Pick's Theorem

Date: 2/8/96 at 15:19:3
From: JimLindsay
Subject: Pick's theorem

Dear Dr. Math,

     I was wondering what Pick's theorem is? Can you tell me? 
Thanks so much!

                                Sincerely,
                                     Brian Lindsay

Date: 2/8/96 at 16:22:33
From: Doctor Sarah
Subject: Re: Pick's theorem

Hi there -

There are some useful Web pages on Pick's theorem.  I'll include some 
excerpts but you'll want to go look up the images yourself.  
You can find more pages by using Alta Vista 
http://altavista.digital.com/   
to search the Web.

(1)  
http://www.mcs.drexel.edu/~crorres/Archimedes/Stomachion/Pick.html   

"Pick's theorem provides an elegant formula for the area of a simple 
lattice polygon: a lattice polygon whose boundary consists of a sequence 
of connected nonintersecting straight-line segments. The formula is 

   Area = I +B /2 - 1 
     where 
   I = number of interior lattice points
     and 
   B = number of boundary lattice points

For example, the area of simple lattice polygon in the figure is 

       Area = 31 + 15 /2 - 1 = 37.5 

The original result can be found in 

       Georg Pick
       "Geometrisches zur Zahlenlehre"
       Sitzungber. Lotos, Naturwissen Zeitschrift
       Prague, Volume 19 (1899) pages 311-319. 

Recent proofs and extensions of Pick's theorem can be found in 

       W. W. Funkenbusch
       "From Euler's Formula to Pick's Formula using an Edge Theorem"
       The American Mathematical Monthly
       Volume 81 (1974) pages 647-648 

       Dale E. Varberg
       "Pick's Theorem Revisited"
       The American Mathematical Monthly
       Volume 92 (1985) pages 584-587 

       Branko Grunbaum and G. C. Shephard
       "Pick's Theorem"
       The American Mathematical Monthly
       Volume 100 (1993) pages 150-161"

(2)  http://134.121.112.29/fair_95/gym/hm022.html   

"Pick's Theorem states that the area of any simple polygon whose 
vertices are lattice points in the plane is given by this formula:

   A=Vi + 1/2 Vb - 1

where Vi denotes the number of lattice points in the interior of the 
polygon and Vb denotes the number of lattice points which occur in the 
boundary between the vertices. See figure..."

There's more good information on methods in this paper.

Does this help?

-Doctor Sarah,  The Math Forum