Definition:Hilbert 23/3
Hilbert $23$: Problem $3$
Finite Dissection of Polyhedra
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?
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 $B$.
Historical Note
The Hilbert 23 were delivered by David Hilbert in a famous address at Paris in $1900$.
He considered them to be the oustanding challenges to mathematicians in the future.
There was originally going to be a $24$th problem, on a criterion for simplicity and general methods in proof theory, but Hilbert decided not to include it, as it was (like numbers $4$, $6$, $16$ and $23$) too vague to ever be described as "solved".
Sources
- 1902: David Hilbert: Mathematical Problems (Bull. Amer. Math. Soc. Vol. 8, no. 10: pp. 437 – 479)
- (translated by Mary Winston Newson from "Mathematische Probleme")
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): dissection proof