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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.
    
Associated Topics:
High School Coordinate Plane Geometry
High School Definitions
High School Discrete Mathematics
High School Geometry

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