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. |
|
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
Famous quotes containing the words construction, system and/or lines:
“Striving toward a goal puts a more pleasing construction on our advance toward death.”
—Mason Cooley (b. 1927)
“Exploitation and oppression is not a matter of race. It is the system, the apparatus of world-wide brigandage called imperialism, which made the Powers behave the way they did. I have no illusions on this score, nor do I believe that any Asian nation or African nation, in the same state of dominance, and with the same system of colonial profit-amassing and plunder, would have behaved otherwise.”
—Han Suyin (b. 1917)
“Every living language, like the perspiring bodies of living creatures, is in perpetual motion and alteration; some words go off, and become obsolete; others are taken in, and by degrees grow into common use; or the same word is inverted to a new sense or notion, which in tract of time makes an observable change in the air and features of a language, as age makes in the lines and mien of a face.”
—Richard Bentley (1662–1742)