Articles tagged with "geometry"

The Dehn Invariant, or, Tangrams In Space

\(\newcommand{\ZZ}{\Bbb Z} \newcommand{\QQ}{\Bbb Q} \newcommand{\RR}{\Bbb R}\)

Fans of wooden children’s toys may remember tangrams, a puzzle composed of 7 flat pieces that can be rearranged into numerous different configurations.

Tangrams in square and cat configurations

As mathematicians, we’re interested in shapes that are slightly simpler than cats or houses.

Monsky's Theorem

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For which \(n\) can you cut a square into \(n\) triangles of equal area?

This question appears quite simple; it could have been posed to the Ancient Greeks. But like many good puzzles, it is a remarkably stubborn one.

It was first solved in 1970, by Paul Monsky. Despite the completely geometric nature of the question, his proof relies primarily on number theory and combinatorics! There is a small amount of algebraic machinery involved, but his proof is quite accessible, and we will describe it below.