Notes on Geometry
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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
Predict 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
Predict 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!