Representing Four Dimensions

Mazes for Programmers — by Jamis Buck (93 / 121)

👈 Displaying a 3D Maze | TOC | Your Turn 👉

You’ve now seen how the pattern comes together. Zero-dimensional cells make one-dimensional rows (or columns), which come together to make two-dimensional levels, which stack to make three-dimensional worlds.

So what happens when you take those three-dimensional worlds and stack them along another axis, perpendicular to the three we already have? (Ignore for the moment that it’s rather difficult to imagine a direction that is simultaneously perpendicular to the three axes we’ve already got.) You can probably guess that we’re going to wind up with a four-dimensional grid, and you’ll be exactly right.

To illustrate stacking in the fourth dimension, we can take those three-dimensional grids and line them up vertically on the page, as shown in the figure.

Figure 7. Representing a 4D Stack

This grid is 2×4×4×4 — two sets of three-dimensional grids (or worlds), each of which is composed of four two-dimensional grids (levels), and each of those two-dimensional grids is a 4×4 grid of cells (rows and columns). Four coordinates, four dimensions. Movement between adjacent three-dimensional grids is via the hither/yon axis, analogous to up/down but in the fourth dimension.

Adjacency works just as it did with the three-dimensional grids, but also connects the hither cells (those that are in the 3D grid above the current one) and yon cells (those that are in the 3D grid below). The following figure shows a 3×3×3×3 grid with a single cell drawn in green, and all of the cells adjacent to it in yellow.

Implementing such a grid is as easy as it was for the third dimension — adding another dimension to the array that holds the cells, adding new neighbor accessors (hither and yon) to the Cell class, and updating the various Grid methods to accommodate this new dimension. Once you’ve made those changes, you can go truly nuts and generate mind-bending four-dimensional mazes, like Figure 8, ​A 4D Maze​.

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