Maybe I'm just being ignorant here, but how could people not have realized the need for curved plane geometry, when they were already sailing on a curved ocean? I suppose they could have assumed the earth was flat if they did not take a long voyage, but didn't they ever look up at the sky and think about that dome? Or look at a piece of fruit, like an apple, and wonder about geometry for that? It just strikes me as odd historically that curved surface geometry took so long to develop when people were already thinking about flat plane geometry.
The question of spherical mapping onto a surface reminds me of the contour mapping we studied earlier in the semester. My brother, Gus, was telling me about this mesa top near Philmont, New Mexico. The Native Americans named it something meaning approximately "skull mountain". They must have had an understanding of contour geometry, because the skull shape is not readily apparent until one looks at a contour map. Then the forehead comes out in sharp relief, and the orbits and nasal cavities sink into depths. The chin is basically a curved cliff. The whole thing made me realize that maybe Gauss wasn't the first to catch on to this curved plane idea, and that possibly he wasn't even the first to document it.
I found the idea of a real projective plane at once confusing and interesting. I understand the Mobius strip. and the Klein bottle, but for some reason the real projective plane is more difficult. The idea of having to try to find a vantage point from which we could see all five edges of the simplex pentagon reminds me of the three house, three utilities problem. In other words, I can see why it can't be done in three space, but I do not understand the shape that would happen were we to attach them in four space.
An appropriate exercise for this chapter would be to build a couple of Mobius strips and draw continuous lines or multiples of figures on each, and then cut them down the middle to see where things end up.>