Curvature of Polyhedra (2021) (opens in new tab)

It's definitely not immediately obvious that this number should be an invariant for any shape topologically equivalent to a sphere.

However, it is easy to see why it should hold for a dihedron.

For a general convex shape (to start with) I think I would try to show people how the planar face orientations change at the corners, and point out that these must sweep out the whole sphere in the 3-dimensional space of planar orientations. This is similar to the way in which the linear orientations of the sides of a polygon must sweep out a whole circle (exterior angles sum to 2π). Non-convex corners can "subtract" some of the areas we already swept out, which can be a little tricky to reason about, but could probably be made clearish with a nice interactive computer diagram.

We just need to look at the corners because on the face there is no orientation change, and at the edges the orientation change is 1 dimensional and therefore has no "area" relative to the 2-dimensional quantity we are measuring. This might be easier if you look at the linear orientations normal to the faces.

It might be easier still to just start with tetrahedra.

* * *

I have a much harder problem for everyone: come up with some way of calculating the sum of angle deficit at the corners of a polyhedron constructed of great-spherical faces on a 3-sphere and relating it to the size of the polyhedron. Does this generalize the pattern for spherical polygons on the 2-sphere? (This is a genuine question: I'm not sure quite what is known about this.)

Drop perpendiculars from the center of the circle through each side. Pairs of adjacent perpendiculars each subtend an arc of the circle, of length equal to the "exterior angle" between the corresponding sides. Those arcs sum to 2π radians. The intersection points with the circle from a sort of "dual" polygon to the original, exchanging sides with vertices, and exchanging central angles with exterior angles. (But this dual construction is not uniquely reversible back to a non cyclic polygon or not using the same center)

Now take a polyhedron and choose a center and draw a sphere around it. Drop perpendiculars to faces, intersecting with the sphere to create new "dual" vertices from faces. This time, draw edges between adjacent-face perpendiculars , so each original edge generates a new dual (perpendicular!) edge. This now creates a new face for each vertex of the original shape. Now we have a dual polyhedron inscribed in a sphere, where each face subtends the 2D "exterior angle" of an vertex of the original polygon.

Note that an exterior (hyper!)angle of a hyperpolyhedron in N-dimensional space is an N-1 dimensional measure (face-dimensional dimensional measure) of a vertex (with its cluster of incident faces). In 2D, there is an isomorphism between pairs-of-edges and ordinary 1D angles. (and edges are 2D hyper-faces). But in higher dimensions we need use N-1 dimensional dual hyperfaces to measure N-1 dimensional exterior angles

I'm not actually sure this really works in n>3 dimensions.

Thinking about HN favorite Geometric Algebra is a little helpful for the intuition here.

For concave shapes, some of these angles are negative, as the relevant side/face is oriented the opposite way from its neighbors, with respect to the enclosing circle/sphere/hypersphere.

These N-1 dimensional angles in N dimensional are called "solid" angles. The unit is the "steradian", which is usually considered dimensionless, but cool cats know that its dimension is really the dimension of normalized hypersurface hyperarea. (Similar to how a https://en.m.wikipedia.org/wiki/Solid_angle

So the polygon has 4pi total exterior angle (vertex angle defect) is you consider it as a degenerate 3D object.

https://en.wikipedia.org/wiki/Chern%E2%80%93Gauss%E2%80%93Bo... (the History section is pretty interesting)

https://en.wikipedia.org/wiki/Gauss%E2%80%93Bonnet_gravity

As briefly touched on there, Einstein-Gauss-Bonnet (EGB) gravity turned out to require higher dimensions (the start of this abstract is worth the click :-) <https://link.springer.com/article/10.1140/epjc/s10052-020-82...> ) so the idea that EGB might sweep up higher order curvature in the self-interaction of physical gravitation is mostly dead.

Mostly. There's still a trickle in the literature. For example, <https://doi.org/10.1103/PhysRevD.104.044026> ("General relativity from Einstein-Gauss-Bonnet gravity", 2021, open access; ref [12] therein corresponds to <https://arxiv.org/abs/0805.3575> (no singularity in EGB black holes -- lots of Ricci calculus there, nothing like soccer balls as far as I can see)). The references in <https://arxiv.org/abs/1904.00260v1> (no singularity in Ef(G)B cosmology, applying a correcting function on the Gauss-Bonnet term) can takes an interested reader to years of discussions about the accelerated expansion of the universe. As always the first two digits of an arXiv identifier are the submission year.