Can Number Theory be visualized? So I was thinking about a hard euclidean geometry problem, when it hit me just how much more difficult it would become without the aid of a diagram. This got me thinking: Wouldn't it be great if we could somehow find corresponding diagrams for something like number theory? It doesn't have to be euclidean geometry diagrams like the Greeks did with Algebra (which actually made it harder than without diagrams, as we all know), but instead we need to find a natural diagrammatical representation. Something like Ferrers diagram seems to be a tiny step in that direction, and I can imagine similar things being done for all of number theory.
Is this idea at all feasible? Please explain why.
EDIT: I will leave the above to make answers more interesting. However, a related question (as suggested by Omnomnomnom) that is perhaps more useful is the following: What kinds of diagrams already exist in number theory?
Eagerly awaiting any responses!
 A: Clifford Algebra, a.k.a. Geometric Algebra, is a most extraordinary synergistic confluence of a diverse range of specialized mathematical fields, each with its own methods and formalisms, all of which find a single unified formalism under Clifford Algebra. It is a unifying language for mathematics, and a revealing language for physics.
Clifford Algebra: A Visual Introduction
A: Not an answer, but perhaps a good contribution to the discussion. This is from page 261 of Siobhan Robert's brilliant Genius at Play, a biography of John H. Conway. She's quoting Conway:

When we were first working on the ATLAS [of finite groups], we didn't
  quite appreciate it. So you won't. I think it's best to get away from
  explaining things with numbers. I use numbers reluctantly. It's the
  only way I can work out the beautiful things about these groups. I
  would do something else - draw pictures if I could - but I can't draw
  beautifully symmetric things in 7-dimensional spaces, ... For me,
  numbers are a substitute for touch, feel, sight, everything else. With
  high dimensional space I can't touch it, can't feel it, can't see it.
  I can calculate it, but the calculation isn't the point. The numbers
  are a set of instructions. A set of instructions isn't beautiful, but
  that's what the numbers are, a set of instructions, point by point.

https://en.wikipedia.org/wiki/ATLAS_of_Finite_Groups
http://www.amazon.com/Atlas-Finite-Groups-Subgroups-Characters/dp/0198531990
http://www.amazon.com/Genius-At-Play-Curious-Horton/dp/1620405938
A: There is an area called arithmetic geoemetry that exploits links between arithmetic and algebro-geometric questions. 
For example, Fermat's famous equations $X^n + Y^n = Z^n$ can be thought of as a curve in projective space, called Fermat curves, and one can use geometric tools to study it. 
The affine part, so $X^n + Y^n = 1$ is somewhere between a circle and a square; for small $n$ close to a circle (well for $n=2$ it is of course a circle, but this is not relevant for FLT) an for large $n$ it approaches a square-like form. 
