# Dudeney Puzzle: A Tale from Pythagoras to Dehn - Part II

## Let me provide the math and measurements of one angle as it consists of one parameter only.

$\alpha =$arc$\sin (\frac{\sqrt{\sqrt{3}}}{2}) \approx 41.150335^\circ$﻿

## Now the beauty of something outside of Human Imagination comes into the question:

Can you decompose the circle into a finite number of pieces and reassemble them to get square?[equidecomposability, it means it may not be cut be scissors but yes it can be done somehow]

## Hilbert’s 3rd Problem exactly deals with this problem:

Given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces which can be reassembled to yield the second?

## Given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball. Indeed, the reassembly process involves only moving the pieces around and rotating them without changing their shape. However, the pieces themselves are not "solids" in the usual sense, but infinite scatterings of points. The reconstruction can work with as few as five pieces.

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