Finite Dissection of Polyhedra

Theorem

Let $A$ and $B$ be polyhedra with the same volume.

Then it is not necessarily the case that there exists a dissection of $A$ into finitely many components that may be reassembled to form $B$.

Proof

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Historical Note

The question of the Finite Dissection of Polyhedra was the $3$rd of the Hilbert 23.

It was settled in $1900$ by Max Wilhelm Dehn, who provided a counterexample: two polyhedra of the same volume for which there exists no dissection of one into the other.

This was the first of the Hilbert 23 to be solved.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): equidecomposable
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): dissection proof
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): equidecomposable