# How small can the average dihedral angle be in a large polyhedron?

Given a non-degenerate non-self-intersecting polyhedron $$P$$, consider the average of the dihedral angles at each edge in $$P$$. For small polyhedra, this average can be fairly small; for instance, a regular tetrahedron has an average dihedral angle of $$70.53^\circ$$, and the tetrahedron with vertices at $$(1,0,\epsilon),(-1,0,\epsilon),(0,1,-\epsilon),(0,-1,-\epsilon)$$ has an average dihedral angle approaching $$60^\circ$$.

I am interested in the limiting value of this average for large polyhedra, i.e. as the number of edges, faces, and vertices go to infinity. (If any one of these measures goes to infinity, so do the others.)

By taking an $$N$$-gonal pyramid with the vertex extremely close to the base, one can get an average angle approaching $$90^\circ$$ (half the dihedral angles are negligible, half are extremely close to $$180^\circ$$). Is it possible to get any smaller than this in the limit?

So far as I can tell, it seems like no polyhedron of any size can have an average dihedral angle less than $$60^\circ$$ without being self-intersecting; a proof or counterexample to this assertion would be welcome.

• I am afraid that the very meaning of "average dihedral angle" is lying on sand as one can create for example 1000 new tiny dihedral angles out of an old one... Commented May 19, 2021 at 22:43
• @JeanMarie: How do you propose to do this? Given two faces, the associated planes intersect in a single line, so they can form at most one edge between them if the polyhedron is not degenerate. Can you provide an example? Commented May 19, 2021 at 22:47
• @JeanMarie: But a curved edge cannot be the meeting point of two planar faces - planes only intersect along a unique line. Try generating an explicit example of such a polyhedron and I think you will find that the curving of an edge splits one face into many, producing lots of new edges with very large dihedral angles. Commented May 19, 2021 at 23:04
• Without having performed any computations to check this, what about the following idea: take any tetrahedron and a vertex thereof an dent in the vertex (the vertex is now pointy towards the interior of the polyhedron, which is no longer convex). I hope you have an idea what I mean. Does this lower the average angle? You can do this to all vertices and you can also repeat the process with the same vertex again (a double dent, tripe dent etc.). Commented May 26, 2021 at 13:47
• According to a formula from one of my comments there, $$\frac{\sum_e\theta_e}{\sum_e1}\geq\pi-\frac{\sum_v(2\pi-\Omega_v)}{3\sum_v1},$$ the average dihedral angle is at least $\pi/3=60^\circ$ because solid angles are positive. (At least that works for convex polyhedra.) Commented May 3, 2022 at 23:49

It is possible to get at least as low as $$\arccos(\sqrt{5}/3)$$, or about $$41.8$$ degrees, since that is the average dihedral angle in a great icosahedron.