Elements
A hypercube of dimension n has 2n "sides" (a 1-dimensional line has 2 end points; a 2-dimensional square has 4 sides or edges; a 3-dimensional cube has 6 2-dimensional faces; a 4-dimensional tesseract has 8 cells). The number of vertices (points) of a hypercube is 2n (a cube has 23 vertices, for instance).
A simple formula to calculate the number of "n-2"-faces in an n-dimensional hypercube is:
The number of m-dimensional hypercubes (just referred to as m-cube from here on) on the boundary of an n-cube is
- , where and n! denotes the factorial of n.
For example, the boundary of a 4-cube (n=4) contains 8 cubes (3-cubes), 24 squares (2-cubes), 32 lines (1-cubes) and 16 vertices (0-cubes).
This identity can be proved by combinatorial arguments; each of the vertices defines a vertex in a -dimensional boundary. There are ways of choosing which lines ("sides") that defines the subspace that the boundary is in. But, each side is counted times since it has that many vertices, we need to divide with this number. Hence the identity above.
These numbers can also be generated by the linear recurrence relation
- , with, and undefined elements = 0.
For example, extending a square via its 4 vertices adds one extra line (edge) per vertex, and also adds the final second square, to form a cube, giving = 12 lines in total.
| m | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| n | γn | n-cube | Names Schläfli symbol Coxeter-Dynkin |
Vertices | Edges | Faces | Cells (3-faces) |
4-faces | 5-faces | 6-faces | 7-faces | 8-faces | 9-faces | 10-faces |
| 0 | γ0 | 0-cube | Point - |
1 | ||||||||||
| 1 | γ1 | 1-cube | Line segment {} |
2 | 1 | |||||||||
| 2 | γ2 | 2-cube | Square Tetragon {4} |
4 | 4 | 1 | ||||||||
| 3 | γ3 | 3-cube | Cube Hexahedron {4,3} |
8 | 12 | 6 | 1 | |||||||
| 4 | γ4 | 4-cube | Tesseract Octachoron {4,3,3} |
16 | 32 | 24 | 8 | 1 | ||||||
| 5 | γ5 | 5-cube | Penteract Decateron {4,3,3,3} |
32 | 80 | 80 | 40 | 10 | 1 | |||||
| 6 | γ6 | 6-cube | Hexeract Dodecapeton {4,3,3,3,3} |
64 | 192 | 240 | 160 | 60 | 12 | 1 | ||||
| 7 | γ7 | 7-cube | Hepteract Tetradeca-7-tope {4,3,3,3,3,3} |
128 | 448 | 672 | 560 | 280 | 84 | 14 | 1 | |||
| 8 | γ8 | 8-cube | Octeract Hexadeca-8-tope {4,3,3,3,3,3,3} |
256 | 1024 | 1792 | 1792 | 1120 | 448 | 112 | 16 | 1 | ||
| 9 | γ9 | 9-cube | Enneract Octadeca-9-tope {4,3,3,3,3,3,3,3} |
512 | 2304 | 4608 | 5376 | 4032 | 2016 | 672 | 144 | 18 | 1 | |
| 10 | γ10 | 10-cube | Dekeract icosa-10-tope {4,3,3,3,3,3,3,3,3} |
1024 | 5120 | 11520 | 15360 | 13440 | 8064 | 3360 | 960 | 180 | 20 | 1 |
Read more about this topic: Hypercube
Famous quotes containing the word elements:
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—Thomas Campbell (1777–1844)
“psychologist
It is through friendships that teenagers learn to take responsibility, provide support, and give their loyalty to non- family members. It is also in teenage friendships that young people find confidants with whom to share thoughts and feelings that they are not comfortable sharing with their parents. Such sharing becomes one of the elements of true intimacy, which will be established later.”
—David Elkind (20th century)
“The two elements the traveler first captures in the big city are extrahuman architecture and furious rhythm. Geometry and anguish. At first glance, the rhythm may be confused with gaiety, but when you look more closely at the mechanism of social life and the painful slavery of both men and machines, you see that it is nothing but a kind of typical, empty anguish that makes even crime and gangs forgivable means of escape.”
—Federico García Lorca (1898–1936)