Look around the Graphics Lab
Regular, Semi-Regular Polyhedra, and thier Duals (first page)
Prisms, Anti-prisms, Pryamids, and related Polyhedra
Miscellanous Polyhedra: Deltahedra
Johnson Solids -- The other convex polyhedra with regular faces
Why I studied polyhedra, and Image Generation Techniques
Known Polyhedral Mathematical Formula
Data Sources and links for Polyhedral Data
Generally if you set the points as defined below, then find the polyhedra's convex hull, you will produce a mathematically exact polyhedra, to however many decimal places you need. I myself generated the OFF (Object File Format, See my Details Page) files for the objects listed below to 13 decimal places (overkill I know). The data source for the file if generated in this way is given as "Exact Mathematics".
Name Vertices defining polyhedra's convex hull cube (1,1,1) all permutations [8] cuboctahedron (0,1,1) all permutations [12] octahedron (0,0,1) all permutations [6] truncated octahedron (0,1,2) all permutations [24] tetrahedron (1,1,1) all permutations with odd -ve counts [4] truncated tetrahedron (3,1,1) all permutations with odd -ve counts [12] icosahedron z=(sqrt(5)-1)/2 (golden ratio) (1,0,z) ordered permutations [12] rombic dodecahedron v=1/2 (1,0,0)[6] (v,v,v)[8] kite icositetrahedron u=1/sqrt(2) v=1/(2*sqrt(2)-1) (1,0,0)[6] (0,u,u)[12] (v,v,v)[8] disdyakis-dodecahedron u=1/sqrt(2) v=1/sqrt(3) (or hexakis-octahedron) (1,0,0)[6] (0,u,u)[12] (v,v,v)[8] dodecahedron a=1/sqrt(3) b=sqrt((3-sqrt(5))/6) c=sqrt((3+sqrt(5))/6) (a,a,a)[8] (0,b,c) ordered permutations [12] hexagonal prism v=sqrt(3) (0,2,1) signed permutations [4] (r,1,1) signed permutations [8] Syntax Notes: [n] Number of permutations (vertices) that results "all permutations" All permutations of the three axis components and all possible +/- sign changes of each axis component. "ordered permutations" As above but sequence order retained. EG: 3 rolls of the vertices and all sign changes in each case. "signed permutations" Order is left as given, with +/- sign changes "with odd -ve counts" The number on negative values in vertices is odd IE: vertices (1,1,-1) allowed, but (-1,1,-1) is NOT allowed |
Errors in the data from these alternative sources, specifically points in a face not being co-planer, has caused me some problems in ray-tracing the figures. This required a study into triangulation of the polygons, to resolve the issue. For more information on polyhedra generation see my Ray-tracing Details page.
Name F V E angle cos(a) tan(a) tetrahedron 4 4 6 70.53 1/3 2*sqrt(2) cube 6 8 12 90 0 infi octahedron 8 6 12 109.47 -1/3 -2*sqrt(2) rombic dodecahedron 12 14 24 120 -1/2 -sqrt(3) cuboctahedron 14 12 24 125.26 -sqrt(3)/3 -sqrt(2) dodecahedron 12 20 30 116.57 -sqrt(5)/5 -2 icosahedron 20 12 30 138.19 -sqrt(5)/3 -2*sqrt(2)/5 sin(a) = 2/3 icosidodecahedron 32 30 60 142.62 -sqrt((5+2*sqrt(5))/15) sqrt(5)-3 triacontahedron 30 32 60 144 -(sqrt(5)-1)/4 -sqrt((5-2*sqrt(5))) |
Given that 'l' is the length of the edge, then....
Sin of angle at edge: | 2 * sqrt(2) / 3 |
Surface area: | sqrt(3) * l^2 |
Volume: | sqrt(2) / 12 * l^3 |
Circumscribed radius: | sqrt(6) / 4 * l |
Inscribed radius: | sqrt(6) / 12 * l |
Sin of angle at edge: | 2 * sqrt(2) / 3 |
Surface area: | 2 * sqrt(3) * l^2 |
Volume: | sqrt(2) / 3 * l^3 |
Circumscribed radius: | sqrt(2) / 2 * l |
Inscribed radius: | sqrt(6) / 6 * l |
Sin of angle at edge: | 1 |
Surface area: | 6 * l^2 |
Volume: | l^3 |
Circumscribed radius: | sqrt(3) / 2 * l |
Inscribed radius: | 1 / 2 * l |
Sin of angle at edge: | 2 / 3 |
Surface area: | 5 * sqrt(3) * l^2 |
Volume: | 5 * (3 + sqrt(5)) / 12 * l^3 |
Circumscribed radius: | sqrt(10 + 2 * sqrt(5)) / 4 * l |
Inscribed radius: | sqrt(42 + 18 * sqrt(5)) / 12 * l |
Sin of angle at edge: | 2 / sqrt(5) |
Surface area: | 3 * sqrt(25 + 10 * sqrt(5)) * l^2 |
Volume: | (15 + 7 * sqrt(5)) / 4 * l^3 |
Circumscribed radius: | (sqrt(15) + sqrt(3)) / 4 * l |
Inscribed radius: | sqrt(250 + 110 * sqrt(5)) / 20 * l |
For example, this is the result for a "Kite Hexacontrahedron", which I used to create a Wind inflated ball.
https://antofthy.gitlab.io/graphics/polyhedra/