Pick's theorem

Given a simple polygon constructed on a grid of equal-distanced points (i.e., points with integer coordinates) such that all the polygon's vertices are grid points, Pick's theorem provides a simple formula for calculating the area "A" of this polygon in terms of the number "i" of "interior points" located in the polygon and the number "b" of "boundary points" placed on the polygon's perimeter::$A\; =\; i\; +\; frac\{b\}\{2\}\; -\; 1.$

In the example shown, we have "i" = 39 and "b" = 14, so the area is A = 39 + 14/2 − 1 = 39 + 7 − 1 = 45 (square units).

Note that the theorem as stated above is only valid for "simple" polygons, i.e., ones that consist of a single piece and do not contain "holes". For more general polygons, the "−1" of the formula has to be replaced with "$-chi(P)$", where $chi(P)$ is the Euler characteristic of "P".

The result was first described by Georg Alexander Pick in 1899. The Reeve tetrahedron shows that there is no analogue of Pick's theorem in three dimensions that expresses the volume of a polytope by counting its interior and boundary points. However, there is a generalization in higher dimensions via Ehrhart polynomials. The formula also generalizes to surfaces of polyhedra.

Proof

Consider a polygon "P" and a triangle "T", with one edge in common with "P". Assume Pick's theorem is true for "P"; we want to show that it is also true to the polygon "PT" obtained by adding "T" to "P". Since "P" and "T" share an edge, all the boundary points along the edge in common are merged to interior points, except for the two endpoints of the edge, which are merged to boundary points. So, calling the number of boundary points in common "c", we have

:$i\_\{PT\}\; =\; (i\_P\; +\; i\_T)\; +\; (c\; -\; 2)$

and

:$b\_\{PT\}\; =\; (b\_P\; +\; b\_T)\; -\; 2(c\; -\; 2)\; -\; 2.$

From the above follows

:$(i\_P\; +\; i\_T)\; =\; i\_\{PT\}\; -\; (c\; -\; 2)$

and

:$(b\_P\; +\; b\_T)\; =\; b\_\{PT\}\; +\; 2(c\; -\; 2)\; +\; 2.$

Since we are assuming the theorem for "P" and for "T" separately,

:

Therefore, if the theorem is true for polygons constructed from "n" triangles, the theorem is also true for polygons constructed from "n" + 1 triangles. For general polytopes, it is well known that they can always be triangulated. That this is true in dimension 2 is an easy fact.To finish the proof by mathematical induction, it remains to show that the theorem is true for triangles. The verification for this case can be done in these short steps:

* observe that the formula holds for any unit square (with vertices having integer coordinates);
* deduce from this that the formula is correct for any rectangle with sides parallel to the axes;
* deduce it, now, for right-angled triangles obtained by cutting such rectangles along a diagonal;
* now any triangle can be turned into a rectangle by attaching (at most three) such right triangles; since the formula is correct for the right triangles and for the rectangle, it also follows for the original triangle.

The last step uses the fact that if the theorem is true for the polygon "PT" and for the triangle "T", then it's also true for "P"; this can be seen by a calculation very much similar to the one shown above.

ee also

* Ehrhart polynomials — the generalization of Pick's theorem in higher dimensions

External links

* [http://www.cut-the-knot.org/ctk/Pick.shtml Pick's Theorem (Java)] at cut-the-knot
* [http://www.mcs.drexel.edu/~crorres/Archimedes/Stomachion/Pick.html Pick's Theorem]
* [http://www.geometer.org/mathcircles/pick.pdf Pick's Theorem proof] by Tom Davis
* [http://demonstrations.wolfram.com/PicksTheorem/ Pick's Theorem] by Ed Pegg, Jr., The Wolfram Demonstrations Project.
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