A dodecahedron is a solid figure contained by twelve equal, equilateral, and equiangular pentagons.
A dodecahedron is (the only)1 regular three-dimensional polyhedron with \(12\) faces, \(30\) edges, and \(20\) vertices.
Using the golden ratio
\[\phi:=\frac{1+\sqrt 5}{2},\]
the Cartesian coordinates \((x,y,z)\) of all \(20\) vertices of a dodecahedron centered at the origin are given by
\[\begin{array}{lrrr} \text{Vertex}&x&y&z\\ v_1&1&1&1\\ v_2&1&1&-1\\ v_3&1&-1&1\\ v_4&1&-1&-1\\ v_5&1&1&1\\ v_6&-1&1&-1\\ v_7&-1&-1&1\\ v_8&-1&-1&-1\\ v_9&0&\phi&1/\phi\\ v_{10}&0&\phi&-1/\phi\\ v_{11}&0&-\phi&1/\phi\\ v_{12}&0&-\phi&-1/\phi\\ v_{13}&1/\phi&\phi&0\\ v_{14}&1/\phi&-\phi&0\\ v_{15}&-1/\phi&\phi&0\\ v_{16}&-1/\phi&-\phi&0\\ v_{17}&\phi&0&1/\phi\\ v_{18}&\phi&0&-1/\phi\\ v_{19}&-\phi&0&1/\phi\\ v_{20}&-\phi&0&-1/\phi \end{array}\]
The \(12\) faces of the dodecahedron are equiangular pentagons with the following vertices:
\[\begin{array}{lccccc} \text{Face}\\ f_1&v_1&v_{17}&v_3&v_{11}&v_9\\ f_2&v_1&v_9&v_5&v_{15}&v_{13}\\ f_3&v_{17}&v_{18}&v_2&v_{13}&v_1\\ f_4&v_2&v_{10}&v_{12}&v_4&v_{18}\\ f_5&v_2&v_{13}&v_{15}&v_6&v_{10}\\ f_6&v_3&v_{14}&v_{16}&v_7&v_{11}\\ f_7&v_{14}&v_4&v_{18}&v_{17}&v_3\\ f_8&v_4&v_{12}&v_8&v_{16}&v_{14}\\ f_9&v_5&v_9&v_{11}&v_7&v_{19}\\ f_{10}&v_{15}&v_6&v_{20}&v_{19}&v_5\\ f_{11}&v_6&v_{20}&v_8&v_{12}&v_{10}\\ f_{12}&v_{16}&v_8&v_{20}&v_{19}&v_7 \end{array}\]
Corollaries: 1
Proofs: 2 3
Propositions: 4 5
Sections: 6
This will be proven in the Prop. 18 of Book 13, thus the dodecahedron is well-defined. ↩
