Beyond the Third Dimension--chapter 9
It is interesting how controversial the new kind of geometry was at the time it was developed. The statement that "the suggestion that some new system of statements deserved to be called geometry was a threat" shows that the accepted mathematics of a time is more than just a way of looking at number problems. It is, to some, a way of analyzing the world.
The idea of a noncommutative algebra was introduced when we talked about the idea of rotating in the fourth dimension and returning to 3-space looking like your mirror image. We had also touched on the concepts of intrinsic and extrinsic geometry, but never so explicitly. I found the flatworm example (in the grand tradition of lower-dimensional analogies) to be very helpful in understanding better how the shape of space affects geometry directly.
Is anyone still searching for a triangle whose angles do not add up to 180?