6-cube
{{Short description|6-dimensional hypercube}}
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!bgcolor=#e7dcc3 colspan=2|6-cube | |
| bgcolor=#ffffff align=center colspan=2|280px Orthogonal projection inside Petrie polygon Orange vertices are doubled, and the center yellow has 4 vertices | |
| bgcolor=#e7dcc3|Type | Regular 6-polytope |
| bgcolor=#e7dcc3|Family | hypercube |
| bgcolor=#e7dcc3|Schläfli symbol | {4,34} |
| bgcolor=#e7dcc3|Coxeter diagram | {{CDD|node_1|4|node|3|node|3|node|3|node|3|node}} |
| bgcolor=#e7dcc3|5-faces | 12 {4,3,3,3} 25px |
| bgcolor=#e7dcc3|4-faces | 60 {4,3,3} 25px |
| bgcolor=#e7dcc3|Cells | 160 {4,3} 25px |
| bgcolor=#e7dcc3|Faces | 240 {4} 25px |
| bgcolor=#e7dcc3|Edges | 192 |
| bgcolor=#e7dcc3|Vertices | 64 |
| bgcolor=#e7dcc3|Vertex figure | 5-simplex |
| bgcolor=#e7dcc3|Petrie polygon | dodecagon |
| bgcolor=#e7dcc3|Coxeter group | B6, [34,4] |
| bgcolor=#e7dcc3|Dual | 6-orthoplex 25px |
| bgcolor=#e7dcc3|Properties | convex, Hanner polytope |
In geometry, a 6-cube is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube 5-faces.
It has Schläfli symbol {4,34}, being composed of 3 5-cubes around each 4-face. It can be called a hexeract, a portmanteau of tesseract (the 4-cube) with hex for six (dimensions) in Greek. It can also be called a regular dodeca-6-tope or dodecapeton, being a 6-dimensional polytope constructed from 12 regular facets.
Related polytopes
It is a part of an infinite family of polytopes, called hypercubes. The dual of a 6-cube can be called a 6-orthoplex, and is a part of the infinite family of cross-polytopes. It is composed of various 5-cubes, at perpendicular angles on the u-axis, forming coordinates (x,y,z,w,v,u).{{Cite web|url=https://www.researchgate.net/publication/361688598|title=A New Six-Dimensional Hyper-Chaotic System}}{{Cite journal|url=https://www.sciencedirect.com/science/article/pii/S0747717188800105|title=An improved projection operation for cylindrical algebraic decomposition of three-dimensional space - ScienceDirect|journal=Journal of Symbolic Computation |date=February 1988 |volume=5 |issue=1 |pages=141–161 |doi=10.1016/S0747-7171(88)80010-5 |last1=McCallum |first1=Scott }}
Applying an alternation operation, deleting alternating vertices of the 6-cube, creates another uniform polytope, called a 6-demicube, (part of an infinite family called demihypercubes), which has 12 5-demicube and 32 5-simplex facets.
As a configuration
This configuration matrix represents the 6-cube. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole 6-cube. The nondiagonal numbers say how many of the column's element occur in or at the row's element.Coxeter, Regular Polytopes, sec 1.8 ConfigurationsCoxeter, Complex Regular Polytopes, p.117
Cartesian coordinates
Cartesian coordinates for the vertices of a 6-cube centered at the origin and edge length 2 are
: (±1,±1,±1,±1,±1,±1)
while the interior of the same consists of all points (x0, x1, x2, x3, x4, x5) with −1 < xi < 1.
Construction
There are three Coxeter groups associated with the 6-cube, one regular, with the C6 or [4,3,3,3,3] Coxeter group, and a half symmetry (D6) or [33,1,1] Coxeter group. The lowest symmetry construction is based on hyperrectangles or proprisms, cartesian products of lower dimensional hypercubes.
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!Name !Order | |||
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!Regular 6-cube |{{CDD|node_1|4|node|3|node|3|node|3|node|3|node}} |{4,3,3,3,3} |[4,3,3,3,3] | 46080 | ||
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!Quasiregular 6-cube |{{CDD|node_f1|3|node|3|node|3|node|split1|nodes}} | |[3,3,3,31,1] | 23040 | ||
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!rowspan=10|hyperrectangle |{{CDD|node_1|4|node|3|node|3|node|3|node|2|node_1}} | {4,3,3,3}×{} | [4,3,3,3,2] | 7680 |
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|{{CDD|node_1|4|node|3|node|3|node|2|node_1|4|node}} | {4,3,3}×{4} | [4,3,3,2,4] | 3072 |
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|{{CDD|node_1|4|node|3|node|2|node_1|4|node|3|node}} | {4,3}2 | [4,3,2,4,3] | 2304 |
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|{{CDD|node_1|4|node|3|node|3|node|2|node_1|2|node_1}} | {4,3,3}×{}2 | [4,3,3,2,2] | 1536 |
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|{{CDD|node_1|4|node|3|node|2|node_1|4|node|2|node_1}} | {4,3}×{4}×{} | [4,3,2,4,2] | 768 |
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|{{CDD|node_1|4|node|2|node_1|4|node|2|node_1|4|node}} | {4}3 | [4,2,4,2,4] | 512 |
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|{{CDD|node_1|4|node|3|node|2|node_1|2|node_1|2|node_1}} | {4,3}×{}3 | [4,3,2,2,2] | 384 |
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|{{CDD|node_1|4|node|2|node_1|4|node|2|node_1|2|node_1}} | {4}2×{}2 | [4,2,4,2,2] | 256 |
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|{{CDD|node_1|4|node|2|node_1|2|node_1|2|node_1|2|node_1}} | {4}×{}4 | [4,2,2,2,2] | 128 |
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|{{CDD|node_1|2|node_1|2|node_1|2|node_1|2|node_1|2|node_1}} |{}6 |[2,2,2,2,2] | 64 |
Projections
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!B6 !B5 !B4 |
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!Graph |
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|[12] |[10] |[8] |
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!Coxeter plane !Other !B3 !B2 |
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!Graph |
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!Dihedral symmetry |[2] |[6] |[4] |
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!Coxeter plane ! !A5 !A3 |
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!Graph | |
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!Dihedral symmetry | |[6] |[4] |
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|colspan=2 valign=top align=center|3D Projections |
| 280px 6-cube 6D simple rotation through 2Pi with 6D perspective projection to 3D. |280px |
| 280px A 3D perspective projection of a hexeract undergoing a triple rotation about the X-W1, Y-W2 and Z-W3 orthogonal planes. |
Related polytopes
The 64 vertices of a 6-cube also represent a regular skew 4-polytope {4,3,4 | 4}. Its net can be seen as a 4×4×4 matrix of 64 cubes, a periodic subset of the cubic honeycomb, {4,3,4}, in 3-dimensions. It has 192 edges, and 192 square faces. Opposite faces fold together into a 4-cycle. Each fold direction adds 1 dimension, raising it into 6-space.
The 6-cube is 6th in a series of hypercube:
{{Hypercube polytopes}}
This polytope is one of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.
{{Hexeract family}}
References
- Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, {{ISBN|0-486-61480-8}} p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n>=5)
- {{KlitzingPolytopes|polypeta.htm|6D uniform polytopes (polypeta)|o3o3o3o3o4x - ax}}
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
{{Polytopes}}