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(30Oct13: Now solved; see Addendum.) Define a curve, a ratchet spiral, $S(r_0,\epsilon)$ as follows, where $r_0 > 0$ and $\epsilon < 1$.     $S(r_0,\epsilon)$ begins with the arc ...

Q. Do shortest paths walking between rational points of height $h$ ever properly cross themselves? Explaining this question takes a bit of definitional exposition. First, I copy definitions from ...