Steve Witham shows how even the integer (floor) version of Minsky's circle hack is invertible/reversible: http://www.tiac.net/~sw/2005/03/Mandala/index.html The Minsky steps are: x := x - floor(delta*y) y := y + floor(epsilon*x) This means, starting with (x0,y0): x1 = x0 - floor(delta*y0) y1 = y0 + floor(epsilon*x1) Suppose now that we have (x1,y1): Since y1 = y0 + floor(epsilon*x1), we can solve for y0: y0 = y1 - floor(epsilon*x1) Since x1 = x0 - floor(delta*y0), we can solve for x0: x0 = x1 + floor(delta*y0) Notice that "+"/"-" doesn't even have to be the usual integers, but could be some group -- e.g., Z_2^n, Z_p, or something even wilder. --- Neil Bickford comments that when (delta,epsilon)=(9/17,15/2), the initial point (1,0) goes through at least 104 _trillion_ points without looping. http://nbickford.wordpress.com/2011/04/03/the-minsky-circle-algorithm/ Given delta, epsilon, let us define an equivalence relation for lattice points on the X-Y plane: (x1,y1) ~ (x0,y0) if (x1,y1) is a finite number of forward or backwards Minsky steps from (x0,y0). Since the Minsky algorithm is reversible, each of the equivalence classes is either a finite cycle or an infinite chain (isomorphic to Z). We want to place some bounds on the sizes of these equivalence classes. Under "normal" conditions, the Minsky circle algorithm starting with (x,y) (x,y integers) should explore no more than 2*pi*sqrt(x^2+y^2)*C points before looping, where C>0 is some small constant. Rationale: this _is_, after all, a _circle_ drawing algorithm, so if it colors very many more points than are on the circumference of the circle, it can't possibly be a circle drawing algorithm. We already know that some difference equation approximations to differential equations don't converge to the differential equation if the "step size" is too big. Something analogous to this behavior seems to be going on here. 1. Can we get tight bounds on C, if delta & epsilon are bounded in some way? 2. For the less bounded case: Is the Collatz Conjecture hiding in the Minsky algorithm somewhere?