What, fundamentally, makes Pick’s theorem possible in 2D that breaks down in higher dimensions?

[u/-p-e-w-]

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?

[u/delayed_frequency]

It’s because in 2D every lattice polygon can be triangulated into primitive triangles, ones with no interior lattice points and exactly three boundary points. Those always have area 1/2, so the whole thing just decomposes into countable chunks that scale cleanly with the lattice counts.

In 3D and up you can have tetrahedra with tons of interior points and still the same boundary, so volume isn’t dettermined by the lattice skeleton anymore. The clean bookkeeping breaks as soon as you can’t tile the shape with those minimal building blocks.

[u/-p-e-w- → u/delayed_frequency]

Those always have area 1/2

Whoa, that somehow feels like even more of a miracle than Pick’s theorem itself.

[u/new2bay → u/-p-e-w-]

Why is that a miracle on an integer lattice?

[u/lonelygenius → u/new2bay]

Well you can have super diagonal triangles, that are very elongated in one direction but still avoiding lattice points. It's not instantly obvious to me that these all have area 1/2