Comments by "EebstertheGreat" (@EebstertheGreat) on "g-conjecture - Numberphile" video.

  1. All simplicial ("triangulated") spheres in this video are polytopal, but Wikipedia claims that the g-conjecture is only open for non-polytopal simplicial spheres (and was proved for all polytopal simplicial spheres in 1979 as the "g-theorem"). What is a "non-polytopal" simplicial sphere? Wikipedia says that a "simplicial d-sphere" is defined as a "simplicial complex homeomorphic to the d-sphere." It's easy to see that for d≤3, the only simplicial complexes homeomorphic to the sphere are polytopes (specifically, polyhedra for d=2, polygons for d=1, 0-spheres for d=0). But Wikipedia says "most" simplicial spheres in d≥4 are not polytopal. How is that possible? This is very difficult to imagine.
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  2. I don't understand.
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  3. I'm asking a serious question that some people on here are likely to know the answer to.
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