Is Pick's formula true for a general (non-integer-vertices) lattice?

Asked Apr 16 '19 at 10:17

Active Apr 16 '19 at 10:17

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Does Pick's formula hold for non-integer lattices (nodes with non-integer coordinates)?

I heard that it holds for any lattice (given lattice is: we take two groups of parallel lines and intersect them - so we have a lattice based by a fundamental parallelogramm) - nodes of such lattice need not be integer numbers. The formula is the same but if it gives number 7 it means "7 times the area of a fundamental parallelogramm of that lattice"

asked Apr 16 '19 at 10:17

Code Complete

1

Seems reasonable. You could take a proof of Pick for the usual lattice, and see whether it goes through for the general lattice. Earlier related question: https://math.stackexchange.com/questions/768/picks-theorem-on-a-triangular-or-hex-grid – Gerry Myerson Apr 16 '19 at 10:21
I think the general lattice is done in https://arxiv.org/abs/1707.04808 although all the diagrams are for the square lattice. Why not have a look, and report back? – Gerry Myerson Apr 16 '19 at 10:33
Had a look yet? – Gerry Myerson Apr 18 '19 at 00:45
@Gerry Myerson It didn't clarify anything for me. – Code Complete Apr 20 '19 at 15:23
Sorry to hear that. I'll try to have another look, though it won't be for a couple of days. – Gerry Myerson Apr 20 '19 at 22:19

Is Pick's formula true for a general (non-integer-vertices) lattice?

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