Zwiebach on Lattices: What Does He Mean?

[ehrenfest]

Homework Statement

On this page Zwiebach says that "It is a known fact about lattices that the torus C-hat contains I copies of each point on the unit torus".

I am confused about what this means and what this has to do with lattices. Any compact, 2-dimensional region in a plane contains the same number of points as any other compact 2-dimensional region in the plane since you can find a bijection between the two. So what is Zwiebach really saying here?

Homework Equations

The Attempt at a Solution

 

[Jimmy Snyder]

So what is Zwiebach really saying here?

He means that each point in the unit square is mapped to I points in the parallelogram by the indentification x ~ x + 1, y ~ y + 1. (I don't have the book in front of me right now, so I forget exactly what the identification is.)

Look at the unit square in the bottom lefthand corner of the figure, imagine that it is repeated in every square. Consider the intersection points of the oblique lines in the unit square. Now imagine those intersection points also repeated in every square. How many such intersection points will be covered by the parallelogram? Don't forget to identify points that lie on the edges of the parallelogram.