooo3ooqoo3oooqo&#xt (A3×A1 axial, edge-first)
• oqoooo3ooqooo3oooqoo3ooooqo&#xt (A4 axial, vertex-first)
• qo3oo3oq *b3oo3oo&#zx (D5 symmetry)

Gallery

• 

  A rotating penteract, with two opposite tesseracts highlighted.

• 

  A penteract projected into a rhombic icosahedron envelope.

• 

  Stereographic projection of a penteract.

External links

• Bowers, Jonathan. "Category 1: Primary Polytera" (#2).

• Klitzing, Richard. "pent".

• Wikipedia Contributors. "5-cube".
• Hi.gher.Space Wiki Contributors. "Geoteron".