Things to Make and Do in the Fourth Dimension

by Matt Parker

What is the largest number you can count to on your fingers? Well, most people stop counting on their fingers when they hit ten, mainly because they’ve run out of fingers. But not everyone uses the rather limited system of raising their fingers to count on and not putting them back down. If you do allow fingers to go back down, then you can count to 3 using only your first two fingers. Lift your first finger for 1, your second finger for 2 and both for 3. Now your third finger is available to be raised up, alone, for 4; then your third and first fingers for 5, and so on. You can get all the way to 16 before needing to use even the fifth finger on your first hand. Using this system, you can count all the way from 0 to 1,023 just with your fingers.

Of the Platonic solids, only the cube can fill a 3D space. People often claim that the tetrahedron is also space-filling (including Aristotle, in his work On the Heavens) but, despite the fact that this, er, ‘fact’, is repeated again and again, it isn’t, in fact, a fact: it isn’t true. If feels like it should work but, if you stack tetrahedrons, there will always be some small gaps between them.

Thankfully, mathematicians didn’t have to categorize every single possible knot you can tie in a piece of string: they already knew that there are prime knots. The area of maths dedicated to investigating knots is known as knot theory (with the mathematicians involved called knot theorists) and, much as the area of number theory uses prime numbers to learn about all the other numbers, knot theory is only really worried about prime knots, because, if we understand them, we are within reach of understanding all knots.

We can explain the disjointed nature of the 4D monster by looking at what would happen if we, as 3D creatures, attacked a 2D creature. Being 3D means that the space we operate in extends in three different directions: side to side, backwards and forwards, and up and down. A 2D creature can move only in two directions: it’s constrained to a flat surface. Let’s imagine a hypothetical creature who is completely flat – a hypoflatical, say – living in a completely thin universe, so thin that it would appear as a piece of paper does to us. We could loom as close to it in an up or down direction as we want and, because the hypoflatical has no concept of a third dimension, it would have no idea we were there. The third dimension provides perfect cover. Time to mount our own terrifying attack on a lesser-dimensional being – and all it takes is a move in the third direction into its two-dimensional world. The 4D monster which came from Higher Space! Imagine it from the hypoflatical’s point of view. As our fingertips pierce its 2D reality, they appear as individual floating circles which grow, move and then merge together as our palm reaches the 2D world. If we pass our fingers through their flat universe, all the hypoflatical sees are shape-shifting 2D cross sections, like flying meat pancakes. Trying to hide won’t do it any good either: as 3D creatures, we can see the entire 2D world set out in front of us like a blueprint – which is exactly how the 4D creature in the Alan Moore st...

Finding a life partner is a delicate balance. When you first start dating people, you don’t know, on average, how romantically well matched other people could be to you, and without that base-line you cannot ascertain if someone is an above-average catch and someone you should settle down with. This makes permanently partnering up with the first person you date a bit of a gamble: you should date a few people to get the lay of the land. That said, if you take too long dating people, you run the risk of missing your ideal partner and being forced to make do with whoever is available at the end. It’s a tricky one. The ideal thing to do would be to date just the right number of people to gain the best sense of your options while leaving the highest probability of not missing your ideal partner. Luckily, maths has made it easy for us: that right number of people is the square root of the total number of people you could date in your life. How you estimate the size of your possible dating population is entirely up to your statistical skills and the level of your self-confidence, as is how you then collect your sample.

Along with working on the Basel problem, Euler realized that adding an infinite sequence of reciprocal powers for all whole numbers will give you the same answer as multiplying together an infinite sequence of fractions which use only the prime numbers. So the zeta function can be written as two different equations, one of which relies only on the prime numbers. The one which uses all the whole numbers gives the same result as the prime fractions, but it’s easier to work with. We know what all the whole numbers are, but we don’t know what all the primes are. So we can substitute one for the other. This link to the primes was the springboard for Riemann’s paper. Investigating the sum of reciprocal powers gives insights into the product of all the primes (in fraction form).

I find it fascinating that regular shapes can go from there being infinitely many in 2D, to five and six options in 3D and 4D respectively, and then suddenly only three for all other dimensions. The problem with 2D is that there aren’t enough options to place any significant constraints on regular shapes: they’re allowed to run wild. 2D is too limited to have any nuanced behaviour. 3D and 4D are the sweet-spot for enough freedom to build something interesting but there not being too many options to ruin everything. From 5D onwards, there’s too much freedom to be able to lock anything down. For me, this explains why we live in a universe with around three to four dimensions: it’s enough to make the options interesting, but not too many that any attempt at anything too complex falls apart. Throw in the lack of knots and orbits in 4D and it tips the scales to why we experience a 3D reality.

If I give you the set of numbers {8,10,12,9,10,11}, and we agree where each edge goes, they specify an exact tetrahedron which you could build. They could also be coordinates which specify an exact point in 6D space. So, each 3D tetrahedron is equivalent to a point in 6D space; exploring different distorted tetrahedrons is like moving around in 6D.

The compact disc may now be outdated technology, but a lot of music albums are still released on CD. A standard 700-megabyte CD is actually 703.125 megabytes (a rare case of the music industry giving something extra away for free), which is a total of 5,898,240,000 1s and 0s. By my calculations, the number of possible different CDs in base-10 would have 1,775,547,162 digits. Which is also the number of corners a hypercube would have in 5,898,240,000 dimensions. So whenever a musician claims they have written a new album, all they have really done is choose a corner on a very high-dimension hypercube!

The classic example of a number which everyone desperately wants to be normal is π. We’ve checked the first nearly 30 million digits, and they seem to be normal, with every string of digits being equally likely. But we don’t know for sure. If it is, then any number at all will be in there somewhere. My name, Matt, turned into numbers as 13012020 (m = 13, a = 01, etc.) appears in π, starting at the 291,496,384th digit. If π is normal, your name, your favourite song lyric, the complete description of what you will have for lunch tomorrow, is in there somewhere. As it will be in any normal number. ‘Matt’ also starts at the 301,480,410th digit of , the 312,366,242nd digit of e and the 137,673,084th digit of the golden ratio. Any string of digits you choose, however long, will appear in almost all numbers. More numbers contain the complete works of William Shakespeare than don’t.

This paradox – that there are always more balls in the box, but yet they end up in the drawer – is caused by our expectation that infinity behaves like a really, really big number. But infinity is not a number. Infinity is nowhere to be found on the number line. People seem to have the idea that if you keep counting up along the number line past bigger and bigger numbers, in the end, the numbers just give up. They can’t be bothered to go on, and there’ll be an infinity sign (∞) there to mark the end of the number line. This is not the case. There is always a bigger number. Infinity is not safely contained at the end of the number line. Infinity is not a big number. Infinity is actually a measure of how many numbers there are. The number line never stops, so we say it is infinitely long. Just like we said before that the number ‘five’ is the size of any set of five things, infinity is the size of a never-ending set of things.