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Pick's and Euler's Theorems


Date: 05/06/99 at 06:44:12
From: Charlie
Subject: Pick's theorem

What is Pick's theorem and how can it be linked with Euler's theorem? 
                                                      

Thank you.


Date: 05/06/99 at 10:50:06
From: Doctor Anthony
Subject: Re: Pick's theorem

Hi Charlie,

Pick's Theorem says that the area of a simple lattice polygon P is 
given by  I + B/2 - 1, where I is the number of lattice points in the 
Interior of P and B is the number of lattice points on the Boundary 
of P.

Pick's Theorem is equivalent to Euler's formula and closely connected
to the Farey series; it may be generalized to non-simple polygons and 
to higher dimensions.

To help remember the correct formula, you can check it on easy cases 
(unit square, small rectangles, etc) or, better, you can view how it 
arises from additivity of area. One can view Pick's formula as 
weighting each interior point by 1, and each boundary point by 1/2, 
except that two boundary points are omitted. Now suppose we are 
adjoining two polygons along an edge as in the diagram below. Let's 
check that Pick's formula gives the same result for the union as it 
does for the sum of the parts (and thus it gives an additive formula 
for area, as required).

           1/2           1/2                        1/2   1/2
      ... - @             @ - ...              ... - @     @ - ...
    /         \         /         \          /         \ /         \
             0 @       @ 0                              @ 0 
               |       |                                
   .       1/2 @       @ 1/2       .        .           @ 1         . 
   .           |   +   |           .   =>   .                       . 
   .       1/2 @       @ 1/2       .        .           @ 1         .
               |       |                                
             0 @       @ 0                              @ 0 
    \         /         \         /          \         / \         /
      ... - @             @ - ...              ... - @     @ - ...
           1/2           1/2                        1/2   1/2

The edge endpoints we choose as the two omitted boundary points. The 
inside points on the edge were each weighted 1/2 + 1/2 on the left, 
but are weighted 1 on the right since they become interior. All other 
points stay interior or stay boundary points, so their weight remains 
the same on both sides. So Pick's formula is additive.

- Doctor Anthony, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Discrete Mathematics
High School Geometry
High School Triangles and Other Polygons

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