PICK'S THEOREM

How can you find the area of the shapes below?

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Well, we could, but it would be tiring without the Pick's Theorem. Let's find out what the Pick's Theorem is.

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Some History

The theorem was first stated by Georg Alexander Pick, an Austrian mathematician, in 1899. However, it was not popularized until Polish mathematician Hugo Steinhaus published it in 1969, citing Pick. Georg Pick was born in Vienna in 1859 and attended the University of Vienna when he was just 16, publishing his first mathematical paper at only 17 (The History Behind Pick's Theorem). He later traveled to Prague where he became the Dean of Philosophy at the University of Prague. Pick was actually the driving force to the appointment of an up-and-coming mathematician, Albert Einstein, to a chair of mathematical physics at the university in 1911 (O'Connor). Pick himself ultimately published almost 70 papers covering a wide range of topics in math such as linear algebra, integral calculus, and, of course, geometry. His name still frequently comes up in studies of complex differential equations and differential geometry with terms like ‘Pick matrices,’ ‘Pick-Nevanlinna interpolation,’ and the ‘Schwarz-Pick lemma.’ He is, however, most remembered for Pick’s Theorem, which he published in his 1899 paper, “Geometrisches zur Zahlenlehre” (The Geometric Theory of Numbers), in Sitzungber. Lotos, Naturwissen Zeitschrift (Sitzungber. Lotus, Natural Science Journal). Pick retired in 1927 and returned to Vienna, but fled to Prague in 1938 when the Nazis invaded Austria. Tragically, after the Nazis invaded Czechoslovakia in 1939, Pick was sent to Theresienstadt concentration camp in 1942 where he finally passed away at 82 years old. Steinhaus included Pick’s Theorem in his famous book, Kalejdoskop matematyczny (Mathematical Snapshots), published in 1969, at which point the theorem garnered much more attention than it did during Pick’s lifetime.

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The Formula

Pick’s Theorem provides a simple formula for the area of any lattice polygon. A lattice polygon is a simple polygon embedded on a grid, or lattice, whose vertices have integer coordinates, otherwise known as grid or lattice points. Given a lattice polygon P, the formula involves simply adding the number of lattice points on the boundary, b, dividing b by 2, and adding the number of lattice points in the interior of the polygon, i, and subtracting 1 from i. Then the area of P is

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Proof:

Consider a polygon P and a triangle T, with one edge in common with P. Assume Pick's theorem is true for both P and T separately; we want to show that it is also true for 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.

 

 

 

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 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.

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