You are right; my rational rationale was wrong. But I still don't want to extend to Q(sqrt(3)), because therein lies madness (or at least a slippery slope). --- Someone over the years _must_ have looked at the "orbits" of the Minsky algorithm (including floor(x)). I hope this doesn't degenerate into something like the 3n+1 problem (Collatz conjecture)... At 03:22 PM 7/5/2011, Andy Latto wrote:
On Tue, Jul 5, 2011 at 3:14 PM, Henry Baker <hbaker1@pipeline.com> wrote:
I should have pointed out that the Trinsky analysis of Minsky essentially produces Pythagorean triples/Gaussian primes for certain rational numbers;
e.g., if d=1, delta*eps = 4/(1+4*d^2) = 4/(1+4*1^2)=4/5,
and w = 2*asin(sqrt(4/5)/2) = 2*asin(1/sqrt(5)) = asin(4/5)
-- i.e., we get a 3,4,5 triangle.
When we compute arbitrarily high powers ((3+4*i)/5)^n, we conserve the length (i.e., energy) of the system, but we don't actually get an "orbit", since Q(i) is a field of characteristic 0.
While I agree we don't get an orbit, because tan(3/4)/tau is not rational, I don't see the connection with the characteristic of the field. Q((sqrt(3)) is a field of characteristic 0, but it includes (sqrt(3) + i)/2, and the twelfth power of this is 1.
Andy