If, instead of defining the number of dimensions according to the degrees of freedom in movement, we instead define them according to the visual appearance of the space in which the game is played, then Wolf 3D was 3D, this game is 1D, and a game with more than 1 pixel would be 2D.
And Doom was 2.5D - you could be at different heights, but could never be at the same x-y at two different z.
The reason is that R^n has the same cardinality as R. You could create a bijection between the set of 2-dimensional real coordinates to the set of 1-dimensional reals by using every even positioned digit to represent the X value and odd positioned digits for the Y.
The issue with this is that the intuitive metric you were using in the space breaks down and you need a totally unintuitive one to treat this 1-dimensional version as a metric space. It wouldn't be fun to play a game in.
You're missing the fact that several different definitions of dimension exist. If we're talking about topological dimension, then yes, the fact that you need exactly one parameter to define a point is what makes it one-dimensional.
It is easy to view the board as 64 consecutive spaces (left-to-right through the ranks, then snaking through the ranks, and then a circular representation. The circle was cool because the moves became rotations (a rook can stay in its octant or move an exact multiple of 45 degrees...). I did some 2D transformations with a one space skew on each rank (this made the bishop act simultaneously like a rook and knight which was interesting).
So it is easy to preserve the rules of chess and create a different visual representation of the board. Of course, every one I tried just made the game more difficult to understand. I wonder if there is a transform that could aid in understanding chess.
If you want me to describe the curve itself, you are essentially asking for an embedding - there are infinitely many of those, and they will depend on the dimension and coordinate system of the superspace.
Of course it's impossible to explicitly describe an embedding without reference to the superspace, but that only goes for one level - you can ignore said superspace when describing the interior of the subspace.
I play the game very similarly to you. Have you tried the self—imposed permadeath route? I did one play that way—didn't end up beating it, of course—and it was a hell of an experience.
So I don't like CoD and other linear shooters which present you with a series of scripted shooting galleries and a quota of targets to hit before allowing progress (it's utterly mindless); nor do I like (any more) the Quake / Doom style of horror "surprises", where long straight corridors and shadowy alcoves are near certain to contain unpleasant ambushes (it almost punishes foresight); and nor do I like Half-Life 2 and its episodes, which do not reward exploration or observation - every possible "bad outcome" is blatantly and repeatedly telegraphed, and almost every interesting alcove turns out to actually be the way forward because the obvious way forward turns out to be blocked: HL2+ are over-designed.
With more and more games focused towards consoles, lacking first person (I would have enjoyed GTA4 more if it were first person outside vehicles), and with "crouch in cover mode" features activated by dedicated controls (I thought ducking in and out of cover was part of the challenge?), I sometimes fear the depth I seek is getting harder to find. But I'm not sure. Games like the Thief series, Deus Ex, Far Cry 1 & 2, and modern RPGs like Oblivion and Fallout, are probably actually coming out with roughly the same regularity as always, just that memory compresses them.
Just for the sake of argument, don't 3D games also include the 4th dimension? Surely a marketing guy from a games must have tried "NEW 4D ACTION YEAH!"-sloganed advertising by now?
also, check out Braid (braidgame.com) for a fun puzzle game that requires you to consider time travel.
Even that price estimate turned out to be optimistic.
It's a great concept for a joke, but It'd be even cooler if it were playable, more than just move-right, shoot stuff ;)
edit: if so, I wonder what the Jackson Pollock FPS of our current era would be. Or Mondrian Tournament?
BTW, Castle Wolfenstein's scenery is based on Wewelsburg  which housed a SS 'school' before WW2.
The latter was a game almost 30 years ago on Apple ][ (and other 8 bit machines?).
By "original wolfenstein" I understood "wolf3d" (as opposed to the new wolf1d).
A buddy in college updated the old Trek73 game with a 3d version. It was colossally confusing to play. In response I wrote Trek1D which was played very like this game. My universe had stars, which had to be gotten around by firing up the warp engines and entering the 2nd dimension!
EDIT: also isn't it 2d, seeing as it has to include the dimension of time?
EDIT : I would say the number of dimensions is roughly the number of coordinates you need to store the position of a sprit in the game. (In this sens, Wolf3D is not real 3D since the third dimension (up-down) is just "emulated".)
There is no movement or freedom along the axis you're talking about.
(Sorry just got up from studying signal processing, couldn't help but point that out).
Anyho, a little sense of humor would not have killed you you know.