Construction By A System of Equiangular Lines
Icosahedron H3 Coxeter plane |
6-orthoplex D6 Coxeter plane |
| This construction can be geometrically seen as the 12 vertices of the 6-orthoplex projected to 3 dimensions. This represents a geometric folding of the D6 to H3 Coxeter groups:
Seen by these 2D Coxeter plane orthogonal projections, the two overlapping central vertices define the third axis in this mapping. |
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The following construction of the icosahedron avoids tedious computations in the number field necessary in more elementary approaches.
The existence of the icosahedron amounts to the existence of six equiangular lines in . Indeed, intersecting such a system of equiangular lines with a Euclidean sphere centered at their common intersection yields the twelve vertices of a regular icosahedron as can easily be checked. Conversely, supposing the existence of a regular icosahedron, lines defined by its six pairs of opposite vertices form an equiangular system.
In order to construct such an equiangular system, we start with this 6×6 square matrix:
A straightforward computation yields A2 = 5I (where I is the 6×6 identity matrix). This implies that A has eigenvalues and, both with multiplicity 3 since A is symmetric and of trace zero.
The matrix induces thus a Euclidean structure on the quotient space which is isomorphic to since the kernel of has dimension 3. The image under the projection of the six coordinate axes in forms thus a system of six equiangular lines in intersecting pairwise at a common acute angle of . Orthogonal projection of ±v1, ..., ±v6 onto the -eigenspace of A yields thus the twelve vertices of the icosahedron.
A second straightforward construction of the icosahedron uses representation theory of the alternating group A5 acting by direct isometries on the icosahedron.
Read more about this topic: Icosahedron
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