Formulas for shapes

Last revised 2003-03-14

n-Faces

n-cubes

 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!))

n-Volumes

n-cubes

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

n-spheres

 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

Rotation matrices, using the angle θ.

Planespace/2-space

 cosθ sinθ -sinθ cosθ

Realmspace/3-space

XY plane

 cosθ sinθ 0 -sinθ cosθ 0 0 0 1

YZ plane

 1 0 0 0 cosθ sinθ 0 -sinθ cosθ

ZX Plane

 cosθ 0 -sinθ 0 1 0 sinθ 0 cosθ

Tetraspace/4-space

XY plane

 cos θ sin θ 0 0 -sin θ cos θ 0 0 0 0 1 0 0 0 0 1

YZ plane

 1 0 0 0 0 cos θ sin θ 0 0 -sin θ cos θ 0 0 0 0 1

ZX Plane

 cos θ 0 -sin θ 0 0 1 0 0 sin θ 0 cos θ 0 0 0 0 1

XW plane

 cos θ 0 0 sin θ 0 1 0 0 0 0 1 0 -sin θ 0 0 cos θ

YW plane

 1 0 0 0 0 cos θ 0 -sin θ 0 0 1 0 0 sin θ 0 cos θ

ZW Plane

 1 0 0 0 0 1 0 0 0 0 cos θ -sin θ 0 0 sin θ cos θ