Sydler pi/4 polyhedron: Step 1
We start with a prism over an isosceles right triangle. This prism is scissors-congruent to a cube but has two π/4 dihedral angles (labeled 1 and 2). After this step, we are left with only one edge with angle π/4. However, we introduced two angles ζ and π-ζ (labeled 3, respectively, 4).
From now on, we always call the angles of a polyhedron which are not right or π/4 the extra angles.