Notes on Geometry
-----------------

d = diagonals
e = edges
f = faces
v = vertices

Euler's Formula            f + v = e + 2

Area of a circle           pi * r^2

Circumference of a circle  c = 2 * pi * r

Equation of a circle       x^2 + y^2 = r^2

Volume of a sphere         4/3 * pi * r^3

Surface area of a sphere   4 * pi * r^2

Lateral Surface area 
of a Cylinder =            2 * pi * r * h 

Golden Ratio               1/2 * ( 1 + sqrt(5) ) = 1.618033989
Also the last two terms of the Fibonacci series converge on the Golden Ratio:
 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89....  34/21 = 1.6190
                                           55/34 = 1.617647
                                           89/55 = 1.6181818...

Long Diagonal of a Cube    d = Sqrt(3*Length^2) = Length * Sqrt(3)

Any 3 points are co-planar, since 3 points define a plane.

X^4      = Hypercubes
X^3      = Cubes
X^2      = Faces
X        = Edges
Integer  = Vertices

The expansion of this expression predicts attributes of the N-cube:
( x + 2)^N

Point     (x + 2)^0 = 1

Line      (x + 2)^1 = x + 2

Square    (x + 2)^2 = x^2 + 4*x + 4

Cube      (x + 2)^3 = x^3 + 6*x^2 + 12*x + 8

Hypercube (x + 2)^4 = x^4 + 8*x^3 + 24*x^2 + 32*x + 16

The expansion of this expression predicts attributes of the N-tetrahedron:
(( x + 1)^ (N + 1) - 1) / x

Line              = x + 2

Triangle          = x^2 + 3*x + 3

Tetrahedron       = x^3 + 4*x^2 + 6*x + 4

Hyper Tetrahedron = x^4 + 5*x^3 + 10*x^2 + 10*x + 5 

There are 5 Platonic Solids, and 13 Archimedian Solids

All 5 Platonic solids can be inscribed inside a sphere
The tetrahedron and the octahedron can be inscribed inside a cube
A tetrahedron inscribed inside a cube would have 1/3 of the cube's volume

Name                 Faces         Vertices       Edges     Orientations   Internal Diagonals
----                 -----         --------       -----     ------------   ------------------
 
Tetrahedron           4 triangles      4              6        12                  0
Cube                  6 squares        8             12        24                  4     
Octahedron            8 triangles      6             12        24                  3
Dodecahedron         12 pentagons     20             30        60                100
Icosahedron          20 triangles     12             30        60                 36

To calculate Internal Diagonals:
(V choose 2) - E - face diagonals, e.g. (20 choose 2) - 30 - 60 = 100 for the dodecahedron

The orientations in space of any platonic solid with N faces, with each face 
having E edges, is simply N*E.

Truncated Icosahedron 32              60             90        60
Also called icosadodecahedron or Bucky Ball
12 pentagons + 20 hexagons

Tetradecahedron (A)  14 =             12             24        24
Also called Cuboctahedron, or truncated cube
6 squares + 8 triangles

Tetradecahedron (B)  14 =             24             36        24
Also called truncated octahedron
6 squares + 8 hexagons

Tetradecahedron (C)  14 =             18             30        12
2 hexagons + 12 trapezoids

The edge graph of a cube does not contain an Eulerian circuit.

The vertex graph of a cube contains a Hamiltonian circuit but is 
not itself a Hamiltonian circuit.

The edge graph of a hypercube does contain an Eulerian circuit!