Some structures like the donut are not homeomorphic to a sphere. According to this link (https://en.wikipedia.org/wiki/Angular_defect#Positive_defects_on_non-convex_figures) the basis of the Descartes' theorem seems to be a polyhedron that is homeomorphic to a sphere. Why can't a non-homeomorphic structure like a ring shaped polyhedron(like an angled donut) satisfy Descartes' theorem?
2026-03-25 07:41:36.1774424496
Non-homeomorphic structures and the Descartes' theorem
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Read the second paragraph of that link, which explains how Descartes' theorem may be generalized.
In the case of the angled donut, the Euler characteristic is equal to zero, and so in this case the generalized theorem implies that angle defect is equal to zero.