Three Dollar Quill

I'm writing this while waiting for the train. I don't have my usual topics handy, so I'm going to write about something that I don't have anything related to on my laptop anyway.

So, surreal numbers are an extension of the real numbers, and of various other extensions as well. One thing that I find really fascinating about them is that there's a representation of (some) surreal numbers in terms of discrete shapes: by drawing shapes on a square grid, it's possible to represent, not just natural numbers, but negative numbers and some simple fractions.

(One notable thing about this representation is that it actually represents some mathematical objects that don't "act like numbers".)

Now, there's a straightforward extension of the surreal numbers to what's called the "surcomplex" numbers: just define "i" as normal, and carry on. What bothers me about this is that, unless I'm missing something, it loses the geometric representation. Can complex numbers be represented with similar discrete grid figures? I haven't made much progress yet, so my answer thus far is a resounding "maybe".

Here's the background for the approach I've been looking into. The canonical way to represent the natural numbers using sets is to identify each number with a single set: the set of all smaller numbers. The surreal numbers are each identified with an ordered pair of sets. (And rather than containing all simpler numbers, the sets just need to establish a bound from above or below, depending.) This enables representing negative numbers and fractions.

My idea was to think of the number of sets per number as relating to the roots of unity. With one set/root, we have access to the natural numbers. With two, we have the real integers. I therefore regard it as plausible, that with three sets/roots, the Eisenstein integers should be accessible.

Now, I haven't written anything down, but the rules for addition and multiplication of surreal numbers seem to generalize cleanly to these more complicated objects. And the geometric representation should be shapes filled out on a hex grid instead of a square grid.

There are, however, shockingly many problems with this idea, especially given that I've only just started to think about this in earnest. (I was thinking about some of these ideas twenty years ago; I just didn't get anywhere at the time, and put it down until now.)

• Surreal numbers have a procedure for identifying a finite surreal number with a real number. My attempts at a corresponding procedure have broken down on very simple cases.
• Relatedly, the surreal numbers have a procedure for determining whether or not a pair of sets represent a number. I know of some sets that should and shouldn't qualify, but I don't believe I have a general procedure.
• The geometric representation of surreal numbers is interpreted through the medium of a simple two-player game. I haven't figured out what the game should look like, on a hex grid.

I suppose one issue is, while it's straightforward to interpret addition in the context of grids—you just put two shapes next to each other—I'm not aware if there's a nice representation of multiplication.

It's late; I'll try going into more detail later.

Good night.