* Veit Elser <ve10@cornell.edu> [Jul 16. 2015 08:15]:
There’s something to be said for having conventions, but you also lose sight of interesting things when you are rigid in your ways.
The Galois automorphism sqrt(5) <-> -sqrt(5) has geometrical consequences. Perhaps you’ve encountered the construction of the 2D Penrose tiling starting from a lattice in 4D? Or its counterpart in 3D derived from a lattice in 6D? Basic to those constructions is an orthogonal decomposition of the 4D or 6D space by a pair of irrational subspaces. The automorphism has the effect of swapping those subspaces. So in addition to those obvious symmetries of the tiling, e.g. the pentagon (or icosahedron), there is the less obvious symmetry associated with changing the sign of sqrt(5). I’m actually surprised the term “Penrose involution” doesn’t bring up any hits.
Could you point me to any reference regarding this (especially 6D-->3D)?
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