I have a puzzle to construct a polyhedron where the measure of every dihedral angle is $\frac\pi4$ radians or prove that none can exist. The polyhedron may self-intersect however it must be finite, bounded, and dyadic as well as have planar faces.
Here's a quick glossary of terms as I use them:
- Polyhedron: A polytope in Euclidean 3-space.
- Finite: There are only a finite number of elements in the polyhedron.
- Bounded: There exists a sphere of finite radius which encloses every vertex of the polyhedron.
- Dyadic: Each edge is incident on exactly two faces.
- Planar faces: For every face all vertices of that face lie on some plane.
These are all pretty standard when talking about polyhedra.
As an example, if we were to imagine a 2-dimensional version of this problem, a regular octagram (Schläfli symbol $\{8/3\}$) would be a solution. A square bowtie would be another solution (illustration). Both are polygons with all interior angles measuring $\frac\pi4$ radians.
Of course a polyhedron is tougher since there is more freedom to arrange things in 3 space.