An Ancient Geometry Problem Falls to New Mathematical Techniques

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An Ancient Geometry Problem Falls to New Mathematical Techniques
Quanta Magazine

Summary

The article describes a major advance on the ancient “squaring the circle” problem. While it was proven in the 19th century that a circle cannot be transformed into an equal-area square using only a compass and straightedge, mathematicians later asked whether a circle could instead be cut into finitely many pieces and rearranged into a square. Building on decades of work, three mathematicians—András Máthé, Oleg Pikhurko, and Jonathan Noel—developed a new constructive solution that uses pieces simple enough to be visualized, improving on earlier proofs that relied on highly abstract, nonconstructive, or difficult-to-describe pieces. 
Their work shows that a circle and a square of equal area can be decomposed and reassembled into one another using finitely many well-defined pieces, representing a significant step forward in understanding geometric decomposition and the relationship between shape and area. 

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