Division of Space

[Keiji]

After playing VVVVVV, I started wondering whether a 3D equivalent would be worthwhile.

I soon shot that idea in the head with the note that one of the reasons 2D games work so well is because of the "division symmetry" in 2D worlds. The frontal dimension takes on an opposing role to the lateral dimension, and the vertical dimension is a vector with (possibly zero-valued) components in each. Both tunnels and walls are fundamentally one-dimensional (the other dimension is their "thickness"), and each surround the other.

In 3D, you can either have a one-dimensional tunnel surrounded by two-dimensional polar mass, or a one-dimensional wall surrounded by two-dimensional polar space. These two possibilities are dual to each other, and lead to two different types of games. The first case gives you a game where your movement is limited to a network of tunnels, which can give the player a claustrophobic feeling which is not felt from playing 2D games. The second case gives you a wide-open game, more akin to the real world, where the player is theoretically free to explore a vast area, and while this can certainly look very nice, it often makes players unsure where they are supposed to go next and makes developers' work much harder since they need to either spend a lot of time map-making for very little gain, or introduce invisible walls (or Insurmountable Waist-Height Fences) to unrealistically limit the player's movement. So both leave room for improvement.

In 4D, as well as extrapolating the cases of one-dimensional tunnel/wall surrounded by three-dimensional polar mass/space, there should theoretically be a new option, with two dimensions each, which should exhibit the same type of "symmetry" as 2D worlds. Both tunnels and walls would be two-dimensional polar, each surrounding the other. If you want to imagine it in 3D, first imagine a 2D polar space (as your tunnel/wall), add some 3D thickness to it, discard one of the first two dimensions to make it appear as an infinite cylinder, and then the mass/space occupies the remaining two dimensions around the cylinder. I have briefly touched on this note before.

If we were to make a 4D "Metroidvania" type game (which we could actually play on our 2D screens, and without making it something as simple as the various hypercube-cell maze games that've been popping up around here for a while now), I suspect the easiest way to do this would be to discard two dimensions at the interface (i.e. input control interpretation, and output video rendering), choosing which dimensions to discard based either on game scripting, user selection or a combination of both. For rendering, the discarded dimensions could factor into brightness/color and parallax/stereoscopic information. To help see both extra dimensions at once, the renderer could apply a rotation in the ZW plane (assuming WLOG that the screen is the XY plane) prior to discarding, based on some superfluous information such as the position of the player modulo some factor, or a periodic time counter, etc.

I imagine that in such a 4D game, duocylinders and toric surfaces would make natural appearances all over the place and we would probably have to come up with some sci-fi explanation for 2D polar gravity and oddly-shaped planets...

Any thoughts? Would this be at all worthwhile, or would it be better to stick with 2.5D with partially controllable (and partially scripted) 3D rotations?

[quickfur]

Did you mean 3.5D? I would say that 3.5D would be closer to how 4D beings would perceive their space, though 2+2D does give you very interesting effects. E.g., the ridge of the duocylinder would give something akin to 2D asteroids, where leaving one edge of the screen brings you back to the opposite edge. But other surfaces are possible, including RPL, and other such weird stuff.

But we need not be restricted to locally-Euclidean manifolds. We could have arbitrarily complex configurations of intersecting 2D planes, or even arbitrary intersecting 2-manifolds, which in general only intersect in a point, and have those points be "portals" or "aleph points" in which you can rotate your perspective to a different set of axes.

[Keiji]

Did you mean 3.5D? I would say that 3.5D would be closer to how 4D beings would perceive their space

No, I didn't. I think you might have misunderstood my whole post. The point I was trying to make is that (I hypothesize) there is a "symmetry" common to even dimensions such as 2D and 4D that does not occur in odd dimensions such as 3D. 2.5D exhibits the same symmetry as 2D because for all intents and purposes it is 2D, it's just rendered as if it were 3D.

though 2+2D does give you very interesting effects. E.g., the ridge of the duocylinder would give something akin to 2D asteroids, where leaving one edge of the screen brings you back to the opposite edge. But other surfaces are possible, including RPL, and other such weird stuff.

That's not what I was attempting to describe. You refer to various manifolds which have a net space of 2D. The space I am describing has a net space of 4D. It is split into 2+2 by requiring both tunnels and walls to have 2 major dimensions, just like 2D is split into 1+1 by requiring tunnels and walls to have 1 major dimension. This can't be achieved with odd dimensions, since you can't have half a dimension as fractal dimensions aren't involved here.

But we need not be restricted to locally-Euclidean manifolds. We could have arbitrarily complex configurations of intersecting 2D planes, or even arbitrary intersecting 2-manifolds, which in general only intersect in a point, and have those points be "portals" or "aleph points" in which you can rotate your perspective to a different set of axes.

This is no better than 2.5D. Hence why I said it may just be better to stick with that.

[quickfur]

Ahhh I get it.

So you're proposing to build a 4D maze out of 2D corridors and 2D walls. Of course, the 2D walls will have 3D and 4D thickness, but they are not extended except in 2 of the dimensions, ditto for the corridors.

I thought about this briefly, and I can see two cases:

1) because a 2D corridor necessarily assumes two walls on either side, you will have sections of the maze which is essentially the extrusion of a 2D maze.

2) however, this is not always the case, since at junctions you can change the orientation, so you can have two extruded 2D mazes that meet at a junction at 90°. A single corridor can therefore interface with many layers of 2D extrusions along its edge (all of them parallel to each other, and at 90° to the first corridor). Other more complex combinations are also possible.

I suspect, but I'm not 100% sure, that every such maze can be decomposed into N extruded 2D mazes of type (1) connected by complex junctions of type (2).

[Keiji]

Yep, I think you've got it now



Above is a possible segment of a 2D maze. You have segments that are just parallel tunnels and walls, which occur at the red lines. You also have the junction segments, such as the interior of the green rectangle.

I imagine that your cases are analogous, but you seem to be saying that you'd extrude a 2D maze itself to make the equivalent of a red line segment (which seems a little odd), and I'm not sure what you mean by a single corridor "interfacing" with "layers" of 2D extrusions.

[quickfur]

[...] I imagine that your cases are analogous, but you seem to be saying that you'd extrude a 2D maze itself to make the equivalent of a red line segment (which seems a little odd), and I'm not sure what you mean by a single corridor "interfacing" with "layers" of 2D extrusions.

The basic observation is that the tunnels extend in 2 dimensions, and so do the walls. So we may approximate that by saying walls and passages are both like squares (or rather, 4-cubes flattened in 2 dimensions). Now given any particular square passage, it obviously is bounded by two parallel walls, and on the other side of either wall must, in general, be another corridor parallel to the first one. Of course, the walls may have junctions on the other side, so this isn't always true. But think about how a bunch of parallel alternating squares can be made to tile 4-space. Eventually you will end up with a lot of parallel squares, i.e., parallel passages.

If you now look orthogonally at these parallel passages, they are approximately extrusions of 2D mazes. The entire 4D maze then consists of pieces of extruded 2D mazes fitted together at non-parallel angles, roughly speaking.