What, fundamentally, makes Pick’s theorem possible in 2D…
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What, fundamentally, makes Pick’s theorem possible in 2D that breaks down in higher dimensions?
Pick's theorem https://en.wikipedia.org/wiki/Pick's_theorem allows calculating the area of **any** 2D polygon (including nonconvex polygons) whose vertices lie on an integer lattice from only the number of lattice points within it and on its boundary.
This feels like a minor miracle, and indeed there is no equivalent formula for the volume of polytopes in any higher dimension, even when restricted to convex polytopes.
What geometric/topological property of 2D space makes this magic possible that somehow fails in every other dimension?
#science
Pick's theorem https://en.wikipedia.org/wiki/Pick's_theorem allows calculating the area of **any** 2D polygon (including nonconvex polygons) whose vertices lie on an integer lattice from only the number of lattice points within it and on its boundary.
This feels like a minor miracle, and indeed there is no equivalent formula for the volume of polytopes in any higher dimension, even when restricted to convex polytopes.
What geometric/topological property of 2D space makes this magic possible that somehow fails in every other dimension?
#science
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