[sp332]

Any discrete set of points can be losslessly represented in any positive number of dimensions. The 4d version has 16 spaces, so you could just have them all in a line for a 1d representation of the same game. There are lots of ways to lay them out in 2d, or 3d, or 4d as well. So I would say that you certainly have developed an intuition for four dimensions! You're not cheating or anything just by rearranging the points into 2d.

(Continuous shapes, like solid hypercubes, are tricker ;)

[tiger10guy]

That makes sense.

I think my intuitions for the discrete and continuous are similar for 2D and 3D. Are they fundamentally similar? How do they differ, even if only a little? It seems that discrete 4D intuition should somehow help with continuous 4D intuition.

[sp332]

It's all about how the points are connected to each other. In 1d obviously you can only have 2 connections per node - one to the left and one to the right. In 2d you have a lot more flexibility. You can curve lines around to make much more connected graphs.

In 2d there are still restrictions though, mainly the "3 utilities problem" https://en.wikipedia.org/wiki/3_utilities_problem But in 3d, you don't have restrictions on connections anymore. You can always weave a new connection around the existing ones.

In a square, each corner is connected to 2 neighbors. In a cube, each corner has 3 connections. In a 4d hypercube, the corners have 4 connections. So to make a "4d" hypercube in 3d, just draw 4 connections between corners :) https://en.wikipedia.org/wiki/File:Hypercube.svg Some supercomputers have nodes connected in a hyper-torus configuration. It doesn't matter how you arrange the nodes physically, they just run cables around to recreate the interconnections that would exist in a 4d torus.

For 4d, it's easy enough to do the corners. Each one has 4 connections - one each for up/down dimension, left/right, forward/back, and... +W/-W direction. You can see this in the game. Each corner can move two directions in its own square, or to two other squares, total of 4 directions. If you want to do more than just corners though, it gets more complex. Think of a 3d cube http://joppi.github.io/2048-3D/?utm_source=hn A piece on the edge can move to two corners, or the center of two faces. Actually, it's easier to think of it as lots of little cubes, stuck together so you only have corners :) A 3x3x3 cube is really 8 2x2x2 cubes that share some nodes. So the block in the very center has lots of options. Up, down, left, right, forward, or backward - 6 options. That's twice as many as the corners! That's because the corners are as far as you can go in one direction, so they can only move back the other way. Pieces in the middle can move either way - exactly twice as many options.

Let's make an even bigger hypercube, 3x3x3x3. That's 64 little 2x2x2x2 cubes stacked in 4d. That's 8 3x3x3 cubes that make up the "faces" of the hypercube, and each cube has 8 2x2x2 smaller cubes in it. But remember most of those nodes are shared by more than one cube/hypercube. So the corners of our hypercube can only move to 4 other spaces. A piece in an edge can move to either corner, or into one of the 3 faces. (Look again if you can't imagine all 3 faces https://en.wikipedia.org/wiki/File:Hypercube.svg) So that's 5 options for a piece on the edge. The middle of a face can move (say) up and down and left and right, as well as into the middle of either cube that it's on. For a space on a face, that image is actually awful. Remember, that outer cube is a whole other cube, it's overlapping all the other cubes in that picture. Try this page http://eusebeia.dyndns.org/4d/8-cell.html Anyway, a piece in the middle of a face can move (say) up, down, left, right, or into the center of either of the two cubes that it's on - one in the forward direction, the other in the +W direction. Try visualizing this for several different faces until your intuition kicks in :) Finally, the piece in the very center can move either direction in each dimension, for a total of 8 directions, exactly twice as many as the corners. It will always end up in the center of a face, because those are the only pieces it's connected to.