# Matthias Goerner

Senior Software Engineer

Pixar Animation Studios

Emeryville, California, USA

I am a software engineer by day and mathematician by night interested in low-dimensional topology, hyperbolic geometry, knots and links.

Sydler pi/4 polyhedron

4 x 6 x 6 cm

3d print

2017

Can you find a polyhedron where all but one dihedral angle are right angles?

Jean-Pierre Sydler is a forgotten master of polyhedral engineering with a special skill to tackle such questions. The construction of the above polyhedron was outlined in the final paper of his series of papers proving the Dehn-Sydler theorem, a stronger version of Hilbert's 3rd problem: two polyhedra of equal volume are scissors-congruent (one can be obtained from the other by cutting along planes and moving the pieces around) if and only if certain algebraic conditions on the edge lengths and dihedral angles are met.

The modern proof of this result is non-constructive. But can modern tools (3d printing) revive Sydler's constructive ideas?

Jean-Pierre Sydler is a forgotten master of polyhedral engineering with a special skill to tackle such questions. The construction of the above polyhedron was outlined in the final paper of his series of papers proving the Dehn-Sydler theorem, a stronger version of Hilbert's 3rd problem: two polyhedra of equal volume are scissors-congruent (one can be obtained from the other by cutting along planes and moving the pieces around) if and only if certain algebraic conditions on the edge lengths and dihedral angles are met.

The modern proof of this result is non-constructive. But can modern tools (3d printing) revive Sydler's constructive ideas?