Can you think of a single line of reasoning that explains both facts?
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.)
Comments
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.)