What is the Hypercube?
The hypercube is the cube with four dimensions.
Our imagination is not sufficient enough to understand the fourth dimension
and the hypercube.
You can approach the hypercube through analogy to the 3-dimensional
cube from different sides. So you can become familiar with it.
Cubes in Perspective top
||If you move a square parallel in space and join the corresponding corners,
you get the perspective sight of the cube.
The hypercube has 16 corners (derived from 2 cubes) and 32 edges (2 cubes
and joining lines).
||If you move a cube parallel in space and join the corresponding corners,
you get the perspective sight of the hypercube.
The hypercube has 24 squares.
The numbers 134, 124, 234, 123 indicate the base vectors (declared below).
||The cube is covered by six squares.
In the same way eight cubes
form the hypercube.
If you know the 3D view, you can look at the hypercube three-dimensionally,
Central Projections top
||The cube is distorted in a central projection. 4 of the 6 squares appear
as trapeziums, which lie between the small and the big square.
||A representation of the hypercube has been developed of this.
(Viktor Schlegel, 1888)
4 cubes, 6 squares, and 4 edges meet at each corner.
||6 of the 8 cubes appear as pyramid stumps, which lie between the small
and the big cube.
3 cubes and 3 squares meet at each edge.
2 cubes meet at each square.
||If you spread out the cube, you get its net. Together the six squares
have 6x4=24 sides. 2x5=10 sides (red) are bound. If you build a cube, you
have to stick the remaining 14 sides in pairs.
There are 11 nets.
||If you spread out the hypercube, you get its net as an arrangement
of 8 cubes. Together the eight cubes have 8x6=48 squares. 2x7=14 squares
are bound. If you "build" a hypercube, you have to stick the remaining
34 squares in pairs.
How many nets are there?
Peter Turney and Dan Hoey counted 261 cases.
||A cube (more exact: a cube with the edges 1) is produced by three unit
vectors (red) perpendicular to each other.
They form a coordinate system.
||A triplet formed by the numbers 0 or 1 describes the corners. The triplet
(011) belongs to the point P. You reach P by going from the origin O first
in x2 direction and then in x3 direction. This way is fixed by 011.
||You describe all the 8 corners by coordinates in this manner. All combinations
of three numbers using 0 or 1 occur as coordinates.
If you write x1+x2+x3=a and substitute all numbers between 0 and 3 with
a, you find to every value another plane. The hexagon corresponding to
a=1.5 is famous.
||If you add the coordinates of one point, you get the sums 0,1,2, or
3. The sums 0 and 3 belong to opposite corners. They are ending points
of a diagonal (green). If you join the points with the sums 1 or 2, you
get triangles (red).
||Corresponding to the cube four basic vectors (red) produce the hypercube.
All combinations of four numbers using 0 or 1 occur as coordinates.
If you add the coordinates of one point, you get the sums 0,1,2,3,
or 4. The sums 0 and 4 belong to opposite corners. They are ending points
of a diagonal (green). If you join the points with the sums 1 or 3, you
get two tetrahedra (red).
If you join the points with the sum 2, you get an octohedron (blue).
If you write x1+x2+x3+x4=a and substitute all numbers between 0 and
4 with a, you find another body as a section to every value.
The section is perpendicular to the diagonal from (0000) to (1111).
More Drawings in Perspective
The n-dimensional Cube top
The datas of the hypercube might follow in the next line. Dimension=4 and
corners=16 are clear.
The hypercube is a construct of ideas. You recieve a plausible explanation
for its features by the "permanent principle", which often is used in mathematics
to get from the "known to the unknown".
Cubes with the dimensions 1, 2 and 3 have the properties as follows.
There are formulas for continuing the sequence for the edges and squares.
If you take n=4, you get the datas of the hypercube.
Encore: Datas of the 5-dimensional cube
The Hypercube on the
Hyper-Dimensional References (collection of links)
(1) Martin Gardner: Mathematischer Karneval, Frankfurt am Main, 1975
(2) R.Thiele, K.Haase: Der verzauberte Raum, Leipzig, 1991
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