I had a grumpy day on Monday (it started with the fact that I was obliged to go to work on a public holiday and went downhill from there) and in the afternoon I decided to give up doing serious work and spend some time building a tesseract to hang outside the office door. I think nearly every office I have occupied since I was school teaching 25 (gulp) years ago has had one hanging from the ceiling. The only exception that I am certain about is my last Melbourne office — the new one — which had ceilings that were not only very high, but had all sorts of plumbing and ducting running all over them as part of the building’s faux industrial el cheapo decor.

Let’s see if I can explain what is going on, but we’ll be arguing by analogy, starting with a shape we know well.

The good old cube is a 3D object. You can hold one in your hands and you’ll see that it is made up of six nice square faces (like the “sides” of a 6-sided die). Each corner (technically “vertex”) of the cube has three edges meeting at it and they are all mutually perpendicular … which just means that each edge meets each of the other two edges at right angles. This means that the corners of a cube actually show the three mutually perpendicular directions that make up three dimensional space: left-right, up-down, and towards-awayfrom.

I am now going to show you a picture of a cube.

[Oh, good, I think WordPress coped with having two pictures near each other but with different justifications.]

Now a PICTURE of a cube (which is what you see above) is not a cube at all. It’s flat, for starters. It tries to show three dimensions — kind of in perspective — but it has to cheat a bit. It can show the left-right and up-down directions successfully, but the towards-awayfrom direction it has to show by drawing slanty lines (the ones in blue in the diagram above).

Let’s think about how I drew the cube above. Basically I drew a square (shown in green) to be the face “nearest” the viewer, and then an overlapping square (in red) to be the face at the “back” of the cube. I then connected the corresponding vertices of these squares (with the light blue lines) to get the remaining edges of the cube. This means that four of the faces of the cube — the top, the bottom, and the left and right sides — are not actually shaped as squares in the diagram at all; instead, they are parallelograms. This is what we have to do in order to show 3 dimensions in a 2D picture. The third dimension — towards-awayfrom — doesn’t (can’t) meet the other two dimensions in a perpendicular way (i.e., with a right angle) because flat 2D doesn’t have the depth to do it (such a shallow place to be!). In the real 3D world, a cube has a square at the front, and a square at the back, and they are connected to each other by edges that are perpendicular to the edges of the front and back squares, and these new edges make squares on the top, bottom and sides.

Okay, let’s take a deep breath.

It would be great if I could give you a real 4D tesseract to look at but I can’t. If there is a fourth spatial dimension we can’t perceive it (just as Edwin Abbott’s square in the book *Flatland* was stuck in a 2D world and had no idea about the third dimension). So, all I can do at the moment, is give you a 3D picture of one (well, actually, the 3D picture is actually the object outside my office door; the *photo* you’re seeing here is actually a *2D* picture of this 3D object, which makes things even more complicated!)

Just as squares are the building blocks for the cube (if you like), cubes are the building blocks for a tesseract, and the tesseract is the 4D analogue of the cube. So, think back to what I did with the cube picture above: I had a front square, a back square, and I showed the third dimension with sloping lines.

For the tesseract, there is a “front” CUBE (shown in green in the picture at right, with some distortion added due to the fact that the whole thing is made out of straws and cotton and is somewhat wonky (!) plus the extra perspective distortion that comes with my shooting angle), and an intersecting “back” cube (shown in red). In the model outside my door these are REAL cubes, with proper right angles in all the right places.

Now we get to the fourth dimension — the extra one that we can’t show with proper right angles because they don’t exist where we are in 3D. This is depicted by the blue sloping lines that join corresponding vertices/corners of the green and red cubes. Having got this far, the fun bit from here is to realise that some of those blue, red and green lines also make parallelopipeds (slanty cubes, if I can abuse language and maths in one fell swoop), analogous to the parallelograms in the cube picture earlier (take, for example, the square at the bottom of the green cube, and the square at the bottom of the red cube, and the blue lines that join them). In the real 4D world these parallelopipeds would actually be cubes exactly like the red and green cubes (just as the parallelograms in my 2D cube picture are really squares on a 3D cube).

Note, too, that each vertex has FOUR edges meeting at it: and these are the four mutually perpendicular directions of 4D space (but, of course, we’ve had to show the fourth one at a slant).

So now you know (I hope!). Anything you still can’t cope with is therefore your own problem* 🙂 !

I have written more about the fourth dimension previously and I also have photos of some other 4D shapes with extra explanations if you’re interested or want to revisit this.

* Douglas Adams, *Hitchhiker’s Guide to the Galaxy*.

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