5cube
5cube penteract (pent) 


Type  uniform 5polytope  
Schläfli symbol  {4,3,3,3} {4,3,3}×{ } {4,3}×{4} {4,3}×{ }×{ } {4}×{4}×{ } {4}×{ }×{ }×{ } { }×{ }×{ }×{ }×{ } 

CoxeterDynkin diagram  
4faces  10  tesseracts 
Cells  40  cubes 
Faces  80  squares 
Edges  80  
Vertices  32  
Vertex figure  5cell 

Coxeter group  BC_{5}, [4,3^{3}, order 3840 [4,3,3,2], order 768 [4,3,2,4], order 384 [4,3,2,2], order 192 [4,2,4,2], order 128 [4,2,2,2], order 64 [2,2,2,2], order 32 

Dual  5orthoplex  
Base point  (1,1,1,1,1,1)  
Circumradius  sqrt(5)/2 = 1.118034  
Properties  convex, isogonal regular 
In fivedimensional geometry, a 5cube is a name for a fivedimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4faces.
It is represented by Schläfli symbol {4,3^{3}}, constructed as 3 tesseracts, {4,3,3}, around each cubic ridge. It can be called a penteract, a portmanteau of tesseract (the 4cube) and pente for five (dimensions) in Greek. It can also be called a regular deca5tope or decateron, being a 5dimensional polytope constructed from 10 regular facets.
Contents
Related polytopes
It is a part of an infinite hypercube family. The dual of a 5cube is the 5orthoplex, of the infinite family of orthoplexes.
Applying an alternation operation, deleting alternating vertices of the 5cube, creates another uniform 5polytope, called a 5demicube, which is also part of an infinite family called the demihypercubes.
The 5cube can be seen as an order3 tesseractic honeycomb on a 4sphere. It is related to the Euclidean 4space (order4) tesseractic honeycomb and paracompact hyperbolic honeycomb order5 tesseractic honeycomb.
Cartesian coordinates
Cartesian coordinates for the vertices of a 5cube centered at the origin and edge length 2 are
 (±1,±1,±1,±1,±1)
while the interior of the same consists of all points (x_{0}, x_{1}, x_{2}, x_{3}, x_{4}) with 1 < x_{i} < 1.
Images
ncube Coxeter plane projections in the B_{k} Coxeter groups project into kcube graphs, with power of two vertices overlapping in the projective graphs.
Coxeter plane  B_{5}  B_{4} / D_{5}  B_{3} / D_{4} / A_{2} 

Graph  
Dihedral symmetry  [10]  [8]  [6] 
Coxeter plane  Other  B_{2}  A_{3} 
Graph  
Dihedral symmetry  [2]  [4]  [4] 
Wireframe skew direction 
B5 Coxeter plane 
Vertexedge 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 5cube perspective projection to 3D. 
Related polytopes
This polytope is one of 31 uniform polytera generated from the regular 5cube or 5orthoplex.
References
 H.S.M. Coxeter:
 Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0486614808, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in ndimensions (n≥5)
 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, WileyInterscience Publication, 1995, ISBN 9780471010036 [1]
 (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380407, MR 2,10]
 (Paper 23) H.S.M. Coxeter, Regular and SemiRegular Polytopes II, [Math. Zeit. 188 (1985) 559591]
 (Paper 24) H.S.M. Coxeter, Regular and SemiRegular Polytopes III, [Math. Zeit. 200 (1988) 345]
 Norman Johnson Uniform Polytopes, Manuscript (1991)
 N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
 Richard Klitzing, 5D uniform polytopes (polytera), o3o3o3o4x  pent
External links
 Weisstein, Eric W., "Hypercube", MathWorld.
 Olshevsky, George, Measure polytope at Glossary for Hyperspace.
 Multidimensional Glossary: hypercube Garrett Jones
Fundamental convex regular and uniform polytopes in dimensions 2–10  

Family  A_{n}  BC_{n}  I_{2}(p) / D_{n}  E_{6} / E_{7} / E_{8} / F_{4} / G_{2}  H_{n}  
Regular polygon  Triangle  Square  pgon  Hexagon  Pentagon  
Uniform polyhedron  Tetrahedron  Octahedron • Cube  Demicube  Dodecahedron • Icosahedron  
Uniform polychoron  5cell  16cell • Tesseract  Demitesseract  24cell  120cell • 600cell  
Uniform 5polytope  5simplex  5orthoplex • 5cube  5demicube  
Uniform 6polytope  6simplex  6orthoplex • 6cube  6demicube  1_{22} • 2_{21}  
Uniform 7polytope  7simplex  7orthoplex • 7cube  7demicube  1_{32} • 2_{31} • 3_{21}  
Uniform 8polytope  8simplex  8orthoplex • 8cube  8demicube  1_{42} • 2_{41} • 4_{21}  
Uniform 9polytope  9simplex  9orthoplex • 9cube  9demicube  
Uniform 10polytope  10simplex  10orthoplex • 10cube  10demicube  
Uniform npolytope  nsimplex  northoplex • ncube  ndemicube  1_{k2} • 2_{k1} • k_{21}  npentagonal polytope  
Topics: Polytope families • Regular polytope • List of regular polytopes 
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