Hypercube

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This article refers to the mathematical concept. For the movie of the same name, see Cube 2: Hypercube.

A projection of a cube (into a two-dimensional image)

A projection of a hypercube (into a two-dimensional image)

In geometry, a hypercube is an n-dimensional analogue of a square (n = 2) and a cube (n = 3). It is a closed, compact, convex figure consisting of groups of opposite parallel line segments aligned in each of the space's dimensions, at right angles to each other.

An n-dimensional hypercube is also called an n-cube. The term "measure polytope" (which is apparently due to Coxeter; see Coxeter 1973) is also used but it is rare.

A unit hypercube is a hypercube whose side has length one unit. Often, the hypercube whose corners (or vertices) are the 2ⁿ points in Rⁿ with coordinates equal to 0 or 1 is called "the" unit hypercube.

A point is a hypercube of dimension zero. If one moves this point one unit length, it will sweep out a line segment, which is a unit hypercube of dimension one. If one moves this line segment its length in a perpendicular direction from itself; it sweeps out a two-dimensional square. If one moves the square one unit length in the direction perpendicular to the plane it lies on, it will generate a three-dimensional cube. This can be generalized to any number of dimensions. For example, if one moves the cube one unit length into the fourth dimension, it generates a 4-dimensional unit hypercube (a unit tesseract).

The hypercubes are one of the few families of regular polytopes that are represented in any number of dimensions. The dual polytope of a hypercube is called a cross-polytope. The 1-skeleton of a hypercube is a hypercube graph.

1 Elements
2 n-cube rotation
3 References
4 External links

[edit] Elements

A hypercube of dimension n has 2n "sides" (a 1-dimensional line has 2 end points; a 2-dimensional square has 4 sides or edges; a 3-dimensional cube has 6 2-dimensional faces; a 4-dimensional tesseract has 8 cells). The number of vertices (points) of a hypercube is 2ⁿ (a cube has 2³ vertices, for instance).

The number of m-dimensional hypercubes (just referred to as m-cube from here on) on the boundary of an n-cube is

$2^{n-m}{n \choose m}.$

For example, the boundary of a 4-cube contains 8 cubes, 24 squares, 32 lines and 16 vertices.

Hypercube elements
n-cube	Names Schläfli symbol Coxeter-Dynkin	Vertices (0-faces)	Edges (1-faces)	Faces (2-faces)	Cells (3-faces)	(4-faces)	(5-faces)	(6-faces)	(7-faces)	(8-faces)
0-cube	Point -	1
1-cube	Digon {} or {2}	2	1
2-cube	Square Tetragon {4}	4	4	1
3-cube	Cube Hexahedron {4,3}	8	12	6	1
4-cube	Tesseract octachoron {4,3,3}	16	32	24	8	1
5-cube	Penteract decateron {4,3,3,3}	32	80	80	40	10	1
6-cube	Hexeract dodecapeton {4,3,3,3,3}	64	192	240	160	60	12	1
7-cube	Hepteract tetradeca-7-tope {4,3,3,3,3,3}	128	448	672	560	280	84	14	1
8-cube	Octeract hexadeca-8-tope {4,3,3,3,3,3,3}	256	1024	1792	1792	1120	448	112	16	1
9-cube	Enneract octadeca-9-tope {4,3,3,3,3,3,3,3}	512	2304	4608	5376	4032	2016	672	144	18

[edit] n-cube rotation

Based on our observations of how 1-, 2-, and 3-dimensional objects can rotate, we can hypothesize how objects with n dimensions can rotate. A 3-dimensional object can rotate in 2 different ways, about 3 axes. The first way it can rotate is about an edge. A cube (as an example) can rotate about an entire edge, meaning that everything but that edge changes position. The second is about a single point. It's possible to rotate a cube around a single point, without that point changing its position. Similarly, a 2-dimensional object can rotate about a single point, but that's the only way it can rotate. So, a 3D cube can rotate about the 1st dimension or the 0th dimension, and a 2D plane can only rotate about the 0th dimension. So what about the 1st dimension? According to our theory, it would be able to revolve around the −1st dimension, which is nothingness, or nonexistent, which makes sense, because it can't rotate. This supports our theory for the first through third dimensions, so we can therefore assume that it will apply for all the other dimensions. This means that 4-dimensional hypercube can rotate about a whole face, and a 5-cube can rotate about a whole cube.

[edit] References

Bowen, J. P., Hypercubes, Practical Computing, 5(4):97–99, April 1982.
Coxeter, H. S. M., Regular Polytopes. 3rd edition, Dover, 1973, p. 123. ISBN 0-486-61480-8. p.296, Table I (iii): Regular Polytopes, three regular polytopes in n dimensions (n ≥ 5)