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. And there’s no better way of turning Conway’s “deflation” into “inflation”. In physics the Galois involution i <-> -i is also not exactly trivial, in the sense that it corresponds to “charge conjugation symmetry” (invariance upon changing the signs of all electrical charges). -Veit
On Jul 15, 2015, at 3:05 PM, Dan Asimov <dasimov@earthlink.net> wrote:
What Jim says is true, but a large majority of the literature agrees that the golden ratio is greater than 1:
phi = (1+sqrt(5))/2) : 1.
Namely the ratio of the sides of a rectangle such that if you cut off a square from it, what's left is the same ratio scaled and rotated 90 degrees. Which leads to
phi = 1/(phi-1)
and the same polynomial x^2 - x - 1 = 0.
And it all agrees that phi s positive.
—Dan