« Site Map « Shapes of the dimensions
n-cubes > k-cubes \/ |
0 point |
1 line |
2 square |
3 cube |
4 tetracube |
5 pentacube |
formula |
vertices | 1 | 2 | 4 | 8 | 16 | 32 | 2n |
edges | 0 | 1 | 4 | 12 | 32 | 80 | n2n-1 |
faces | 0 | 0 | 1 | 6 | 24 | 80 | (1/2)n(n-1)2n-2 |
volumes | 0 | 0 | 0 | 1 | 8 | 40 | (1/6)n(n-1)(n-2)2n-3 |
bulks | 0 | 0 | 0 | 0 | 1 | 10 | (1/24)n(n-1)(n-2)(n-3)2n-4 |
pentabulks | 0 | 0 | 0 | 0 | 0 | 1 | - |
sum | 1 | 3 | 9 | 27 | 81 | 243 | 3n |
number of k-cubes in an n-cube: 2n-kn!/(k!(n-k!))
a = side length
n-cubes | 1 line |
2 square |
3 cube |
4 tetracube |
5 pentacube |
perimeter | a | 4a | 12a | 32a | 80a |
area | - | a2 | 6a2 | 24a2 | 80a2 |
volume | - | - | a3 | 8a3 | 40a3 |
bulk | - | - | - | a4 | 10a4 |
pentabulk | - | - | - | - | a5 |
r = radius
n-spheres | 1 line |
2 circle |
3 sphere |
4 glome |
5 pentaglome |
circumference | 2r | 2πr | ? | ? | ? |
area | - | πr2 | 4πr2 | ? | ? |
volume | - | - | (4/3)πr3 | 2π2r3 | ? |
bulk | - | - | - | (1/2)π2r4 | (8/3)π2r4 |
pentabulk | - | - | - | - | (8/15)π2r5 |
surface area of n-sphere:
n = odd: (2(n+1)/2π(n-1)/2rn-1)/(n-2)!!
*note: n!! = double factorial - see below*
n = even: (2πn/2rn-1)/((1/2)n-1)!
volume of n-sphere:
n = odd: (πn/2rn)/(n/2)!
n = even: (2n((n-1)/2)!π(n-1)/2rn))/n!
double factorial (n!!):
n odd: n×(n-2)×(n-4)...5×3×1
n even: n×(n-2)×(n-4)...6×4×2
n = 0: 1
Rotation matrices, using the angle θ.
cosθ | sinθ |
-sinθ | cosθ |
XY plane
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YZ plane
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ZX Plane
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XY plane
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YZ plane
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ZX Plane
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