Chiral squaring

Approximate Squaring

We study the " approximate squaring " map f (x) := x⌈x⌉ and its behavior when iterated. We conjecture that if f is repeatedly applied to a rational number r = l/d > 1 then eventually an integer will be reached. We prove this when d = 2, and provide evidence that it is true in general by giving an upper bound on the density of the " exceptional set " of numbers which fail to reach an integer. We...

Squaring and Not Squaring One or More Planes

A set of natural numbers tiles the plane if a square-tiling of the plane exists using exactly one square of sidelength n for every n in the set. In [9] it is shown that N, the set of all natural numbers, tiles the plane. We answer here a number of questions from that paper. We show that there is a simple tiling of the plane (no nontrivial subset of squares forms a rectangle). We show that neith...

Squaring the Plane

Borel circle squaring

We give a completely constructive solution to Tarski’s circle squaring problem. More generally, we prove a Borel version of an equidecomposition theorem due to Laczkovich. If k ≥ 1 and A,B ⊆ Rk are bounded Borel sets with the same positive Lebesgue measure whose boundaries have upper Minkowski dimension less than k, then A and B are equidecomposable by translations using Borel pieces. This answ...

Squaring Rectangles for Dumbbells

The theorem on squaring a rectangle (see Schramm [6] and CannonFloyd-Parry [1]) gives a combinatorial version of the Riemann mapping theorem. We elucidate by example (the dumbbell) some of the limitations of rectangle-squaring as an approximation to the classical Riemamnn mapping.