Hypercube

Definition

If E is an Euclidean space of dimension N provided with an orthonormal base, one can define a hypercube unit like the hypercube of which the 2^N points in Rⁿ with coordinates equal to 0 or 1. The hypercubes are the figures obtained starting from the hypercube unit by Similitude S.

To represent a hypercube dimension N

To represent a hypercube dimension N , one proceeds as follows:

Dimension 1: A not is a hypercube of dimension zero. If one moves this point a length unit, it will sweep a segment of right-hand side, which is a hypercube unit of dimension a

Dimension 2: If one moves this segment a length unit in a direction Perpendiculaire from itself; it sweeps a two-dimensional square.

Dimension 3: If one moves the square a length unit in the direction perpendicular to the site of this one, it will generate a three-dimensional cube.

Dimension 4: If one moves the cube a length unit in the fourth dimension, it will generate a four-dimensional hypercube unit (a Tesseract unit).

…

Dimension N > 3: One traces a hypercube dimension n-1 , one reproduces his image and one binds the items two to two

In short, the construction of a cube is done by the translation of the cube of lower size according to an axis perpendicular to dimensions of this cube.

The hypercubes are one of some regular families of polytopes which are represented in an unspecified number of dimensions. The dual Polytope of a hypercube is called a cross Polytope. the 1 - skeleton of a hypercube is a Graphe hypercube.

A generalization of the cube to dimensions larger than three is called a “Hypercube”, “N-cubic” or “polytope measurement”. The tesseract is the four-dimensional or 4-cubic hypercube. It is a regular Polytope. It is also a particular case of Parallélotope: a hypercube is a right parallélotope whose edges are of the same length.

4 dimensions

The hypercube with four dimensions is also called tesseract in English, according to Charles Howard Hinton.

According to the formula of Gardner, one can find the properties of the tesseract by developing (2x + 1) ⁴:

(2x + 1) ⁴ = 16x⁴ + 32x³ + 24x² + 8x + 1

Thus the hypercube is composed of:

16 tops;
32 edges;
8 cubic faces (either 24 plane faces): each face of the tesseract is a Cube.

The intersection of a hypercube with a Hyperplan gives the Cartesian equation:

ax + by + cz + dw = e

With the four coordinates of the hyperspace of dimension 4, namely X, there, Z, and W. Actually, a hyperplane in four dimensions can be compared with three-dimensional space, i.e. the intersection of a hypercube with a plan is in fact a projection 3D of this hypercube.

Volume: c⁴, with C the side of the hypercube.
total Surface: 24c²

The faces of a hypercube are:

Before/Back
Left/Right
High/Low
Anna/Kata

N dimensions

A hypercube with N dimensions has:

V_n = 2ⁿ tops;
S_n = 2 × S_n-1 + V_n-1 edges; (or N × 2^n-1)
F_n = 2 × F_n-1 + S_n-1 plane faces;
HF_n = 2 × HF_n-1 + F_n-1 hyperfaces (cubic or cubic faces);
It goes from there in the same way for the hyperfaces in 5 dimensions (hypercubic faces) etc
In a general way, the number of faces to K dimensions of a hypercube with N dimension is equal to

f_k (H_n) = {N \ choose K} 2^ {n-k}

the full number of faces of a hypercube is of $3^n-1$
Volume = cⁿ with C the side of the hypercube
total Aire = F_nc² with F_n the number of faces

Elements

A hypercube of dimension N has 2 N " cotés" (a 1-dimensional segment has two points at the ends; a 2-dimensional square has four edges; a 3-dimensional cube has 6 2-dimensional faces; a 4-dimensional hypercube (tesseract) has 8 cells). The number of tops (points) of a hypercube is 2^N (a cube has 2³ tops, for example).

The number of hypercubes m - dimensional (as indicated under the m-cubic name above) on the border of a N - cubic is

$2^ {Nm} {N \ choose m}.$

For example, the border of 4-cubic contains 8 cubes, 24 squares, 32 segments and 16 tops.

Rotation of a N - cubic

Based on our observations in the way in which the objects 1,2 and 3 dimensional can turn, we can put forth an assumption on the way in which the objects turn to N dimensions. A 3-dimensional object can turn in two different ways out of 3 axes. By the first, it can turn on an edge. A cube (for example) can turn on a whole edge, which means that any exchange of position except this edge. By the second, it can turn on a single point. It is possible to make turn a cube around a single point, without this point not changing a position. In a similar way, a 2-dimensional object can turn on a single point, but it is the only way in which it can turn. Therefore, a 3-dimensional cube can turn on its 1st dimension or the 0e dimension, and a 2-dimensional plan can only turn on the 0e dimension. Thus, that is it the first dimension? According to our theory, it could turn around dimension -1, negative dimension which is nothing or non-existence, which has a direction, because it cannot turn. This consolidates our assumption since the first until three dimensions, therefore, we can consequently suppose that will apply for all other dimensions. This means that 4-dimensional hypercube can turn around a whole face, that a 5-dimensional hypercube can turn around a whole cube, etc…

Artistic representations

In film of Science fiction Cubic ²: Hypercube , the heroes are locked up in a tesseract, or at least they evolve/move while moving from one cube to another among the faces of the hypercube. From one cube to another, the orientation of gravity can vary (in any case the characters feel it when they pass from a cube to the other) time can dilate or contract, and the characters are brought to meet doubles of themselves because of the superposition of possible futures. But the bond between these properties and the fact that the history proceeds in a tesseract are not explicit.

structures about It, the Arche of Defense in Paris in France, is a projection in three dimensions of a hypercube.

See too

magic Hypercube
magic Tesseract
Hypercube OLAP in decisional Informatique
the Arche of Defense can be regarded as a representation partial of a hypercube

References

Bowen, J.P., Hypercubes, Practical Computing , 5 (4): 97-99, April 1982.
Coxeter, H.S. Mr., 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)

External bonds

HyperSolids : a program open-source for Macintosh (Mac OS X and more) which can generate 6 hypersolides in four dimensions.
an illustration (the plugin Java requires)
Hypercube 98: a Windows program which post animated hypercubes, created by Rudy Rucker
Démonstration and hypercube interactive (the plugin Shockwave requires)
Ken Perlin: how to visualize a hypercube, by Ken Perlin.
Magic Cubes 4D: the Rubik' S Cubes in four dimensions
Cut The Knot! : An interactive explanation, requires the plugin Java. Created by Alex Bogomolny.
4dimensions: Explanation in French of the concept of a space with four dimensions by the hypercube. Created by Mounier Florian
Images of hypercubes (2D-15D)
Animation of a hypercube

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