Cartesian Coordinates
The following Cartesian coordinates define the vertices of an icosahedron with edge-length 2, centered at the origin:
- (0, ±1, ±φ)
- (±1, ±φ, 0)
- (±φ, 0, ±1)
where φ = (1 + √5) / 2 is the golden ratio (also written τ). Note that these vertices form five sets of three concentric, mutually orthogonal golden rectangles, whose edges form Borromean rings.
If the original icosahedron has edge length 1, its dual dodecahedron has edge length, one divided by the golden ratio.
The 12 edges of a regular octahedron can be subdivided in the golden ratio so that the resulting vertices define a regular icosahedron. This is done by first placing vectors along the octahedron's edges such that each face is bounded by a cycle, then similarly subdividing each edge into the golden mean along the direction of its vector. The five octahedra defining any given icosahedron form a regular polyhedral compound, while the two icosahedra that can be defined in this way from any given octahedron form a uniform polyhedron compound.
Read more about this topic: Icosahedron