View source for 5-cube

{{Short description|5-dimensional hypercube}}
{{Uniform polyteron db|Uniform polyteron stat table|pent}}
In [[Five-dimensional space|five-dimensional]] [[geometry]], a '''5-cube''' is a name for a five-dimensional [[hypercube]] with 32 [[Vertex (geometry)|vertices]], 80 [[Edge (geometry)|edge]]s,  80 square [[Face (geometry)|faces]], 40 cubic [[Cell (mathematics)|cells]], and 10 [[tesseract]] [[4-face]]s.

It is represented by [[Schläfli symbol]] {4,3,3,3} or {4,3<sup>3</sup>}, constructed as 3 tesseracts, {4,3,3}, around each cubic [[Ridge (geometry)|ridge]].

== Related polytopes==
It is a part of an infinite [[hypercube]] family. The [[Dual polytope|dual]] of a 5-cube is the [[5-orthoplex]], of the infinite family of [[orthoplex]]es.

Applying an ''[[Alternation (geometry)|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 [[demihypercube]]s.

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 [[Regular 4-polytope#As configurations|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.<ref>Coxeter, Regular Polytopes, sec 1.8 Configurations</ref><ref>Coxeter, Complex Regular Polytopes, p.117</ref>

<math>\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}</math>

== 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 (''x''<sub>0</sub>, ''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>, ''x''<sub>4</sub>) with -1 < ''x''<sub>''i''</sub> < 1 for all ''i''.

== Images==

''n''-cube [[Coxeter plane]] projections in the B<sub>k</sub> [[Coxeter group]]s project into k-cube graphs, with power of two vertices overlapping in the projective graphs.

{| class=wikitable
|+ [[Orthographic projection]]s
|- align=center
![[Coxeter plane]]
!B<sub>5</sub>
!B<sub>4</sub> / D<sub>5</sub>
!B<sub>3</sub> / D<sub>4</sub> / A<sub>2</sub>
|- align=center
!Graph
|[[File:5-cube t0.svg|150px]]
|[[File:4-cube t0.svg|150px]]
|[[File:5-cube t0 B3.svg|150px]]
|- align=center
![[Dihedral symmetry]]
|[10]
|[8]
|[6]
|- align=center
!Coxeter plane
!Other
!B<sub>2</sub>
!A<sub>3</sub>
|- align=center
!Graph
|[[File:5-cube column graph.svg|150px]]
|[[File:5-cube t0 B2.svg|150px]]
|[[File:5-cube t0 A3.svg|150px]]
|- align=center
!Dihedral symmetry
|[2]
|[4]
|[4]
|}

{| class="wikitable" width="240"
|+ More [[orthographic projection]]s
|- valign=top align=center
|[[File:2d of 5d 3.svg|240px]]<BR>Wireframe skew direction
|[[File:5-cubePetrie.svg|240px]]<BR>B5 [[Coxeter plane]]
|}

{| class="wikitable" width="200"
|+ Graph
|-
|[[File:Penteract graph.svg|200px]]<BR>Vertex-edge graph.
|}
{| class="wikitable" width="240"
|+ [[Perspective projection]]s
|- valign=top
|[[File:Penteract projected.png|240px]]<BR>A [[perspective projection]] 3D to 2D of [[stereographic projection]] 4D to 3D of [[Schlegel diagram]] 5D to 4D.
|}
{| class="wikitable" width="200"
|+ Net
|-
|[[File:The Net of 5-cube.png|200px]]<BR>4D net of the 5-cube, perspective projected into 3D.
|}

== 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 [[dissection (geometry)|dissect]] the rhombic icosahedron. The projection vectors are u = {1, φ, 0, -1, φ}, v = {φ, 0, 1, φ, 0}, w = {0, 1, φ, 0, -1}, where φ is the [[golden ratio]], <math>\frac{1 + \sqrt{5}}{2}</math>.
{| class=wikitable
!colspan=2|rhombic icosahedron
!5-cube
|-
!Perspective
!colspan=2|orthogonal
|-
|[[File:Rhombic icosahedron.png|160px]]
|[[File:Dual dodecahedron t1 H3.png|160px]]
|[[File:5-cube t0.svg|160px]]
|}
It is also possible to project penteracts into three-dimensional space, similarly to projecting a cube into two-dimensional space.
{| class="wikitable" width="240"
|- valign="top" align="center"
|[[File:Penteract-q4q5.gif|280x280px]]A 3D [[Perspective (graphical)|perspective projection]] of a penteract undergoing a [[Rotation|simple rotation]] about the W1-W2 [[Orthogonal coordinates|orthogonal]] [[Rotation plane|plane]]
|[[File:Penteract-q1q4-q3q5.gif|280x280px]]A 3D [[Perspective (graphical)|perspective projection]] of a penteract undergoing a [[double rotation]] about the X-W1 and Z-W2 [[Orthogonal coordinates|orthogonal]] [[Rotation plane|planes]]
|}

== Symmetry==
The ''5-cube'' has [[Coxeter_group#Properties|Coxeter group]] symmetry B<sub>5</sub>, abstract structure <math>C_{2}\wr S_{5}</math>, order 3840, containing 25 hyperplanes of reflection. The [[Schläfli symbol]] for the 5-cube, {4,3,3,3}, matches the [[Coxeter notation]] symmetry [4,3,3,3]. 

=== Prisms ===
All [[hypercube|hypercubes]] have lower symmetry forms constructed as prisms. The 5-cube has 7 prismatic forms from the lowest 5-[[orthotope]], {&nbsp;}<sup>5</sup>, 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.
{| class=wikitable
!Description||[[Schläfli symbol]]||[[Coxeter-Dynkin diagram]]||Vertices||Edges||[[Coxeter notation]]<BR>Symmetry|||[[Order (group theory)|Order]]
|-
|5-cube||align=center|{4,3,3,3}||align=center|{{CDD|node_1|4|node|3|node|3|node|3|node}}||align=right|32||align=right|80||align=center|[4,3,3,3]||3840
|-
|[[tesseract]]ic prism||align=center|{4,3,3}×{&nbsp;}||align=center|{{CDD|node_1|4|node|3|node|3|node|2|node_1}}||align=right|16×2 = 32||align=right|64 + 16 = 80 ||align=center|[4,3,3,2]||768
|-
|[[cube]]-[[square]] [[duoprism]]||align=center|{4,3}×{4}||align=center|{{CDD|node_1|4|node|3|node|2|node_1|4|node}}||align=right|8×4 = 32||align=right|48 + 32 = 80 ||align=center|[4,3,2,4]||384
|-
|cube-[[rectangle]] duoprism||align=center|{4,3}×{&nbsp;}<sup>2</sup>||align=center|{{CDD|node_1|4|node|3|node|2|node_1|2|node_1}}||align=right|8×2<sup>2</sup> = 32||align=right|48 + 2×16 = 80 ||align=center|[4,3,2,2]||192
|-
|square-square [[duoprism]] prism||align=center|{4}<sup>2</sup>×{&nbsp;}||align=center|{{CDD|node_1|4|node|2|node_1|4|node|2|node_1}}||align=right|4<sup>2</sup>×2 = 32||align=right|2×32 + 16 = 80 ||align=center|[4,2,4,2]||128
|-
|square-[[rectangular parallelepiped]] duoprism||align=center|{4}×{&nbsp;}<sup>3</sup>||align=center|{{CDD|node_1|4|node|2|node_1|2|node_1|2|node_1}}||align=right|4×2<sup>3</sup> = 32||align=right|32 + 3×16 = 80 ||align=center|[4,2,2,2]||64
|-
|5-[[orthotope]]||align=center|{&nbsp;}<sup>5</sup>||align=center|{{CDD|node_1|2|node_1|2|node_1|2|node_1|2|node_1}}||align=right|2<sup>5</sup> = 32||align=right|5×16 = 80||align=center|[2,2,2,2]||32
|}

== Related polytopes==
The ''5-cube'' is 5th in a series of [[hypercube]]:
{{Hypercube polytopes}}

The [[Regular_skew_polyhedron#Finite_regular_skew_polyhedra|regular skew polyhedron]] {4,5| 4} 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-polytope#Uniform 5-polytope|uniform 5-polytope]]s generated from the regular 5-cube or [[5-orthoplex]].

{{Penteract family}}

== References ==
<references />
* [[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]:
** Coxeter, ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, {{ISBN|0-486-61480-8}}, p.&nbsp;296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
** '''Kaleidoscopes: Selected Writings of H.S.M. Coxeter''', edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, {{ISBN|978-0-471-01003-6}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
*** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380-407, MR 2,10]
*** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591]
*** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45]
* [[Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. (1966)
* {{KlitzingPolytopes|polytera.htm|5D uniform polytopes (polytera)|o3o3o3o4x - pent }}

== External links ==
* {{MathWorld|title=Hypercube|urlname=Hypercube}}
*{{GlossaryForHyperspace | anchor=Measure | title=Measure polytope }}
* [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary: hypercube] Garrett Jones
* [https://www.asymptotos.com/wp-content/uploads/2023/07/Cube_5.html Maltsev, Nick E. https://www.asymptotos.com/wp-content/uploads/2023/07/Cube_5.html]

{{Polytopes}}

[[Category:5-polytopes]]
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