Pictures of Borel Circle Squaring

The images below show the first step in the proof there is a Borel solution to Tarski's circle squaring problem. We give three bounded real-valued function defined on the torus. Their sum is the characteristic function of a square. However, after translating these functions by the vectors shown, they sum to the characteristic function of a circle.

(Click on the picture to enlarge)

These functions were calculated using the formulas in Section 4 of our paper "Borel circle squaring". Section 5 of the paper describes how to convert such real-valued functions to integer-valued functions with the same property. From these integer valued functions, it is easy to find an equidecomposition of a circle and a square (see Section 6).

Click here for a video.

Below is a picture of a full equidecomposition using 22 pieces