Pick s Theorem! Finding the area of polygons on a square lattice grid.

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1 Section 3 4A Pick s Theorem! Finding the area of polygons on a square lattice grid. A polygon is a 2 dimensional closed figure whose sides are straight line segments. Each of the sides is connected to it s adjacent sides at a common point called a vertex points. The sides only intersect at their common vertex point. Convex Polygons Every interior angle in a concave polygons have measures less than 80 degrees. Concave Polygons Concave polygons have at least one interior angle that measures more than180 degrees. A square lattice grid. A square lattice grid is composed of rows of dots that form squares. The length of the sides of each square will be 1 unit. The area of each square is 1 square unit. Polygons can be drawn where the vertex points of the polygon are all points on the grid. Pick s Rule is a formula that allows you to find the area of any polygon if the vertex points of the polygon are all points on the square grid. Math A Pick s Theorem! Page 1! 2019 Eitel

2 Pick s Theorem Area of the Polygon = b + i 1 where 2 b = the number of lattice points on the boundary (edges) of the polygon i = the number of lattice points in the interior of the polygon b = 4 red boundary points! b = 8 red boundary points i = 3 blue interior points! i = 5 blue interior points Area of the Polygon = b 2 + i 1 Example 1! Example 2! Example 3 Yellow polygon! Green polygon! Blue polygon b = 5 I = 5! b = 4 I = 4! b = 9 I = 2 Area = Area = 2 1! Area = Area = ! Area = Area = 5 Area = Area = Area = Math A Pick s Theorem! Page 2! 2019 Eitel

3 Drawing Polygons with a Given Area. Draw a polygon that has a area of 5. The polygon cannot contain any right angles and must have more than 4 sides. If the area of the polygon is 5 then we can find an expression in terms of b and i. 5 = b 2 + i 1 6 = b 2 + i 12 = b + 2i Select a value for the number of interior points and then calculate the number of required number of boundary points. Plot the interior points and then try to build a perimeter around those interior points with the required number of boundary points. If 12 = b + 2i and we choose to have 3 interior points then b = 6 We will try to plot 3 interior points and then try to construct a boundary with 6 points around them. b = 6 and i = 3 A = 5 5 sides and no right angles Math A Pick s Theorem! Page 3! 2019 Eitel

4 Who was Georg Pick? Georg Alexander Pick was an Austrian Jew born in He was home-schooled by his father until he was 11. He then entered the Leopoldstaedter Communal Gymnasium (A Gymnasium is like a High School.) He graduated from the Gymnasium in 1875 at 16 years old and enrolled at the University of Vienna (Universit ät Wien). He published his first paper the next year at just17. He earned his Ph.D. in Pick spent the rest of his career at the German University in Prague in Prague except for one year he spend studying with Felix Klein inleipzig, Germany. In 1899 he published an 8 page paper titled Geometrisches zur Zahlenlehre Geometric results for number theory) that contained Pick s Theorem,the theorem he is best known for today. In 1910, Pick was on the committee that considered Albert Einstein s application to join the faculty at the German University in Prague. Pick s support was a strong factor in Einstein s appointment as chair of mathematical physics in Pick was the driving force behind the appointment and Einstein was appointed to a chair of mathematical physics at the German University of Prague in He held this post until 1913 and during these years the two were close friends. Suggestions have been made that he played a direct role in the development of Einstein s general relativity theory as Pick introduced him to some of the essential work in differential geometry of the time. While Einstein was able to emigrate to the United States through a position at Princeton University, Pick could not avoid the Nazis. He was sent to the Theresienstadt concentration camp where we perished on 26 July 1942 at the age of 82. Pick is largely remembered for a beautiful geometric result he discovered in 1899, but was not widely known until it was publicized in the book Mathematical Snapshots by Steinhaus three-quarters of a century later. Math A Pick s Theorem! Page 4! 2019 Eitel

5 Interesting Fact It is impossible to draw an equilateral triangle as a lattice polygon. Proof: Suppose we could draw an equilateral triangle as a lattice polygon with lattice vertices A.B and C, with side length s. By the distance formula and Pythagorean Theorem, s 2 represents the square of distance between any of these 2 vertices, and must be an integer. Also note that the area of an equilateral triangle can be expressed as A = s2 3 4, thus the area must be irrational. On the other hand, the area of of Triangle ABX must be an integer or half an integer by the previous theorem, which means the area must be rational, a contradiction. Therefore, it is impossible to draw an equilateral triangle as a lattice polygon. The Proof Of Pick s Theorem is a bit complicated. It is done in 2 parts. Part 1. Prove that any polynomial can be broken into a finite number of non overlapping triangles that. This means that the area of the polygon can be found by adding the areas of a finite triangles. It is not that hard to show that for a polygon with n vertex the polygon can be divided into n 2 non overlapping triangles. This proof often is done by induction Part 2. Prove Pick s Rule works for any triangle. This part is a bit more complicated. A Youtube video of the proof is included on my web site (or just search Youtube) Math A Pick s Theorem! Page 5! 2019 Eitel