# Studies into Polyhedra -- Maths

Look around the Graphics Lab

## 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 ```
All the other models displays used data sources that are nowhere nearly as exact, generalialy to 5 decimal places. Such data is also rarely alligned to to the coordinate system. Most commonly the data was extracted from the vrml files published on the WWW by George Hart in his Encyclopedia of Polyhedra.

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))) ```
For Platonic Solids the Geometry Center has a page on Regular Polyhedra Formula, including the dihedral angles between faces, that is very detailed.

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.

Created: 30 April 2001
Updated: 19 September 2003
Author: Anthony Thyssen, <anthony@cit.gu.edu.au>
URL: `http://www.cit.gu.edu.au/~anthony/graphics/polyhedra/`