Pick's theorem 1

Pick's theorem

i = 7, b = 8,A = i + b/2 1 = 10

The triangle with vertices at the lower left, lower right, and upper right points has i= 12 and b = 14, giving by Pick's theorem A = i + b/2 1 = 18; this is confirmed

by the triangle area formula 1/2 base height = 1/2 9 4 = 18.

Given a simple polygon constructed on agrid of equal-distanced points (i.e., pointswith integer coordinates) such that all thepolygon's vertices are grid points, Pick'stheorem provides a simple formula forcalculating the area A of this polygon interms of the number i of lattice points in theinterior located in the polygon and thenumber b of lattice points on the boundaryplaced on the polygon's perimeter:

In the example shown, we have i = 7 interiorpoints and b = 8 boundary points, so thearea is A=7+8/21 =7+41 =10(square units)

Note that the theorem as stated above is onlyvalid for simple polygons, i.e., ones thatconsist of a single piece and do not contain"holes". For a polygon that has h holes, witha boundary in the form of h+1 simpleclosed curves, the slightly more complicatedformula i+b/2+h1 gives the area.

The result was first described by GeorgAlexander Pick in 1899.[1] The Reevetetrahedron shows that there is no analogueof Pick's theorem in three dimensions that expresses the volume of a polytope by counting its interior and boundarypoints. However, there is a generalization in higher dimensions via Ehrhart polynomials. The formula alsogeneralizes to surfaces of polyhedra.

ProofConsider a polygon P and a triangle T, with one edge in common with P. Assume Pick's theorem is true for both Pand T separately; we want to show that it is also true to the polygon PT obtained by adding T to P. Since P and Tshare an edge, all the boundary points along the edge in common are merged to interior points, except for the twoendpoints of the edge, which are merged to boundary points. So, calling the number of boundary points in commonc, we have

and

From the above follows

Pick's theorem 2

and

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 polygonsconstructed from n+1 triangles. For general polytopes, it is well known that they can always be triangulated. Thatthis is true in dimension 2 is an easy fact. To finish the proof by mathematical induction, it remains to show that thetheorem 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 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.

References[1] CiteBank:47270 (http:/ / citebank. org/ node/ 47270)

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

Demonstrations Project. Weisstein, Eric W., " Pick's Theorem (http:/ / mathworld. wolfram. com/ PicksTheorem. html)", MathWorld.

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