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Encyclopedia > Hypercube
A square
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. Image File history File links Square. ... Image File history File links Square. ... Hexahedron (sometimes called cube), rendered by Java applet I wrote. ... Hexahedron (sometimes called cube), rendered by Java applet I wrote. ... Image File history File links Hypercube. ... Image File history File links Hypercube. ... Examples colored by the number of sides on each face. ... Table of Geometry, from the 1728 Cyclopaedia. ... For other uses, see square. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... In topology and related branches of mathematics, a closed set is a set whose complement is open. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... Look up Convex set in Wiktionary, the free dictionary. ... Parallel is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. ... The geometric definition of a line segment In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. ... :For other senses of this word, see dimension (disambiguation). ... This article is about angles in geometry. ...

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.

Image File history File links Dimoffree. ...

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). Fig. ... Stereographic projection In geometry, the tesseract is the four-dimensional analog of the (three-dimensional) cube, where motion along the fourth dimension is often a representation for bounded transformations of the cube through time. ...

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. A dodecahedron, one of the five Platonic solids. ... In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the others. ... In geometry, a cross-polytope, or orthoplex, is a regular, convex polytope that exists in any number of dimensions. ... In mathematics, particularly in algebraic topology, the n-skeleton of a topological space X presented as a simplicial complex, or CW complex, refers to the subspace Xn that is the union of the simplices of X (resp. ... The hypercube graph Q4. ...

## Contents

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). A face is a polygonal component of a higher dimensional polytope. ... Cubic honeycomb - four cubic cells per edge hypercube - three cubic cells per edge In geometry, a cell is a three-dimensional element that is part of a higher-dimensional object. ...

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 Graph 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
{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
deca-5-tope
{4,3,3,3}
32 80 80 40 10 1
6-cube   Hexeract
dodeca-6-tope
{4,3,3,3,3}
64 192 240 160 60 12 1
7-cube   Hepteract
{4,3,3,3,3,3}
128 448 672 560 280 84 14 1
8-cube   Octeract
{4,3,3,3,3,3,3}
256 1024 1792 1792 1120 448 112 16 1
9-cube   Eneneract
{4,3,3,3,3,3,3,3}
512 2304 4608 5376 4032 2016 672 144 18

## 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 was 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.

## 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)

Jonathan P. Bowen FBCS FRSA (born 1956) is a British computer scientist and is Professor of Computing at London South Bank University where he heads the Centre for Applied Formal Methods in the Institute for Computing Research. ... H.S.M. Coxeter. ... Stereographic projection of the 120-cell, a 4-dimensional regular polytope. ...

Results from FactBites:

 Tesseract - Wikipedia, the free encyclopedia (1292 words) In geometry, the tesseract, or hypercube, is a regular convex polychoron with eight cubical cells. In that book she uses the tesseract as a portal, a doorway which you can pass through and emerge far away from the starting point, as if the two distant points were brought together at one intersection (at the tesseract doorway) by the folding of space, enabling near-instantaneous transportation. Hypercubes and all kinds of multi-dimensional space and structures star prominently in many books by Rudy Rucker.
 Cube 2: Hypercube (497 words) A hypercube, or "tesseract" is the four dimensional equivalent of a cube - and an entirely theoretical construct. Hypercube drops hints like it was leaving a trail of crumbs. Hypercube also suffers from weak and sometimes inconsistent characterizations, as well as the occasional head scratching plot twists (one player appears briefly at the film's opening, and shows up toward the end just to be killed without pretense).
More results at FactBites »

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