The celestial image of a polyhedron

We want now to discuss the celestial image of a polyhedron, and use it to get yet another proof of Descartes's angle-defect formula.

Discussion

1. What pattern is traced out on the celestial sphere when you move a flashlight around on the surface of a cube, keeping its tail as flat as possible on the surface? What is the celestial pattern for a dodecahedron?
2. On a convex polyhedron, the celestial image of a region containing a solitary vertex where three faces meet is a triangle. Show that the three angles of this celestial triangle are the supplements of the angles of the three faces that meet at .
3. Show that the area of this celestial triangle is the angle defect at .
4. Show that the total angle defect of a convex polyhedron is .