My thesis talks about “shapes” of “number fields,” but really what we’re looking at is shapes of lattices, and using MATH to go from “number fields” to lattices.
So I thought it would be nice to give a quick intro to what it means for lattices to have shapes and for those shapes to live in a space of shapes.
What is a lattice?
Looking at the picture, I see “repeated pattern” and that repetition is easier to see if you have vectors in a vector space.
In other words, a 2-dimensional lattice is “generated” by two (“linearly independent”) vectors (here linear independence means not on the same line).
What is the shape of a lattice?
If you take your generating vectors and add them together in what I’ll call the natural way, you get a fundamental region. In two dimensions, your two vectors give you a parallelogram.
How many different types of parallelograms are there? Square, rectangle, and rhombus come to mind, along with general parallelogram. We will also give a special name to the rhombus made from two equilateral triangles.
 square |u|=|v|, θ=90° |
 rectangle |u|≠|v|, θ=90° |
 parallelogram
|u|≠|v|, θ≠90° |
 rhombus|u|=|v|, θ≠60° or 90° |
 hexagonal|u|=|v|, θ=60° |
Example If I tell you u is the vector from (0,0) to (1,1) and v is the vector from (0,0) to (-1,0), what is the shape of the lattice generated by u and v?
(space for thinking)
(really, you should be drawing)
Unfortunately, drawing the parallelogram u and v determine isn’t actually the full story. You need all the dots!
We have seen that if we start with dots, we can visually see the shape by choosing the most obvious generating (basis) vectors and drawing the parallelogram they determine. We have seen that if we start with a basis, the parallelogram they determine may not obviously give the shape of the lattice they determine. And so far, we have only used words to describe shape.
If we want to study something called “shapes of lattices,” we need to be able to see shape not just as an element of a set of words, but hopefully as a point in space.
Space of shapes
Two lattices have the same shape if they differ only by scaling, rotating, and reflecting. And if you’ve already picked a basis of generating vectors, changing the basis means keeping the same dots but generating them differently, so this does nothing to the shape of the lattice. As we saw above, though, changing basis may change what kind of parallelogram your basis vectors determine. Part of finding a space of shapes, then, will be figuring out how to chose the right basis so that the generating vectors determine the parallelogram that is the same shape as the lattice. Or else, how to account for the fact that our basis might have been the wrong choice.
In two dimensions we can take what we know about which actions don’t change the shape and explicitly use this to determine a “fundamental domain” for these actions on our lattices. For any given basis, u and v, we can scale, rotate and reflect them until we have that one of the generating vectors (let’s say u, and maybe that required a change of basis) is the unit vector on the x-axis. The shape, then, of a two-dimensional lattice can be described in terms of the end point of v.
Space of shapes of 2-d lattices:
With the normalized u, there is only one way to have a square, thus “square” is represented by (0,1). For v ending on the y-axis with y>1, you get all possible non-square rectangles (do you believe that?). The curvy border gets you shapes where |u|=|v|=1 and 60° < θ < 90° which are certainly rhombi. The point at 60° is hexagonal. Everything in the middle has |u| ≠ |v| and θ ≠ 90°, so just parallelogram. Why does v ending on the line x = 1/2 with y > 1/2 give the other possible rhombi? (Draw the dots!)