Hypercube

This article deals with the dimensional polytopic hypercube; on the communication pattern, see hypercube (communication pattern) .

Projection of a tesseract (four-dimensional hypercube) into the 2nd dimension

Hypercubes or dimensional polytopes are -dimensional analogies to the square ( ) and to the cube ( ). It can be any natural number be. The four-dimensional hypercube is also known as the tesseract . The symmetry group of a hypercube is the hyperoctahedral group . ${\displaystyle n}$ ${\displaystyle n=2}$ ${\displaystyle n=3}$ ${\displaystyle n}$

Construction of regular cubes

Regular cubes of edge length can be created as follows: ${\displaystyle a\neq 0}$

• If a point is shifted in a straight line by the distance , a one-dimensional line is created , mathematically a one-dimensional hypercube. ${\displaystyle a}$
• If this line is shifted by the distance perpendicular to its dimension , a two-dimensional square , a surface, mathematically a two-dimensional hypercube is created. ${\displaystyle a}$
• If this square is shifted by the distance perpendicular to its two dimensions , a three-dimensional cube is created, mathematically corresponding to a three-dimensional hypercube. ${\displaystyle a}$
• General: If a -dimensional cube is shifted by the distance perpendicular to its dimensions , a -dimensional hypercube is created. ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle a}$ ${\displaystyle (n+1)}$

Boundary elements

In a hypercube of the dimension there are exactly edges at every node (corner) . Accordingly, a hypercube is an undirected multigraph (see also: graph theory ). ${\displaystyle n}$ ${\displaystyle n}$

The -dimensional cube is bounded by zero-dimensional, one-dimensional, ..., -dimensional elements. Exemplary: ${\displaystyle n}$ ${\displaystyle (n\!-\!1)}$

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

The number of the individual boundary elements can be derived from the following consideration: Let the dimension be a hypercube . The -dimensional boundary elements of this cube ( ) can be generated from the boundary elements of a -dimensional hypercube as follows: The -dimensional boundary elements ( ) double and all dimensional elements are expanded to -dimensional elements. This results in a total of . ${\displaystyle n\!+\!1}$ ${\displaystyle k}$ ${\displaystyle k_{n+1}}$ ${\displaystyle n}$ ${\displaystyle k}$ ${\displaystyle k_{n}}$ ${\displaystyle k\!-\!1}$ ${\displaystyle (k\!-\!1)_{n}}$ ${\displaystyle k}$ ${\displaystyle k_{n+1}=2k_{n}+(k-1)_{n}}$

example

• The 2-dimensional hypercube is bounded by 1 surface , 4 edges and 4 nodes . ${\displaystyle (k_{n}=2)}$ ${\displaystyle (k_{n}=1)}$ ${\displaystyle (k_{n}=0)}$

• The 3-dimensional cube is bounded by surfaces , by edges and nodes . ${\displaystyle 2+4=6}$ ${\displaystyle (k_{n+1}=2)}$ ${\displaystyle 8+4=12}$ ${\displaystyle (k_{n+1}=1)}$ ${\displaystyle 4+4=8}$ ${\displaystyle (k_{n+1}=0)}$

You can think differently: If you center a -dimensional hypercube in a Cartesian coordinate system around the origin and align it with the coordinate axes, there are coordinate axes for a -dimensional boundary element that are parallel to this boundary element. On the other hand, there is not only one -dimensional boundary element for each selection of coordinate axes, but because the number of boundary elements is doubled through each of the axes perpendicular to the boundary elements (there are the same boundary elements again shifted parallel on the other side of the axis). The number of boundary elements results from the product of the number of possibilities to select axes from the axes with the number of boundary elements for each selection and thus reads (with the binomial coefficient ). ${\displaystyle n}$ ${\displaystyle k}$ ${\displaystyle k}$ ${\displaystyle k}$ ${\displaystyle k}$ ${\displaystyle 2^{n-k},}$ ${\displaystyle n-k}$ ${\displaystyle k}$ ${\displaystyle n}$ ${\displaystyle {\binom {n}{k}}\cdot 2^{n-k}}$ ${\displaystyle {\binom {n}{k}}}$

The way to the hypercube

The 0- to 5-dimensional cubes in the parallel projection

	Schläfli symbol	Number of boundary elements
		0-dim.	1-dim.	2-dim.	3-dim.	4-dim.	${\displaystyle \ldots }$	${\displaystyle (n\!-\!1)}$-dim.	${\displaystyle n}$-dim.
Point	${\displaystyle ()}$	1
route	${\displaystyle \{\}}$	2	1
square	${\displaystyle \{4\}}$	4th	4th	1
3-dim. cube	${\displaystyle \{4,3\}}$	8th	12	6th	1
4-dim. cube	${\displaystyle \{4,3,3\}}$	16	32	24	8th	1
${\displaystyle \vdots }$
${\displaystyle n}$-dim. cube	${\displaystyle \{4,3^{n-2}\}}$	${\displaystyle 2^{n}}$	${\displaystyle n2^{n-1}}$	${\displaystyle {\binom {n}{2}}2^{n-2}}$	${\displaystyle {\binom {n}{3}}2^{n-3}}$	${\displaystyle \ldots }$	${\displaystyle \ldots }$	${\displaystyle {\binom {n}{n-1}}2^{1}=2n}$	${\displaystyle {\binom {n}{n}}2^{0}=1}$

Every -dimensional boundary element of a -dimensional cube of edge length is for a -dimensional cube of the same edge length . A 4-hypercube has 16 corners, an edge network of length , is bounded by a surface network of the total area and by cells with the 3-total volume (the 3-dimensional hypersurface) of and has a 4-volume of . ${\displaystyle k}$ ${\displaystyle n}$ ${\displaystyle a}$ ${\displaystyle 0<k\leq n}$ ${\displaystyle k}$ ${\displaystyle a}$ ${\displaystyle 32a}$ ${\displaystyle 24a^{2}}$ ${\displaystyle 8a^{3}}$ ${\displaystyle a^{4}}$

properties

The construction of the longest diagonals of the square, cube and tesseract

The name Maßpolytop comes from the possibility of aligning the object parallel to all coordinate axes and completely filling the Euclidean space through parallel duplication . It is the only regular polytope that can do this in dimensions . These tiling are self-duplicated with the Schläfli symbol for each dimension ${\displaystyle n>4}$ ${\displaystyle \{4,3^{n-2},4\}.}$

The longest diagonal of a hypercube corresponds to the square root of its dimension multiplied by its edge length.

Dimension polytope (or hypercube) and cross polytope (or hyperoctahedron) are dual to one another . Therefore, their symmetry groups also match.

		conformal projection in			possible operations
dimension	object	2-D	3-D	4-D	push	rotate	squirm	put on
0	Point	+	+	+	-	-	-	-
1	line	+	+	+	+	-	-	-
2	square	+	+	+	+	+	-	-
3	cube	-	+	+	+	+	+	-
4th	Tesseract	-	-	+	+	+	+	+

dimension	edge	node	pages	Degree	diameter	Edge connection	Knot context
1	${\displaystyle 1}$	${\displaystyle 2}$	${\displaystyle 2}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$
2	${\displaystyle 4}$	${\displaystyle 4}$	${\displaystyle 4}$	${\displaystyle 2}$	${\displaystyle 2}$	${\displaystyle 2}$	${\displaystyle 2}$
3	${\displaystyle 12}$	${\displaystyle 8}$	${\displaystyle 6}$	${\displaystyle 3}$	${\displaystyle 3}$	${\displaystyle 3}$	${\displaystyle 3}$
4th	${\displaystyle 32}$	${\displaystyle 16}$	${\displaystyle 8}$	${\displaystyle 4}$	${\displaystyle 4}$	${\displaystyle 4}$	${\displaystyle 4}$
...	...	...	...	...	...	...	...
${\displaystyle n}$	${\displaystyle 2^{(n-1)}\cdot n}$	${\displaystyle 2^{n}}$	${\displaystyle 2n}$	${\displaystyle n}$	${\displaystyle n}$	${\displaystyle n}$	${\displaystyle n}$

Art applications

Visual arts

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

• Tony Robbin - by mirroring and twisting the edges of the cube, Tony Robbin creates situations in drawings and spatial installations that would only be possible in a hyperdimensional world.
• Manfred Mohr - in his compositions illustrates the interactions of lines that follow a spatial logic of more than three degrees of freedom.
• Frank Richter - concretized in graphics, sculptures and room installations according to the specifications of mathematical rules room constellations that go beyond the third dimension.
• In his picture Crucifixion (Corpus Hypercubus) in 1954, Salvador Dalí painted a crucified Jesus on the mesh of a hypercube.

Projection of a rotating hypercube

Hypercube in pop culture

• The film Cube 2: Hypercube is about a hypercube in which the characters move in three spatial dimensions and one temporal dimension and, for example, meet themselves in a different period of time.
• The short story And He Built a Crooked House , in the German version Das 4D-Haus , by Robert A. Heinlein deals with a house that consists of a hypercube.
• The progressive metal band Tesseract has named itself after the 4D Hypercubus (English tesseract ; Tesseract) and uses various projections and animations of it as the band logo.
• The novel Apocalypsis III by the author Mario Giordano uses a tesseract as the figure of Pandora's box

See also

• Euclidean space
• Hilbert cube for the infinite-dimensional case
• Hyper rectangle (aka hyper cube ) - generalization for different edge lengths
• Hyperplane
• Hyper pyramid
• Hyperspace

Web links

• The n-dimensional hypercube (PDF; 3.5 MB)
• Hypercubes and Hyperspheres
• Extended table of boundary elements
• Animated Hypercube (Java)
• 4d-screen.de - four-, five-, six- and seven-dimensional cubes (Java)
• Illustrated construction of a 4D hypercube

Individual evidence

1. ↑ The movements of a point within a hypercube: slide on a straight line; turning as movement on a curved path in a plane; winding as movement on a curved path in three dimensions; evert as a movement on a four-dimensional curved path.
2. ↑ Example of a Dalí painting ( Memento from July 23, 2015 in the Internet Archive )
3. ↑ Tesseract