Hilbert’s Third Problem and Dehn’s Invariant David Hilbert “Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its devel- opment during future cen- turies?” International Congress of Mathematicians, Paris 1900 International Congress of Mathematicians, Paris 1900 “Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its devel- opment during future cen- turies?” Hilbert’s 23 Problems From 1 Hilbert’s 23 Problems From 1 Euclid, c. 300 BC Tetrahedrons with the same triangular base and same height have the same volume From 2 Tetrahedrons with the same triangular base and same height have the same volume From 2 Creating a square out of an equilateral triangle From 3 Creating a square out of an equilateral triangle From 3 Creating a square out of an equilateral triangle From 3 Creating a square out of other polygons From 5 Examples of scissors-congruence From 6 Tangram Puzzle Two squares to one: The Pythagorean Theorem From 4 Two squares to one: The Pythagorean Theorem From 4 Two squares to one: The Pythagorean Theorem From 4 Two squares to one: The Pythagorean Theorem From 4 P ∼ Q and Q ∼ R implies P ∼ R From 4 P ∼ Q and Q ∼ R implies P ∼ R From 4 P ∼ Q and Q ∼ R implies P ∼ R From 4 P ∼ Q and Q ∼ R implies P ∼ R From 4 P ∼ Q and Q ∼ R implies P ∼ R From 4 P ∼ Q and Q ∼ R implies P ∼ R From 4 P ∼ Q and Q ∼ R implies P ∼ R From 4 P ∼ Q and Q ∼ R implies P ∼ R From 4 P ∼ Q and Q ∼ R implies P ∼ R From 4 Polygon to a square From 4 Polygon to a square From 4 Polygon to a square From 4 Polygon to a square From 4 Polygon to a square From 4 Polygon to a square From 4 Polygon to a square From 4 Polygon to a square From 4 Polygon to a square From 4 Polygon to a square From 4 Hill’s Tetrahedron From 5 Max Dehn D(P) = D(P0) + D(P00) From 2 D(P) = D(P0) + D(P00) From 2 D(P) = D(P0) + D(P00) From 2 D(P) = D(P0) + D(P00) From 2 D(P) = D(P0) + D(P00) From 2 D(P) = D(P0) + D(P00) P D(P) = `ef (αe) e2P From 2 A cube and a regular tetrahedron From 3 A cube and a regular tetrahedron From 3 Answer to Gauss’s question regarding Euclid’s proof From 3 Answer to Gauss’s question regarding Euclid’s proof From 3 Answer to Gauss’s question regarding Euclid’s proof From 3 References for pictures 1 D.J. Albers, G.L. Alexanderson, C. Reid, International mathematical congresses. An illustrated history, 18931986. Springer-Verlag, New York, 1987. 2 http://mathcircle.berkeley.edu/sites/default/ files/BMC6/ps0405/dehn.pdf 3 M. Aigner, G.M. Ziegler, Proofs from The Book. Fifth edition. Including illustrations by Karl H. Hofmann. Springer-Verlag, Berlin, 2014. 4 http://math.ucla.edu/˜marks/talks/circle_ squaring_talk.pdf 5 P.R. Cromwell, Polyhedra. “One of the most charming chapters of geometry”. Cambridge University Press, Cambridge, 1997. 6 http://mathworld.wolfram.com/Dissection.html.