Pick's Theorem, Lattice Points, and Area
Date: 08/27/98 at 09:53:09 From: Xuan Sen Subject: Lattice point Dear Dr. Math, It is my second day of school, and my teacher told me to define "lattice point." I've searched everywhere for the solution but I could not find the answer. After defining the meaning of lattice point, I have to state how it relates to the area of a triangle, rectangle, and a circle. Please help me, Dr. Math. I'm new in this school, and don't have any books yet. Thanks a lot for your patience.
Date: 08/27/98 at 10:49:39 From: Doctor Barrus Subject: Re: Lattice point Hi, A lattice point is a point on a coordinate system where two grid lines meet. (In other words, in most cases it's just a point that has integers for its coordinates.) So, to relate it to the areas of triangles, rectangles, and circles, you can think about how you find the area of these shapes on a coordinate graph, you can do a research project on how many lattice points are contained in each of these shapes and how this depends on the area, and so on. You can read more about lattice points at David Eppstein's Geometry Junkyard: http://www.ics.uci.edu/~eppstein/junkyard/lattice.html (You might not be able to understand much of it, but you might give it a try.) I hope this helps. Good luck! - Doctor Barrus, The Math Forum Check out our web site! http://mathforum.org/dr.math/
Date: 08/27/98 at 22:09:06 From: Xuan Sen Subject: Re: Lattice point Thank you for your information. I kind of get it. The CRC Encyclopedia of Mathematics pictures the dots. Are all the dots called lattice points? My teacher gave me a hint to look for interior point boundary. What does that mean? Sorry for asking again but you're the only one here. Thanks for your patience.
Date: 08/28/98 at 16:43:02 From: Doctor Peterson Subject: Re: Lattice point It sounds like your teacher gave you a real challenge. Don't think of it as a test to worry about, but something to exercise your mind and have fun with. You're right, lattice points are the dots in a picture like this: * * * 4* * * * * | * * * 3* * * * * | * * * 2* * * * * | * * * 1* * * * * | *---*---*---*---*---*---*---* -3 -2 -1 0| 1 2 3 4 * * * -1* * * * * | * * * -2* * * * * which are the points that have integer coordinates. As for the relation between lattice points and area, it's something I would never have discovered myself, because I'd never have thought of looking for it, but you can figure it out easily enough, once you have been told it's possible. It turns out that if all the vertices of a polygon are at lattice points, you can calculate its area just by knowing the number of lattice points in the interior of the polygon and the number of lattice points on the boundary (the edges and vertices) of the polygon. So what you can do is to draw some simple polygons and make a table of Interior points, Boundary points, and Area, like this: polygon I B A * / | 0 3 1/2 *---* *---* | | 0 4 1 *---* * / \ 0 4 1 *---*---* *---* | \ 0 5 3/2 *---*---* * | \ * * 0 6 2 | \ *---*---* * / \ / * \ 1 4 2 / \ *---*---* and so on. See if you can see the pattern, and maybe even see why it works. (There are several very different ways to prove it.) At the least, you should find that if I and B are the same for two shapes, so is A. That's fascinating enough by itself. But then try comparing I and A for different shapes with the same B, and comparing B and A for shapes with the same I. The formula you should find relating A to I and B is called Pick's Theorem, and you can find it on the web (in fact, you've almost been there), but see if you can figure it out yourself first. It's fun to discover. (By the way, you mentioned circles in your first message. As far as I know, the relation between circles and lattice is points is much more complicated than for polygons, so I don't know whether you will find anything there.) - Doctor Peterson, The Math Forum Check out our web site! http://mathforum.org/dr.math/
Date: 09/05/98 at 00:30:43 From: Xuan Sen Subject: Re: Lattice point To Dr. Math, Thank you for your help.
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