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Mike Shulman (Apr 09 2022 at 08:04):Today someone claimed to me that "geometers" index spheres by the dimension of their ambient Euclidean space, so that for instance the "2-sphere" would be , which I have always called a 1-sphere. I have never heard of such an execrable thing, even when talking to geometers, but apparently wolfram mathworld thinks the same. Has anyone actually encountered such an indexing convention in published work? Maybe it's restricted to some subclass of "geometers"?
David Michael Roberts (Apr 09 2022 at 09:53):Maybe classical geometers? Those who consider a sphere only as a hypersurface with extrinsic geometry, I would guess. Of the references listed on the mathworld page, only a couple of them look like they'd use this weird indexing, one of which is a Dover book from the 1950s.
David Michael Roberts (Apr 09 2022 at 09:55):The other is Coxeter, but I notice that the "supporting" quote from him doesn't actually justify the "n-sphere" terminology.
John Baez (Apr 09 2022 at 22:04):I think the class of geometers who call a circle a "2-sphere" has measure zero: I've never met one.