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3d would be chess with multiple layers but you can’t move back in time.
4d would be a version of this without parallel timelines or alternatively you could not move between parralel timelines
5d you can move across timelines.
ehh...
There are five dimensions in consideration here, yes.
Width, usually written as X
Height, usually written as Y
Depth, usually written as Z
Linier Time, which is how we experience things in the real world and in the game is represented by the passage of turns, which I'll lable L
Alternate Timelines, referred to as 'multiversal'. I'll lable this M.
Thing is, anything using or existing on/in any pair of them is 2D. XY, XZ, XL, XM, YZ, YL, YM, ZL, ZM, LM, they're all 2D.
Likewise, any 3 of them is 3D
Any 4 of them is 4D
and any/all 5 is 5D.
And, if you were to add more 'dimensions', the pattern continues.
Thing is, in the game, L is actually quite limited. You always move forward along this dimensional axis one turn per turn, and you can only move Backwards, or fowards at a rate faster than 1 turn per turn, if you Also move on M, and even then, you Still move forwards as well!
Meanwhile, the Z axis isn't even present.
So, depending on what criteria you put in place, the game's either "4D" or "3.xD".
Now, the devs did put forward a reasonable reason for why they labled it 5D rather than 4D in one of the other threads (or someone quoted them in said thread, one or the other)... I've quite forgotten what it Was, but it existed. Something to do with causing confusion either way and this variety being less of a problem, or some such.
(interestingly, the mathematics needed to describe the physics of how a number of things in reality work strongly implies (maybe more than that, but that's what I remember) that there's something like another 6 or so dimensions to reality, as well. Maybe more if you don't count time as a dimension, as apparently not doing so is standard in the relevant fields and I'm not sure on the dates of various things. Can't remember the details on that one though, it's been ages.)