Hilbert’s Third Problem and Dehn’s “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 with the same triangular base and same height have the same

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

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 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

From 5 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) e∈P

From 2 A 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 ”. Cambridge University Press, Cambridge, 1997. 6 http://mathworld.wolfram.com/Dissection.html