I'm not sure this works for me; I know what he means, but it's hard to view a rotationally-invariant object with positive curvature as being spikey. The analog

I'm not sure this works for me; I know what he means, but it's hard to view a rotationally-invariant object with positive curvature as being spikey. The analogy captures the 'what happens when you cut off a slice' property at the expense of being a good analogy for other properties. Best not to rely too heavily on analogies IMO. In this case the point of the exercise is to demonstrate that one popular analogy (that of hyperspheres to the spheres and circles we can visualise) is flawed in some important respects. Replacing it with another analogy might not be the best idea, unless you're careful to remember the caveats attached to it.