[HighTek]

Does anyone know a way of computing the number of integer lattice points in a polytope (more specifically in simplices) in 3-space?

One reference we were given was Buraldi's Introductory Combinatorics but the library we have access to does not have the text.

Any kind of help or direction would be wildly helpful.

[HighTek]

Ok, there are two "algorithms" to count the number of lattice points in a polytope. One is for the convex polytope and the other is for a polytope with a fixed dimension.

Neither are definitive. "...there exists no known procedure for counting lattice points at this time." - 2006.

[HighTek]

Upon further inquiry, I need to amend my previous post. The "..no known procedure for counting lattice points..." refers to the GENERAL polytope.

There are numerous methods to find the number of lattice points for special simplices. Our simplex is in the form conv(0, l*e, l*e2, l*e3) in R^3 where l is any integer. Its basically derived from Pick's Theorem and Ehrhart's Polynomial. Its a nice little little equation.

[jakobia]

The paper "The many aspects of counting lattice points in polytopes" provides a survey on this topic.
http://www.math.ucdavis.edu/~deloera/RECENT_WORK/semesterberichte.pdf
You may also look up the book "Computing the continuous discretely", by Matthias Beck and Sinai Robins.
http://math.sfsu.edu/beck/ccd.html

If you are more interested in effective lattice point counting in rational convex polytopes: 
http://www.math.ucdavis.edu/~latte/

[HighTek]

Niiiiiiice. Thanks!!!!