We follow the steps by Jean-Pierre Sydler to construct a polyhedron that is scissors congruent to a cube and where all dihedral angles are right except for one which is 45 degrees.

This polyhedron is used in Sydler's proof of Sydler’s Theorem which is related to Hilbert’s third problem and states that every polyhedron is determined by its volume and Dehn invariant up to scissors congruence.

References: J.-P. Sydler, “Conditions necessaires et suffisantes pour l’equivalence des polyedres de l’espace euclidien a trois dimensions”, Commentarii mathematici Helvetici (1965/66) Volume: 40, page 43-80