I'm working (sort of) desperately on this string theory homework that's due tomorrow; one of the problems involves d-dimensional cylinders and cones, and I just realized that I don't know whether a d-dimensional cylinder is a (d - 1)-dimensional sphere extruded along the dth dimension or a (d - 1)-dimensional hyperplane with its dth dimension curled up into a circle. Or any interpolation between these possibilities. Likewise for the cone. (I hope that's clearly worded, since I can't do an illustration.)
They really ought to call the normal cylinder the (1,1)-cylinder rather than the 2-cylinder.