Hypercube

Hypercube are -dimensional analogies to the square () and to die (). It can be any natural number. The four-dimensional hypercube is also called a tesseract.

Construction of regular cubes

Regular cube of edge length a (≠ 0 ) can be generated as follows:

• If a point by the distance a is moved in a straight line, creates a one-dimensional distance, mathematically one-dimensional hypercube.
• If this route is moved perpendicular to its dimension by the distance a, a two-dimensional square, an area, a two-dimensional mathematical hypercube.
• If this square is moved perpendicular to its two dimensions by the distance a, creates a three-dimensional cube, mathematically according to a three-dimensional hypercube.
• General: Thus, when an n-dimensional cube is moved perpendicular to its dimensions n by the distance A, results in a (n 1) -dimensional hypercube.

In a hypercube of dimension n are located at each node ( corner) exactly n edges. Thus, it is in a hypercube to an undirected multigraph (see also: graph theory).

The n-dimensional cube is (n-1 )-dimensional elements bounded by zero-dimensional, one-dimensional, ....

The example:

The 3-dimensional cube is bounded by nodes ( points ), edges (lines ) and surfaces, ie of elements of dimension 0.1 and 2

The number of individual boundary elements can be derived from the following consideration: Be a hypercube of dimension ( n 1). The k-dimensional boundary elements of this cube ( ) can be produced from the following border elements of an n-dimensional hypercube: the k-dimensional boundary elements () are doubled and all k -1 dimensional elements be extended to k-dimensional. Result is therefore the sum of a number of.

• The 2-dimensional hypercube is bounded by 1 area, 4 edges and 4 nodes.

• The 3-dimensional cube is bounded by faces, edges and nodes.

Otherwise you can consider: If you centered a n-dimensional hypercube in a Cartesian coordinate system around the origin and sets aligned with the coordinate axes, there is a k-dimensional boundary element k coordinate axes that are parallel to this border element. On the other hand, there are on each selection of k coordinate axes, not only a K -dimensional boundary element, but 2n -k, because the vertical through each of the nk to the boundary elements axes doubles the number of the boundary elements (there is the same boundary elements again shifted parallel to the other side of the axis). The number of the boundary elements so results from the product of number of possibilities, K axes out of the N axes auszuwählend ( binomial coefficient ), with the number of interface elements for each selection, and is therefore

• All 0 - to 5 - dimensional cube in parallel projection

Properties

Art applications

Fine Arts

In the visual arts, many artists deal with the hypercube.

• Tony Robbin - by reflections and rotations of cube - edges produced Tony Robbin in drawings, and room installations with situations that would only be possible in a hyper-dimensional world.
• Manfred Mohr - illustrated in his compositions interactions of lines that follow a spatial logic of more than three degrees of freedom.
• Frank Richter - concretized in graphics, sculptures and installations according to the specification of mathematical rules space constellations that go beyond the third dimension.
• Salvador Dalí in his Crucifixion ( Corpus hypercube ) 1954 a crucified Jesus painted on the network of a hypercube.

Hypercube in pop culture

• The movie Cube 2: Hypercube is about a hyper- cube in which the characters in the three spatial dimensions and one temporal dimension, for example, move and encounter themselves in a different time period.
• The short story And He Built a Crooked House, in the German version of the 4D House, by Robert A. Heinlein dealt with a house that consists of a hypercube.
• The progressive metal band Tesseract after the 4D hypercube (English " tesseract "; tesseract ) name and use different projections and animations of it as a band logo.