This reminds me of a solid angle problem that's been bouncing around my visual cortex: Suppose we start with a cone (a single nappe) in 3-space. Remove the conepoint. What's left is topologically a cylinder: Cyl = S^1 x [0, oo). Now elongate the cone and wrap it around the unit circle in the xy-plane, in such a way that it avoids self-intersection. Now, make it wrap more and more around that same circle, always avoiding self- intersection, and finally limiting on the circle as the cone gets pointier and pointier (without the conepoint at all). Here's the question: We can assume this infinitely helical cone is *smooth* (away from the missing conepoint), and so has a well-defined outward unit normal vector at each point. These unit normal vectors define a map G : Cyl = S^1 x [0, oo) —> S^2 by translating each vector to the origin; this is the Gauss map G. Now consider the function f : [0, oo) —> R defined by f(t) = vol_+(G([t, oo)) where vol+ denotes signed area on the unit sphere S^2. Question: --------- What is the limit lim f(t) ??? t—>oo (I believe the answer is independent of choices and is determined by the conditions defining the cylinder in R^3.) —Dan Bill Gosper wrote: ----- I just computed the solid angles to be ... ... -----