Formulas for shapes

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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	n2^n-1
faces	0	0	1	6	24	80	(1/2)n(n-1)2^n-2
volumes	0	0	0	1	8	40	(1/6)n(n-1)(n-2)2^n-3
bulks	0	0	0	0	1	10	(1/24)n(n-1)(n-2)(n-3)2^n-4
pentabulks	0	0	0	0	0	1	^-
sum	1	3	9	27	81	243	3ⁿ

number of k-cubes in an n-cube: 2^n-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

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)/2r^n-1)/(n-2)!! *note: n!! = double factorial - see below*
n = even: (2π^n/2r^n-1)/((1/2)n-1)!

volume of n-sphere:
n = odd: (π^n/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

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