## Gaming in the 4th Dimension 303

Wolf pointed me to a video clip demonstrating this game:

*Nothing to download yet.**"Miegakure*is a platform game where you explore the fourth dimension to solve puzzles. There is no trick; the game is entirely designed and programmed in 4D."

## So Many Questions (Score:5, Interesting)

In Miegakure, it appears that the player is controlling a fourth dimension except it's not too clear what fourth dimension actually represents to me. If Miegakure's fourth dimension was time, we would see some indication of natural decay of the environment to give us visual cues that it's aging. For example, if one ring were made of steel and the other of wood, the wood one would decay as we go to the future and then we would make some action that is "special" (meaning that it is not subjected to our time control) and then move the steel ring into the wood ring and blast back to when the wood ring existed. Our special action could not be undone otherwise you wouldn't get anywhere with being able to control time.

Miegakure seemed to invent non-natural transposed states of the environment that I, for the life of me, could not understand. How did I know which blocks would appear and disappear leaving only shadows? How do I know how far to go in a fourth dimensional direction? Must the player explore the available transposed states before planning their movements along all four dimensions? So that they can construct an interleaved solution?

And what happens with a now block exists in a shadow space and you try to transposition yourself to the point when the shadow space is occupied by another block? Does the game block you from making that transposition? What if you want to transpose to a point beyond that when it is a shadow space again? Is this a blocking mechanism that will add to the difficulty of the puzzle?

As someone ravaged by the Adventures of Lolo series on the NES, I could see a potentially high level of addiction here.

## Re:So Many Questions (Score:5, Interesting)

Today's XKCD [xkcd.com] might help a bit. It's a world that has four spatial dimensions, like a hypercube. [wikipedia.org]

We haven't been able to find any evidence of "real" higher spatial dimensions (though theories abound), but thinking in an extra dimension is an interesting mental exercise nonetheless.

## Re:So Many Questions (Score:4, Interesting)

Well it depends on its rotation as well. For example a cube entering flatland would either pop up, stay the same, disappear, or dot-grow-shrink, depending on whether you are introducing the cube with one of the sides in parallel with the plane, or whether you to so with a vertice entering first.

## Re:So Many Questions (Score:5, Interesting)

This doesn't seem so much like a "fourth dimension" as a form of "subspace" or an alternate 3D reality (then again I haven't played the game and maybe am picking things up wrong from the video).

I don't see how adding another dimension can magically allow two objects to become linked when they were unable to be linked in a lower dimension. Two circles on a piece of paper cannot physically merge with each other if you assume their boundaries are solid and cannot pass through each other. Neither can 2 rings lain on a table, or two cylinders or two spheres be overlapped without breaking them somewhere. So how would adding another dimension allow you to join two 3D objects with a hole in the middle, even if you only moved one of them into this higher dimension?

## Re:So Many Questions (Score:5, Interesting)

Yeah, it's weird. I'm not entriely clear as to what the shadows represent (except, maybe, for a helpful reminder as to what is "next" to you.)

I think that's the idea. It's hard to tell from the short video, but the blocky nature of the world implies to me that the game limits you to arbitrary "jumps" in each dimension. Just like the world could be divided into fixed-width planes in the X, Y, and Z dimensions, it looks like the W dimension is composed of distinct layers. Which would explain the shadows; they represent what would appear if you jumped to the next adjacent "slice" of 4d-space.

## Re:xkcd (Score:3, Interesting)

## Re:So Many Questions (Score:5, Interesting)

## Re:So Many Questions (Score:4, Interesting)

Hmm... well that would similarly work for a sphere containing another sphere.. but a torus or any other object with a hole is surely a different class of object.. I'm not sure what the 2D representation of a torus would be..?

## abstrusegoose.com (Score:3, Interesting)

He copied it from abstrusegoose.com

http://abstrusegoose.com/88 [abstrusegoose.com] ->

http://abstrusegoose.com/secret-archives/across-the-third-dimension [abstrusegoose.com]

## Another 4d game (Score:3, Interesting)

## Re:So Many Questions (Score:3, Interesting)

I've been thinking about trying to make something like this for so long but I've never been able to work out a sensible way of switching dimensions.

Looks like these guys managed to make a decent game out of it.

I've gotta try this this evening.

Original thought was to try for 6 dimensions which you could rotate through but of course the number of points you need to keep track of going exponential- 4 points for a 2D square/rectangle, 8 points for a 3D cube, 16 for 4D, 32 for 5D, 64 for 6D....

this is an area which could potentially make for some really unusual and head bending games

## Re:So Many Questions (Score:3, Interesting)

A cube entering FlatLand at (45,45,45) degrees of rotation would appear at first as a dot, then grow into a triangle. Then the corners of the triangle would split into line segments which would grow while the original segments shrink until it becomes a triangle again, and eventually a dot and nothing.

Interestingly at no point would it possess four sides, so to a flatlander, it would be very hard to conceptualize that this is a construct made up of squares (a concept they would understand).

## Re:So Many Questions (Score:3, Interesting)

It is very much like a spacial dimension, speaking as a physicist; however, it is also very different.

How is it different? why not just consider it indeed being the same as any other spacial dimension? one in which we have a constant velocity that we currently don't, and maybe never will, know how to change. even if 2 objects in our universe have the same coordinates in 3d space they will still miss each other if their 4th dimension of time is different...ie many cars make it through an intersection because they go through at a different times...when their time is the same is when you have a crash...

that we have a velocity, imparted on us by the big bang, in the time dimension that we don't know how to alter, does not mean it isn't a normal 2 way dimension like the other 3 standard spacial ones. at least to me, thinking of it as being the same as the other dimensions makes for a much simpler model of the universe...remember the earth also used to be the center of the universe back when we didn't know any better.

maybe i am getting off base here, maybe Einstein already mentioned something like this, but my hypothesis is that the speed of light isn't just a speed limit, but also a constant. everything the sum of all velocity for every object in all spacial (plus time) dimensions in the universe is always the speed of light. ie assuming your are sitting still at a computer, we are currently moving at the speed of light in the time dimension. as we increase our speed in the 3 spacial dimensions, time moves slower for us (according to Relativity) thus we are reducing our velocity in the time dimension. so our velocity through time plus our velocity in all spacial dimensions equal some constant strongly related to the speed of light.

## Re:So Many Questions (Score:3, Interesting)

Watching the video, it appears (1) that the objects are all cubes, and (2) that all movement (in all dimensions) is in cube-sized jumps (although these are animated smoothly). This makes me think that the underlying representation is as 4d voxels.

If I'm counting cubes right, then the world shown is about 9x4x6 cube-units in the x,y,z-dimensions, and maybe 4 cube-units in the w-dimension. So you need a 9x4x6x4 voxel grid; that's 864 voxels, and can be represented as a bitfield with just 27 32-bit integers. Not bad at all.

From reading the website, it seems that one can simply choose which three dimensions you see on the screen at any time. In the video, the ring they show lives in a 2d slice of the voxel grid. Then there are two axes which are orthogonal to this slice, and it appears that when in the video they "go into the fourth dimension" what they are doing is switching between which of these two dimensions they are appending to the two the rings lives in to get the three dimensions they display.

I wonder if they enforce that you select dimensions in a way that is orientation-preserving (e.g., you can't go (x,y,z) ---[swap z/w]---> (x,y,w)--->[swap y/z]--->(x,z,w)--->[swap w/y]--->(x,z,y) )?

Anyway, it seems that this "toggling dimensions" thing is not animated in any particularly tricky way. What would be absolutely awesome though is if they, as you suggested in your post, actually animated a continuous rotation between the two. Since each exchange only swaps two coordinates, each rotation is really a 2d rotation, and can be represented as a Givens rotation [wikipedia.org]; all they need to do is interpolate the angle. The one issue though would be computing the intersection of your 4d cubes with your 3d subspace as you do this interpolation, since your intermediate shapes won't be cubes...

## OK, lets try this in 1D: (Score:3, Interesting)

______________________

a "ring" called "A":

__A_____A_____________

and another "ring" called "B":

__A_____A___B_____B__

lift "B" into the second dimension:

_____________B_____B__

__A_____A_____________

slide "B" across:

_____B_____B___________

__A_____A_____________

drop "B" back onto the line:

__A__B__A__B_________

"A" and "B" are now "linked" in the 1D universe.