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Lattice Points and Boundary Lattice Points

Date: 08/30/98 at 23:09:52
From: Bernard Doria
Subject: Lattice Points

What is an interior lattice point and a boundary lattice point of a 
given shape (triangle, circle, rectangle, etc.)?

For example, an isosceles right triangle whose legs are 3 units long. 
The teacher wrote that there are roughly 2 interior lattice points and 
8 boundary lattice points, but the teacher isn't here to explain it. 
A friend of mine tried explaining it to me, but I was still confused. 

Can anyone help me on this?

Date: 08/31/98 at 03:03:20
From: Doctor Pat
Subject: Re: Lattice Points


Lattice points are the points on the regular x,y coordinate plane that 
have an integer value for both x and y. When you draw the shape by 
drawing the edges, if the line contains one of these lattice points, 
it is a boundary lattice point. If a lattice point is totally inside 
the shapes edges, it is an interior lattice point. The number of 
lattice points on the boundary and inside depend on where you put the 
shape and how it is rotated. In some positions it may have more 
boundary points and in others, more interior points.  

If you draw the isosceles right triangle whose legs are 3 units long 
with the right angle at the vertex (0,0) and the two legs along the x 
and y axes, then there would be the following boundary lattice points: 

  (0,0), (0,1), (0,2), (0,3), (1,0), (2,0), (3,0), (2,1), and (1,2) 

The only interior lattice point in this case would be (1,1). Plot 
these and it will help.  

But if you slide the triangle to the left 1/2 a unit and down 1/2 a 
unit, suddenly there are no boundary points and 6 interior points. 
Again, try plotting these. I think your teacher is working up to a 
beautiful mathematical idea called Pick's theorem and a way to find 

I hope this helps. Good luck.

- Doctor Pat, The Math Forum
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Associated Topics:
High School Coordinate Plane Geometry
High School Discrete Mathematics
High School Equations, Graphs, Translations
High School Geometry

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