1
$\begingroup$

I recently derived the most general possible version of Pick's Theorem which works for any shape consisting of the union of any number of polygons (these don't have to be simple polygons either) with any number of holes inside of them (the holes also don't have to be simple polygons). The generalization states that:

$A=I+\frac{B}{2}-\frac{P+P_s}{2}+\frac{H+H_s}{2}$

where $I$ is the total number of interior integer coordinate points of the shape, $B$ is the total number of boundary integer coordinate points of the shape, $P$ is the total number of path-connected domains of finite area only containing boundary and interior points of the shape, $P_s$ is the total number of non-intersecting polygons only containing boundary and interior points of the shape, $H$ is the total number of path-connected domains of finite area only containing boundary and exterior points of the shape, and $H_s$ it the total number of non-intersecting polygons only containing boundary and exterior points of the shape.

If the shape in question is itself a single simple polygon then we have $P=1$, $P_s=1$, $H=0$, $H_s=0$, and the generalization reduces to the standard form of Pick's Theorem:

$A=I+\frac{B}{2}-1$

If the shape in question is the union of $P$ non-intersecting polygons with no holes then the generalization reduces to:

$A=I+\frac{B}{2}-P$

If the shape in question is a single polygon with $H$ holes, all of which are non-intersecting polygons, then the generalization reduces to:

$A=I+\frac{B}{2}-1+H$

My question is this: has this generalization been published anywhere? Is there a simpler version of it somewhere to be found?

$\endgroup$
  • $\begingroup$ Could you rewrite your formula in terms of Euler characteristic? $\endgroup$ – Lord Shark the Unknown May 9 '18 at 3:35
  • $\begingroup$ Yes, somebody had already done so back in 1985 here: faculty.math.illinois.edu/~reznick/496-2-6-17.pdf $\endgroup$ – The Riddler May 9 '18 at 14:20
  • $\begingroup$ The 1985 article ("Pick's Theorem revisited" by Varberg) concludes with the remark: "We have recently learned of an even more general form of Pick's theorem". The referenced article (in German) is "Neuere Studien über Gitterpolygone" by Hadwiger and Wills; an online citation is here. (Sadly, the displayed initial page doesn't include the formula.) $\endgroup$ – Blue May 12 '18 at 18:09

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.