#### monikernemo

##### Warlord

- Joined
- May 9, 2020

- Messages
- 242

I do not really get it, but they have figured out a bit more of Anaxagoras of Clazomenae's problem about squaring the circle?

Around 450 BCE, Anaxagoras of Clazomenae had some time to think. The Greek mathematician was in prison for claiming the sun was not a god, but rather an incandescent rock as big as the Peloponnese peninsula. A philosopher who believed that “reason rules the world,” he used his incarceration to grapple with a now-famous math problem known as squaring the circle: Using a compass and a straightedge, can you produce a square of equal area to a given circle?

The exact question posed by Anaxagoras was answered in 1882, when the German mathematician Ferdinand von Lindemann proved that squaring the circle is impossible with classical tools. He showed that pi — the area of a circle with a radius of 1 — is a special kind of number classified as transcendental (a category that also includes Euler’s number, e). Because a previous result had demonstrated that it’s impossible to use a compass and a straightedge to construct a length equal to a transcendental number, it’s also impossible to square a circle that way.

Tarski's Circle Squaring Problem from 1925 asks whether it is possible to partition a disk in the plane into finitely many pieces and reassemble them via isometries to yield a partition of a square of the same area. It was finally resolved by Laczkovich in 1990 in the affirmative. Recently, several new proofs have emerged which achieve circle squaring with better structured pieces: namely, pieces which are Lebesgue measurable and have the property of Baire (Grabowski-Máthé-Pikhurko) or even are Borel (Marks-Unger).

In this paper, we show that circle squaring is possible with Borel pieces of positive Lebesgue measure whose boundaries have upper Minkowski dimension less than 2 (in particular, each piece is Jordan measurable). We also improve the Borel complexity of the pieces: namely, we show that each piece can be taken to be a Boolean combination of Fσ sets. This is a consequence of our more general result that applies to any two bounded subsets of Rk, k≥1, of equal positive measure whose boundaries have upper Minkowski dimension smaller than k.

Writeup Paper

They are answering a different question here. The classical question of squaring the circle (which means to construct a square with same area of circle via compass and straight-edge construction) is resolved and is deemed impossible. The question stated in the paper asks if you can cut a disk into finitely many pieces and reassemble them by rotation and translation to get a square of the same area. But cutting up a disk into pieces in mathematics can mean absolutely ridiculous things and not just a naive slicing and dicing. This does not contradict the classical question of squaring a circle since they are fundementally asking about different things.