I was in a tile shop the other day and it got me thinking about the fourth dimension. What do you mean “what are you talking about?”, I would have thought the connection was obvious. Oh, all right then…
Anyway, I was in a tile shop looking at different tiling patterns. Many of them were very lovely and its impressive to see how a good tile shop can embrace patterns form so many different countries and cultures. This shop had a fine line in British, Spanish, Italian, Moroccan, Arabic, Greek, you name it. Bathroom designs to satisfy even the most demanding and culturally promiscuous time-traveler.
Except after a while you start to notice that all these patterns have something in common: they are all periodic. That is to say, all the patterns repeat. In maths this is known as a translational symmetry – if you slide a repeating pattern around it will eventually line up with the tiles a few feet over. This could be as simple as a checkerboard pattern or something much more complex but typically a tiling that perfectly fills the floor space available repeats (i.e. it doesn’t have any gaps in it) will be repetitive.
There’s actually a whole branch of maths that investigates tilings and symmetries. It’s provable that certain patterns will fill the plane perfectly (like squares or hexagons) and other (like pentagons or octagons) won’t, and that certain combinations of shapes (like octagons and squares) can be used together to perfectly tile the plane. It’s related to the wider study of symmetry and a field called Group Theory, which turns out to be hugely important in physics, cropping up in everything from molecular chemistry and spectroscopy to particle physics. Heady stuff.
Anyway, what’s all this got to do with the 4th dimension. Well, aside from culturally promiscuous time-travel (which I’ll cover in a separate post). The fourth dimension is link to tiling in a rather surprising way. I mentioned that tilings tend to be periodic, but does that mean that EVERY tiling of the plane has to be periodic? The idea is related to a thing called the Domino problem in maths. Specifically, given a set of shapes, is it possible to design an algorithm that will decide if they can tile the plane or not. During the 1960s, a mathematician called Hao Wang suggested that this problem was solvable if all tilings of the plane were in periodic. You’d just need to decide if your set of tiles could be arranged periodically or not and you’d know the answer. Neat, huh?
Except there’s a snag. What if there are tilings of the plane that aren’t periodic? If those existed the test would fail. In 1966 a non-periodic tiling was found: it used over 20000 different tiles (pretty hard to construct, and all the more impressive for being found before powerful computers were widespread). Then anther with 104 tiles was found. Then another with 40 and another with just 13. Finally in 1974 Roger Penrose found an aperiodic tiling that required only 2 tiles. These are pretty interesting patterns: patterns that cover the plane by don’t (quite) repeat. You can’t slide the pattern around and match it to itself, and it only contains two types of tile! It looks like this:
At first it looks regular, but stare at it for a while and you can see that it doesn’t quite repeat. Also notice that this piece doesn’t form a repeating unit on its own either. It’s a pretty cool thing. What turns out to be extra interesting, though, is that tiling patterns are related to packing problems in 3D, and in particular to the arrangement of atoms in crystals. You could ask the same question here: if I want to pack a load of (spherical) atoms into a 3D space, does the pattern have to be repetitive?
Well, no. crystals are periodic, glasses are disordered arrangements of atoms. More organic forms like wood are the result of nonlinear growth processes and many rocks are made up of mixtures of crystals, glasses and dusts all mixed up together but none of these have the same properties as the Penrose Tiling. There are things that do, though: they’re called quasicrystals, and they have a whole field of research attached to them.
A quasi-crystal is a regular but non-periodic packing of atoms. Using electronic force microscopy you can actually make images of the individual atoms and the surfaces turn out to look a bit like this:
But what about the 4th dimension? Well, it turns out that the best way to think about aperiodic tilings is to thing of them as a 3D slice through a regular 4D lattice. If the slice is parallel to one of the regular planes in 4D, it’s periodic. Any non-parallel slice is aperiodic.
What’s a 3D slice? Think of a line. A line is 1D, and you can cut it with a single point. Similarly, a square is 2D and can be cut with a line, which is 1D. A cube is 3D and can be cut with a plane, which is 2D. There’s a pattern here: an nD volume can be sliced with an (n-1)D object. So a 4D object (don’t worry about picturing it!) can be sliced with a 3D volume. Mathematically this a relatively easy thing to do, and so your 3D quasicrystal is a the 3D slice through a 4D object.
This idea of slicing through a 4D volume is something that crops up in my own work. I do a bit of work with 3D graphics. Here is turns out to be convenient to think of your 3D space containing your graphics as a slice through 4D. Why is that, you ask? Well, in 3D I want to move around and also rotate around a point. A rotation in 3D is written as a 3×3 matrix. In fact you can think of any rotation as the result of three other rotations: one about each of the x, y, and z axes. This is provable using – guess what – Group Theory.
The thing is, that once your 3×3 matrix is full of rotations (any other things like scale factors and sheers) there no room for the translations that move things around. The most natural thing to now is to make the matrix bigger to include extra elements for moving around, and this is equivalent to (guess what) using a fourth dimension. So, weirdly, 3D geometry turns out to be easier to think of in 4D. The 3D graphics in your favourite computer game are in fact a slice through a 4D space, just like the tilings and packings for the quasicrystals. The guy in the tiling shop seemed quite interested in this when I mentioned it, although I did steer clear of projective geometry and quasicrystals at the time. It was Sunday lunchtime.
One more thing before I go. I was talking to Mrs of-Science about this later on that day, and she immediately asked if I was talking about time as the 4th dimension. I’m not – all of this is about a 4th space-like dimension, and is very different from the geometry of relativity’s 4D Spacetime, which is fascinating in a completely different way and not at all like regular Euclidean space. Interestingly, though, you can add a 4th space dimension to general relativity. Kaluza ad Klein did it back in the 30s. It turns out that if you do this you get a unification of gravity and classical electromagnetism, which was the start of all the work on grand unification of the the four fundamental interactions that we hear so much about. So the fourth dimension reveal secrets here too.
Life in 4D is pretty cool, and I fully expect my bathroom to look pretty good too.