Surcomplex Numbers 2026-08-16

By Max Woerner Chase

Okay, I'm back on surcomplex numbers for now, and I have some thoughts. Principally, "x - x ≠ 0" should not be regarded as some weird curiosity, but as a demonstration that I defined equality wrong. This means that 0 = * - * = * + (-*) = * + (* + *) = * + * + *. * corresponds to "second player loses", and -* to "third player loses".

(I think I ran into some issues with the whole "optional second move" idea, but I forget what.)

Now, the main thing this gets me is a means of rejecting equality definitions. As of now, I don't have an idea of a way to build up a sensible definition from this constraint.

It's aggravating, because my attempts to comprehend the "game" interpretation of these concepts introduce factorial and exponential scaling, but in imagining simple games, I've come up with a tiny spread of concepts:

Wait... perhaps I can think of "3-nimbers" as generalizing rotations in some way, while "oriented 2-nimbers" generalize reflections, or perhaps three-dimensional rotations...

ABCD -> BADC -> CDAB...

The half-turn symmetries of a cube generate a group with the same structure as the basic addition I've managed of oriented 2-nimbers. This is, of course, the Klein four-group, so it could be corresponding to a lot of things.

To see what else to expect, I need to work out the consequences of stuff like a red/amber edge on top of an amber/blue root. This arrangement clearly maintains the nimber-nature of its components, because the first player will never lose. I should note here that the simple games producing oriented 2-nimbers actually result in games fuzzy with a non-zero number; this number has to be subtracted out to observe the pure nimber-nature. But trying to isolate the number component for observation is a non-starter until I can figure out how to handle the strategic complexity of "two two-segment Hackenbush games added together".

At this point, it feels like my only hope is to exhaustively enumerate game graphs with no regard for turn order.

For now, though, I should get ready for bed.

Good night.