# Geometry in Z/nZ x Z/nZ

1. Oct 19, 2007

### Galileo

I've come upon this situation in physics where I want to 'generalize' (or find some analog) of lines in a the plane R^2, to lines on an NxN grid of points, which would consist of a finite number of points on the grid.
The line should have the property that, if it goes off the boundary on one side, it will continue with the same slope on the opposite side. (Thus the space is like a 'discrete torus').

Im think lots of mathematics has been done on this already, but I don't know what it's called. My first guess was modular geometry, but that didn't look like it. I don't want to waste time reinventing any wheels.

Any mathematician who can help me?

2. Oct 19, 2007

### Hurkyl

Staff Emeritus
Algebraic geometry. Or maybe incidence geometry is enough, depending on your needs.

One of the problems with finite spaces is that there aren't enough "points" For example, in Z/2xZ/2, the functions

f(x) = x
f(x) = x^2

have the same graph.

3. Oct 20, 2007

### Galileo

Thanks.

Well in short, I have some real function defined on these NxN points, which resembles a probability density. I want to 'integrate' (sum) over a line in this space.
In a plane, this is straightforward, but how to define a line on this grid and what the properties are sounds mathematically interesting, but I've never encountered it.
Questions I think are interesting are:
- Given N, how many distinct lines exist?
- Given two distinct points, is there a unique line going through them?

How to characterize these lines and how notions like 'parallel' are carried over to this space may be important. Most things are probably straightforward, but Im hoping there are some really nonintuitive theorems about this that may help me.

4. Nov 9, 2007

### Galileo

It turns out the finite affine plane was what I was looking for.
The axioms are:
(1) Given 2 distinct points there's a unique going through them
(2) Given a line and a point not on the line, there's a unique line parallel to the first
(3) There are 3 noncollinear points
(4) There are a finite number of points

And you can prove that there is a number n such that:
There are n(n+1) lines
Every points has n+1 lines going through it
Every line has n points
There are n^2 points

I want to work the other way around. Im given n, so to say. So I'm interested in knowing for what n does a finite affine plane exist?
If n is a prime power, then you have the unique field of n elements F_n and F_n x F_n will work (like a 2d-vector space).
But what if n is not a prime power? Is there anything known about it?

5. Nov 9, 2007

### Hurkyl

Staff Emeritus
I suspect that you can prove that any finite affine plane is isomorphic to F^2 for some field F, and therefore you have described the entire collection.

If my claim is true, the proof would probably look very much like the proof that the Euclidean plane is isomorphic to R^2. I can give a try at the proof if you don't want to work on it first. (Or try to look it up)

6. Nov 10, 2007

### Galileo

I'd be happy if you want to try it, Hurk. But before you may go on a wild goose chase;
Im not sure about this, but from bits and pieces I've gathered it seems that this is still a very big unsolved problem in mathematics. And I've a hunch it's very deeply related to the problem of finding N+1 mutually unbiased bases in an N-dimensional Hilbert space and finding N-1 orthogonal latin squares of order N. (Very likely it's also related to finding for what order N projective planes exist, but I know next to nothing about those.)

Not only is the existence known when is N a prime power. It is largely unknown when N isn't a prime power. I will try to find more on it and see what the connections are.

I appreciate the input.

7. Nov 10, 2007

### Hurkyl

Staff Emeritus
I find that plausible too -- I remember that coordinates make sense for any affine plane, but not if we can assume the # of points on a line is a prime power.

It's certainly related to projective planes; if you remove a line (and the points on it) from a projective plane, you are left with an affine plane. Conversely, any affine plane can be extended to a projective plane by adjoining a point of intersection for each class of parallel lines, and the line at infinity that passes through all of them them. (And these operations are inverses)