Four-Dimensional Mazes

Before we discuss the concept of a four-dimensional maze, let's first briefly talk about two-dimensional mazes and three-dimensional mazes.

Two-Dimensional Mazes

A maze, in general, is an intricate (and usually confusing) network of interconnecting pathways. You are probably already familiar with two-dimensional mazes, such as the one shown below:

Each "chamber" (or "cell") in such a two-dimensional maze can have up to four nearest neighbors (north, south, east, and west). And each chamber either connects to that neighbor, or else it is blocked from that neighbor by a wall.

Three-Dimensional Mazes

First, let me say that a three-dimensional maze is not simply a two-dimensional maze whose walls "stick up" (even though some authors erroneously think it is! ):

(This is merely a two-dimensional maze --- not a three-dimensional one.)

A three-dimensional maze is one in which each chamber can have up to six  nearest neighbors (north, south, east, west, up, and down) as is shown in the following incomplete illustration:

The completed illustration would look like a solid block of little "cubies" (to borrow from the "Rubik's Cube" terminology) and, without X-ray eyes, you wouldn't be able to see any of the paths.

A "slab" of cubies sliced from such a three-dimensional maze would resemble the previously shown image of the two-dimensional maze with its walls "sticking up".

But there would not be a complete path through the slab. There would be too many walls that need to be "climbed over" (by going into the third dimension).

Finally, instead of referring to a three-dimensional maze's spatial directions as "east", "north", etc., it is customary to refer to them in mathematical terms: "X", "Y", and "Z".

Four-Dimensional Mazes

Extrapolating our discussion about three-dimensional mazes, we find that a four-dimensional maze is simply one in which each chamber can have up to eight  nearest neighbors (north, south, east, west, up, and down), and two that exist at the exact same spatial location... but in "parallel universes".

(The red chamber's nearest neighbors are shown in dark blue.)

In the above illustration, we have shown the separate universes as sitting "side-by-side" in three-dimensional space. In actuality, they all exist in the same  location (in three-dimensional space) but differ in the fourth dimension.

Just as was the case for a two-dimensional slab of cubies sliced out of a three-dimensional maze, a three-dimensional "slab" of cubies (i.e., a single universe) "sliced" from a four-dimensional maze will generally have too many walls that need to be "climbed over" (by going into the fourth dimension).

Conclusion

If all of this sounds a little confusing, let's try looking at from a slightly different perspective. Imagine that you are inside a structure that looks something like a three-dimensional maze . . . but with too many walls blocking the paths. (There is no path that gets you out; they all lead to dead ends.) But some of the chambers have a little sign posted in them that says "PLUS", and some have a little sign that says "MINUS". (And some chambers may have both  signs, and some may have no signs at all.)

You have in your possession a little remote control box with two buttons, one labeled "PLUS" and the other labeled "MINUS". If you press the "PLUS" button whenever you are in one of the chambers marked "PLUS", some of the walls and passages in the entire three-dimensional structure  change. (Certain walls may now swing open to become passages, and vice versa.)

Presto! You have just moved "up" to the next universe!

Each time you press the "PLUS" button (whenever you are in one of the chambers marked "PLUS"), you advance to the next universe from the one that you are currently in. And each time you press the "MINUS" button (whenever you are in one of the chambers marked "MINUS"), you go back "down" one universe.

And that's just about all you need to know about four-dimensional mazes!