From: Joerg Arndt <arndt@jjj.de> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] Pick theorems
Pick's theorem (cf. http://en.wikipedia.org/wiki/Pick's_theorem):
The area A of a simple polygon with all corners on a square grid is A = i + b/2 - 1 where i is the number of lattice points in the interior and b is the number of lattice points on the boundary.
For the following A is the area as number of unit cells covered.
Triangular lattice: A = 2*i + b - 2
Hexagonal lattice: A = i/2 + b/4 - 1/2
Square grid, every second column shifted by a half unit: A = i/2 + b/2 - 1 where b counts only the boundary points whose local neighborhood is not concave (i.e., 90 degrees of outside, 270 degrees of inside).
I obtained these scribbling on a train ride yesterday.
Best, jj
--it seems to me that any "Pick theorem" (in any space-dimension D) can be trivially generalized to work for a lattice L which differs from the plain integer lattice Z^D by just linearly transforming. Whatever lattice L you have, it is just an invertible linear transformation of D-space of the plain Z^D lattice. Wlog L is scales so its voronoi cells have unit D-volume in which case the linear transform is volume-preserving. Now note, the polygon (or polytope) counts of lattice points within, on vertices, most geenrally on k-faces (0<=k<=D), all are preserved by the linear transformation. Also, the volume is preserved (as we just postulated wlog). Hence, any Pick formula should generalize to work for any lattice. QED