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 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!