How Does a Tesseract Fold?

When folding the cubes of a tesseract, how does the 3D net end up as a 4D tesseract? Considering the rotations we know more carefully might help us understand.

Here is a depiction of a tesseract and its net. But how does a net become a tesseract?

Simplifying rotations

Folding the tesseract involves 4D rotations, and four-dimensional space is fairly abstract to us. But there are lots of abstract things that we eventually learn to understand. Perhaps the problem isn't that thinking four-dimensionally is too hard, it's that thinking about rotations is too easy.

When we're young, we get taught about adding and multiplying numbers, but we don't get taught about rotating things, because we all know that already. You see babies learning about rotations, turning objects around and examining the results, but since we have an intuitive understanding of 3D rotations, we might never think about it in detail, and it's easily possible to study mathematics for years without getting much understanding of this fundamental concept.

The tesseract is part of a family of shapes, including the square and the cube, that all have edges the same length, and their angles are all right angles. Having all right angles means we can portray them along each of the axes. And it means we can simplify our study of rotations for now to only 90° rotations between the axes.

Folding a square

So let's first think about folding up a square, since it's the only thing we can view perfectly on a two-dimensional screen. The net of a square is just four line segments. All of these are on the x axis for now. We'll distinguish between the x and y axes by their colours, which will be helpful later.

Another thing that's going to be helpful for later is if we try to think about these lines not so much as being on the x axis, and more as being perpendicular to the y axis.

If we rotate the one on the right, you can see that it changes from the x axis to the y axis. Every 90° rotation switches things between the x and y axes, so that the full shape can go from being entirely on the x axis to having half of its sides on the x axis and half of its sides on the y axis.

Portraying a cube

So now let's move up to a cube. Since we're now interested in a three-dimensional shape, but the screen is only two-dimensional, we have a few choices for how to portray it.

We can draw things in an isometric view, drawing the three axes each 120° apart. Or we can draw a perspective view, where the x and y axes stay perpendicular, and things get smaller as they move back along the z axis.

Notice that in the isometric view, all the squares look distorted, but at least they all look identical. In the perspective view, the front and back squares have no distortion, but the other four squares all look like trapezoids.

But in either view, we can hopefully pick out three pairs of squares, perpendicular to each axis. If a square is perpendicular to the z axis, its edges should be along the x and y axes. We should see a pair of squares like this in both views. Similarly, we should see a pair of squares perpendicular to the y axis, and a pair of squares perpendicular to the x axis.

Folding a cube

Now let's examine what happens if these squares all start off in the xy plane, or perpendicular to the z axis. We'll just focus on three squares here, forming one corner of the cube.

Let's fold the square on the right. The edge joining this square to its neighbour is along the y axis, so that's the axis we'll be folding around. So the y axis won't change in this rotation. The x and z axes, though, are going to change places, like they did in the two-dimensional case. The square's edges that were facing in the y direction kept facing the same way, but those along the x axis now point along the z axis.

Now when the edge being rotated around lies along the x axis, as it does for the top square, we see instead that any edges parallel to the x axis will stay that way, whereas any edges that were parallel to the y axis will now be rotating to be parallel to the z axis. In this case, one edge will be rotating toward the same position that a previous edge already rotated to, and edges will join up as it folds.

Before we move into a four-dimensional case, it's worth taking a moment to consider what we see on the screen, because we're so used to the three-dimensional world that our brains almost make sense of it for us.

Imagine you're only used to seeing two-dimensional images, and that you were trying to recognise what's going on here. Only being used to two-dimensional images, you would have seen four squares meet up at a corner before, but three squares meeting at a corner, like they do in a cube, might seem unreasonable. In the isometric image, the rotation looks like a rhombus just distorting a bit until it ends up as a differently oriented rhombus. And in the perspective view, it would look like two squares turning into trapezoids. But these would just be your 2D view of what was happening in 3D.

Portraying a tesseract

So let's move into 4D. And again, we'll have a few choices for how to portray our extra axis.

Again, we can use perspective. This perspective made our cube look like a smaller square inside a larger square. It will make our tesseract look like a smaller cube inside a larger cube.

Or again, we could try to use an isometric view. We showed the three axes in 2D by putting them each 120° apart, or, if you include the positive and negative axes, 60° apart. Now we can show four axes in 2D, with positive and negative axes spaced 45° apart, which gives a tesseract looking like the flat one below. But this means some pairs of axes will be 90° apart, and not all squares will look the same.

Another option is to show the four axes in 3D, where we can give a genuine isometric view, with all axes about 109° apart, so that all cubes look distorted but identical, just as the squares did in the isometric view of the cube. In fact, just as a cube in the usual isometric view looks like a hexagon, a tesseract in this view looks like a rhombic dodecahedron.

Folding a tesseract

So now let's start to look at the cubes of our tesseract folding up. Our cubes here start off with their edges in the x, y and z directions, so the entire cube is perpendicular to the w axis.

We're not used to a cube being perpendicular to anything, but this is what happens when you have an extra dimension. In the plane, a square couldn't really be perpendicular to anything. It took a third dimension for the square to be perpendicular to an axis, and now we have a fourth dimension, a cube can be perpendicular to an axis.

Noticeably, now that we're in four dimensions, a plane can also be perpendicular to a plane, which is why, instead of rotating around an edge, we're going to rotate around an entire face.

It might also help to notice that in a 2D polygon, an edge was the boundary of an entire shape, whereas in 3D, an edge just becomes where two parts of a shape meet each other and fold. Likewise, in 3D, a face was the boundary of the whole shape, but in 4D, a face will just be where two parts of the shape meet up, and will be the thing that they fold around.

So here's just two cubes sitting next to each other along the x axis. How will they fold? They must fold around the face where they are touching each other, and that face has y and z edges. These must be the axes that are going to stay where they are, so that this face doesn't change. The x and w axes will be the axes for this rotation. And so any edges in this second cube that were pointing along the x axis will now be pointing along the w axis. Since this cube no longer has anything pointing in the x direction, it's now perpendicular to the x axis. And because the two cubes are perpendicular to two different axes, we know that they're at right angles to each other.

Let's go further and add a third cube, this time sitting next to the first along the y axis. It still has to fold around the face it's touching on, but now that face has x and z edges, so those axes have to remain unchanged, and it's the y and w axes that will rotate. As the y edges of this cube rotate to become w edges, we notice that, just as when we joined up three squares, we're going to start rotating some edges to the same place as each other, so that the shape can join up as it folds. This time though, it's four whole edges that will meet each other – in fact, an entire face that's going to meet up and join the face of another cube.

We now have three cubes all joined up face to face around this single edge. Just like it wouldn't be possible in 2D to join three squares together around a vertex, but in 3D that works, it wouldn't be possible in 3D to join three cubes together around an edge. But in 4D, we've managed to make that happen.

But we're not finished at this corner. We can still add a fourth cube, this time joined to the first along the z axis. Again, it will fold around the face that it's joined on, and since that face has x and y edges, which will stay still, the z and w axes will rotate. As the z edges of the cube become w edges, we'll see again that a face that was in the xz plane folds into the xw plane, and so ends up meeting another cube. In fact, this time two of the faces fold to meet the two cubes we've already folded.

Now, if we'd taken a square, put one square on each axis, and folded them up, we would have made one corner of a cube. But strangely, even though it's just one corner, it's half of the cube, and can be put together with an identical corner to make a whole cube.

Likewise, here we've taken a cube, put one cube on each axis, and folded them up. And we now have four cubes making up one corner of the tesseract. And that's a half of the tesseract that could go together with an identical corner to construct a full tesseract. (Tick "Show the other half" in the figures above to see the identical corner that completes each shape.)

More from the fourth dimension

Hopefully this has helped to understand what it means when we talk about shapes folding in four dimensions. If you want to play with tesseracts, you can play Tesseract 261 for free on itch.io.

But there's a lot more to four-dimensional shapes than the tesseract. Just as in three dimensions there's not just the cube, but four other Platonic solids, as well as Archimedean solids and Catalan solids and more, there's a lot more interesting shapes in four dimensions as well. You can examine them and unfold them in the 4D Viewer, and you can try building and folding some shapes yourself in the 4D Builder demo on itch.io.

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