On 14/05/2016 04:55, Keith F. Lynch wrote:
I'm not sure what you mean by modular arithmetic on Gaussian integers.
a = b mod m iff (a-b) = mk where k is an integer. In this case you'd interpret "integer" as "Gaussian integer".
* Modular in both real and imaginary, meaning that the complex plane is tessellated with rectangles of some integer side lengths, and the rectangles are all identified with each other? Equivalently, there's just one rectangle, and its opposite edges are connected, making a torus?
That's what you get if you work mod m in the Gaussian integers and m is a real integer. The rectangles are always squares. If, e.g., m = 2+3i then the squares will no longer be axis-aligned; one of the squares will have its corners at 0, 2+3i, -3+2i, -1+5i.
I think modular in both real and imaginary sounds most interesting. Especially since if you think of it as a tessellation of rectangles, you can generalize to offset rectangles, like bricks in a wall. And if you think of it as a torus, you can generalize to a Klein bottle.
I think that if you do either of those things you're no longer doing modular arithmetic. (Though you might be doing some other interesting thing.)
Long ago I noticed that modulo a number that's one more than a square, negative one has a real square root.
The precise condition is that all the odd prime factors of the modulus should be 1 mod 4, and that there should be at most one factor of 2 in the modulus. (If p|m and m=n^2+1 then -1 is a square mod p and therefore p equals 2 or is 1 mod 4; and you never have 4 | square+1.)
I never played with Eisenstein integers. Would you use a triangular tessellation for them when going modular?
It might be better to think of the fundamental domain as a rhombus (two of your triangles glued together).
Are those the only two interesting ways of generalizing integers to the complex plane?
No. Let D be any square-free positive integer > 1 and consider numbers of the form p+q.sqrt(d) where p,q are rational; this forms a field; the algebraic integers in the field are either the numbers of the form p+q.sqrt(d) where p,q are integers, or the numbers of the form p+q(1+sqrt(d))/2 where p,q are integers, and these are by any reasonable criteria an "interesting way of generalizing integers to the complex plane". The Gaussian and Eisenstein integers are the nicest, though.
What about the vertices of a regular hexagonal tessellation?
I doubt there's any way to do arithmetic on those. Certainly not the "obvious" way -- they don't form a lattice. -- g