Definition: Def. 11.27: Icosahedron

(Definition 27 from Book 11 of Euclid's “Elements”)

An icosahedron is a solid figure contained by twenty equal and equilateral triangles.

Modern Definition

An icosahedron is (the only)1 regular three-dimensional polyhedron with \(20\) faces, \(30\) edges, and \(12\) vertices.

Cartesian Coordinates of the Vertices and Faces of an Icosahedron

Using the golden ratio

\[\phi:=\frac{1+\sqrt 5}{2},\]

the Cartesian coordinates \((x,y,z)\) of all \(12\) vertices of the icosahedron centered at the origin are given by

\[\begin{array}{lrrr} \text{Vertex}&x&y&z\\ v_{1}&-1&0&\phi\\ v_{2}&1&0&\phi\\ v_{3}&-1&0&-\phi\\ v_{4}&1&0&-\phi\\ v_{5}&0&\phi&1\\ v_{6}&0&\phi&-1\\ v_{7}&0&-\phi&1\\ v_{8}&0&-\phi&-1\\ v_{9}&\phi&1&0\\ v_{10}&-\phi&1&0\\ v_{11}&\phi&-1&0\\ v_{12}&-\phi&-1&0 \end{array}\]

The \(20\) faces of the icosahedron are equiangular triangles with the following vertices:

\[\begin{array}{lccccc} \text{Face}\\ f_{1}&v_{1}&v_{2}&v_{5}\\ f_{2}&v_{1}&v_{2}&v_{7}\\ f_{3}&v_{1}&v_{5}&v_{10}\\ f_{4}&v_{1}&v_{7}&v_{12}\\ f_{5}&v_{1}&v_{10}&v_{12}\\ f_{6}&v_{2}&v_{7}&v_{11}\\ f_{7}&v_{2}&v_{5}&v_{9}\\ f_{8}&v_{3}&v_{4}&v_{6}\\ f_{9}&v_{3}&v_{4}&v_{8}\\ f_{10}&v_{3}&v_{6}&v_{10}\\ f_{11}&v_{3}&v_{8}&v_{12}\\ f_{12}&v_{3}&v_{10}&v_{12}\\ f_{13}&v_{4}&v_{6}&v_{9}\\ f_{14}&v_{4}&v_{9}&v_{11}\\ f_{15}&v_{4}&v_{11}&v_{8}\\ f_{16}&v_{5}&v_{6}&v_{9}\\ f_{17}&v_{5}&v_{6}&v_{10}\\ f_{18}&v_{7}&v_{8}&v_{12}\\ f_{19}&v_{7}&v_{8}&v_{11}\\ f_{20}&v_{2}&v_{9}&v_{11}\\ \end{array}\]

Corollaries: 1
Problems: 2
Proofs: 3 4
Propositions: 5 6
Sections: 7

Thank you to the contributors under CC BY-SA 4.0!



Adapted from (subject to copyright, with kind permission)

  1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"

Adapted from CC BY-SA 3.0 Sources:

  1. Prime.mover and others: "Pr∞fWiki",, 2016


  1. This will be proven in the Prop. 18 of Book 13, thus the icosahedron is well-defined.