Pick s theorem pdf files

Picks theorem provides a method to calculate the area of simple. Guess and check if they guess the relationship is linear, then they could notethatonce they get 3 data points they can solve for the variables. You cannot draw an equilateral triangle neatly on graph paper, by placing vertices at grid points. Picks theorem and lattice point geometry 1 lattice. After examining lots of other mathcircle picks theorem explorations, i handed the students the following much simpler version. For example, the red square has a p, i of 4, 0, the grey triangle 3, 1, the green triangle 5, 0 and the blue hexagon 6, 4. Permission is granted to copy, distribute andor modify this document under the terms of the gnu free documentation license, version 1. Prove that all terms of the sequence are divisible. Click on a datetime to view the file as it appeared at that time. Form two 4 digit numbers rabcd and scdab and calculate. Find the area of a p olygon whose v ertices lie on unitary square grid. Picks theorem we consider a grid or \lattice of points.

Next, nd i and b for a latticealigned right triangle with legs m and n. Theorem of the day picks theorem let p be a simple polygon i. Suppose that i lattice points are located in the interior of p and b lattices points lie on the boundary of p. I will use a geoboard to calculate the area of a lattice polygon and compare it with the results of picks theorem. Investigating area using picks theorem teachit maths. In the next subsection we describe these results more fully. If at all possible, please submit assignments as single pdf files. Pick s theorem provides a method to calculate the area of simple polygons whose vertices lie on lattice pointspoints with integer coordinates in the xy plane.

Picky nicky and picks theorem jim wilsons home page. Picks theorem picks theorem gives a simple formula for calculating the area of a lattice polygon, which is a polygon constructed on a grid of evenly spaced points. Despite their different shapes, picks theorem predicts that each will have an area of 4. Media in category picks theorem the following 31 files are in this category, out of 31 total. If we can show that for any such triangle t, ft 0, picks theorem will follow. We will assume our polygons are simple so that edges cannot intersect each other, and there can be no \holes in a polygon. Prove picks theorem for the triangles t of type 2 triangles that only have one horizontal or. The way that the list of theorems is indexed is described here. Beauty, aesthetics, proof, picks theorem, motivation. A formal proof of picks theorem university of cambridge. Lattice polygons and picks theorem chapel hill math circle. Area can be found by counting the lattice points in the inner and boundary of the polygon.

Place a rubber band around several pins to create the figure shown below. Discovering picks theorem n ame nctm illuminations. Picks theorem also implies the following interesting corollaries. Picks theorem provides a method to calculate the area of simple polygons whose vertices lie on lattice pointspoints with integer coordinates in the xy plane. After the preliminary version of this note was disseminated, avi wigderson observed that the proof can be further simpli. The area of a lattice polygon is always an integer or half an integer. Rather than try to do a general proof at the beginning, lets see if we can show that. Rediscovering the patterns in picks theorem national. A simple proof of bazzis theorem university of chicago. If you count all of the points on the boundary or purple line, there are 16. Pick s theorem when the dots on square dotty paper are joined by straight lines the enclosed figures have dots on their perimeter p and often internal i ones as well. Let p be a simple polygon in r2 such that all its vertices have integer coordinates, i. The formula is known as picks theorem and is related to the number.

Picks theorem was first illustrated by georg alexander pick in 1899. Dear picky nicky, i wanted to tell you about this cool activity i did in school this summer. The word simple in simple polygon only means that the polygon has no holes, and that its edges do not intersect. I was assigned to start constructing triangles on a grid. Now, using picks formula, we can calculate the area of the red triangle. Picks theorem tells us that the area of p can be computed solely by counting lattice points. Connecting the dots with picks theorem university of oxford. This theorem is used to find the area of the polygon in terms of square units. A copy of the license is included in the section entitled gnu free documentation license. I know that geometry is your favorite, and i really think you will enjoy this exploration.

Picks theorem based on material found on nctm illuminations webpages adapted by aimee s. Investigating area using picks theorem an investigation to calculate the area of irregular polygons drawn on a lattice using picks theorem download the adaptable word resource. We will discuss picks theorem and minkowskis theorem more after a brief introduc. The formula can be easily understood and used by middle school students. For example, the red square has a p, i of 4, 0, the grey triangle 3, 1, the green triangle 5, 0 and the blue hexagon 6. See, this guy pick thats georg pick, only one e in georg found out that the only thing that matters is the boundary points and the interior points. Pick s theorem in 1899, georg pick found a single, simple formula for calculating the area of many different shapes. Picks theorem provides a simple formula for computing the area of a polygon whose vertices are lattice points. Let a be the area of a lattice polygon, let i be the number of grid.

In 1899 he published an 8 page paper titled \geometrisches zur zahlenlehre geometric results for number theory that contained the theorem he is best known for today. Picks theorem and accompanying video wednesday, april 8th. It is based on three subsidiary results, each of some interest in its own right. An interior lattice point is a point of the lattice that is properly contained in the polygon, and a boundary lattice point is a point of the lattice that lies on the boundary of the polygon. This is the form of picks theorem that holds for any lattice and obvious analogue works in any dimension unlike usual picks formula that has no analogue in 3d even for the cubic lattice. In this series of exercises, you will prove picks theorem using induction. A lattice line segment is a line segment that has 2 distinct lattice points as endpoints, and a lattice polygon is a polygon whose sides are lattice line segmentsthis just means that the. Picks theorem expresses the area of a polygon, all of whose vertices are lattice points in a coordinate plane, in terms of the number of lattice points inside the polygon and the number of lattice points on the sides of the polygon. The area of p is given by, where i number of lattice points in p and b number of lattice points on the boundary of p. By the informal definition, if s is a lattice, the. Explanation and informal proof of picks theorem nctm.

We present the theorem and give a brief inductive proof. Given a polygon with vertices at integer lattice points i. Picks theorem in 1899, georg pick found a single, simple formula for calculating the area of many different shapes. Picks theorem when the dots on square dotty paper are joined by straight lines the enclosed figures have dots on their perimeter p and often internal i ones as well. What are some of the most interesting applications of pick. He is best known for picks theorem which came about in 1899 in an eight page paper. Like in bazzis proof, theorem 1 immediately follows from lemma 2, lemma 6, theorem 7 and lemma 4. The polygons in figure 1 are all simple, but keep in mind. Georg alexander pick this formula allows to find the area s of a polygon with vertices in the knots of a square grid, where v is the number of the grid knots within the polygon and k is the number of the grid knots along its contour, including the polygon vertices. A worksheet to practice picks theorem for calculating areas of 2d shapes. A cute, quick little application of picks theorem is this.

Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. Alternatively, using picks theorem on the green polygon with an interior triangular and exterior pentagonal border and interior points inside the green shaded produces the following calculations. Picks theorem provides a method to calculate the area of simple polygons whose vertices lie on lattice pointsspoints with integer coordinates in the xyplane. A lattice polygon is a polygon all of whose corners or \vertices are at grid points. For example, the lattice polygon in figure 2 has 5 lattice points inside it and 9 lattice points on its sides including the vertices, and so it. Not compelled that a proof is really necessary after all of the evidence supporting picks. At first the theorem wasnt recognized as very important. I wanted to explore picks theorem with our math circle, a group of about 814 middle schoolers mostly 6th graders. However, in 1969 it was included in a book, mathematical snapshots written by a polish mathematician, steinhaus. By question 5, picks theorem holds for r, that is a r f r hence, substituting a r and f r in that last equation, and dividing everything by 2, we get a t f t and picks theorem holds for the triangle t, like we wanted to prove.

Picks theorem gives a way to find the area of a lattice polygon without performing all of these calculations. In this video i will demonstrate that picks theorem of lattice polygons is correct. Planar graphs and eulers formula and accompanying video. Explanation and informal proof of picks theorem date. Pick spent the rest of his career in prague except for one year he spend studying with felix klein in leipzig, germany. Recall that c is the number of lattice points in ps interior. Chandrasekhar, a nobel prize winning physicist, once wrote, a discovery. This is not the order in which the theorem of the day is picked which is more designed to mix up the different areas of mathematics and the level of abstractness or technicality involved.