5-cube

{{Short description|5-dimensional hypercube}} {{Uniform polyteron db|Uniform polyteron stat table|pent}} In five-dimensional geometry, a '''5-cube''' (or '''penteract''') is a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces.

It is represented by Schläfli symbol {4,3,3,3} or {4,3³}, constructed as 3 tesseracts, {4,3,3}, around each cubic ridge.

== Related polytopes == It is a part of an infinite hypercube family. The dual of a penteract 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.

The penteract 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 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''₀, ''x''₁, ''x''₂, ''x''₃, ''x''₄) with -1 < ''x''_''i'' < 1 for all ''i''.

== Images == ''n''-cube Coxeter plane projections in the B_k Coxeter groups project into k-cube graphs, with power of two vertices overlapping in the projective graphs.

{| class="wikitable skin-invert-image" |+ Orthographic projections |- align=center !Coxeter plane !B₅ !B₄ / D₅ !B₃ / D₄ / A₂ |- align=center !Graph |150px |150px |150px |- align=center !Dihedral symmetry |[10] |[8] |[6] |- align=center !Coxeter plane !Other !B₂ !A₃ |- align=center !Graph |150px |150px |150px |- align=center !Dihedral symmetry |[2] |[4] |[4] |}

{| class="wikitable" width="240" |+ More orthographic projections |- valign=top align=center |240px|class=skin-invert<BR>Wireframe skew direction |240px<BR>B5 Coxeter plane |}

{| class="wikitable" width="200" |+ Graph |- |200px|class=skin-invert<BR>Vertex-edge graph. |} {| class="wikitable" width="240" |+ Perspective projections |- valign=top |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 |- |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 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 |- |160px |160px|class=skin-invert |160px|class=skin-invert |} 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" |280x280pxA 3D perspective projection of a penteract undergoing a simple rotation about the W1-W2 orthogonal plane |280x280pxA 3D perspective projection of a penteract undergoing a double rotation about the X-W1 and Z-W2 orthogonal planes |}

== Symmetry == The ''5-cube'' has Coxeter group symmetry B₅, 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 hypercubes have lower symmetry forms constructed as prisms. The 5-cube has 7 prismatic forms from the lowest 5-orthotope, {&nbsp;}⁵, 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 |- |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 |- |tesseractic 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;}²||align=center|{{CDD|node_1|4|node|3|node|2|node_1|2|node_1}}||align=right|8×2² = 32||align=right|48 + 2×16 = 80 ||align=center|[4,3,2,2]||192 |- |square-square duoprism prism||align=center|{4}²×{&nbsp;}||align=center|{{CDD|node_1|4|node|2|node_1|4|node|2|node_1}}||align=right|4²×2 = 32||align=right|2×32 + 16 = 80 ||align=center|[4,2,4,2]||128 |- |square-rectangular parallelepiped duoprism||align=center|{4}×{&nbsp;}³||align=center|{{CDD|node_1|4|node|2|node_1|2|node_1|2|node_1}}||align=right|4×2³ = 32||align=right|32 + 3×16 = 80 ||align=center|[4,2,2,2]||64 |- |5-orthotope||align=center|{&nbsp;}⁵||align=center|{{CDD|node_1|2|node_1|2|node_1|2|node_1|2|node_1}}||align=right|2⁵ = 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 {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-polytopes generated from the regular 5-cube or 5-orthoplex.

{{Penteract family}}

== References == <references /> * H.S.M. Coxeter: ** H.S.M. Coxeter, ''Regular Polytopes'', 1973, 3rd edition, Dover, New York, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in ''n'' dimensions (''n'' ≥ 5), {{isbn|0-486-61480-8}} ** '''Kaleidoscopes: Selected Writings of H.S.M. Coxeter''', edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivić Weiss, Wiley-Interscience Publication, 1995, [https://www.wiley.com/en-us/Kaleidoscopes-p-9780471010036 wiley.com], {{isbn|978-0-471-01003-6}} *** (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 ''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}}

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