超級立方體在數學概念中表示四維空間上的立方體。
如果說是數學上來講，超級立方體就是立方體在多維空間中的推廣，比如在四維空間中，超級立方體是一個每個頂角上有四條棱邊的圖形，其中任意三條邊構成一個三維的立方體，並且這個圖形有16個頂角，32條棱邊。
基本介紹
- 中文名：超級立方體
- 學科：數學
- 詞性：名詞
- 意義：數學概念中表示四維空間上立方體
3D向4D的推導過程,Central Projections,Cross-Sections,The n-dimensional Cube,
3D向4D的推導過程
If you move a square parallel in space and join the corresponding corners, you get the perspective sight of the cube. |
If you move a cube parallel in space and join the corresponding corners, you get the perspective sight of the hypercube. |
The hypercube has 16 corners (derived from 2 cubes) and 32 edges (2 cubes and joining lines).
The hypercube has 24 squares.
The cube is covered by six squares. In the same way eight cubes form the hypercube. |
The numbers 134, 124, 234, 123 indicate the base vectors (declared below). If you know the 3D view, you can look at the hypercube three-dimensionally, too.
Central Projections
...... | 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) |
6 of the 8 cubes appear as pyramid stumps, which lie between the small and the big cube. |
4 cubes, 6 squares, and 4 edges meet at each corner.
3 cubes and 3 squares meet at each edge.
2 cubes meet at each square. Nets top
...... | 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. |
Cross-Sections
..... | 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 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). |
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.
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 top
The n-dimensional Cube
The hypercube is a construct of ideas. You receive 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.
The data of the hypercube might follow in the next line. Dimension=4 and corners=16 are clear.
There are formulas for continuing the sequence for the edges and squares.
If you take n=4, you get the data of the hypercube.
Encore: Data of the 5-dimensional cube
