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Regular, Semi-Regular Polyhedra, and thier Duals (first page)
Why I studied polyhedra, and Image Generation Techniques
Known Mathematics of Polyhedral Generation
Vertix Calculations
The following table defines exactly the locations of vertices for some polyhedra. There is suposed to be a list of the exact formula for all the regular and semi-regular polyhedra, but I have not located a source.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 decial 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 posible +/- 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 negitive values in vertices is odd
IE: vertice (1,1,-1) allowed, but (-1,1,-1) is NOT allowed
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Errors in the data from these alturnitive sources, specifically points in a face not being co-planer, has caused me some problems in raytracing the figures. This required a study in methods to resolve the issue. For more information on polyhedra generation see my Details page.
Mathematical Formula for Polyhedra
The following are the angles between two faces of a polyhedron (the dihedral), for those polygons with only a single such angle.
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)))
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Given that 'l' is the length of the edge, then....
- Tetrahedron
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 - Octahedron
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 - Cube
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 - Icosahedron
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 - Dodecahedron
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
Programs
For my own use I have a perl script to list all the lengths and angles of a polyhedra defined by a OFF file. I can also generate diagrams of the objects or extract the face information for the building of 3D models.