Getting lost in infinite loops
Eugenia Cheng wanders through the wonderful world of infinite mathematics in this extract from her book Beyond Infinity, shortlisted for the 2017 Royal Society Insight Investment Science Book Prize.
As I mentioned at the start, I grew up in a house with a ﬁreplace and chimney in the middle. We did have central heating but the original idea was that the chimney would heat up the centre of the house, and every room in the house was clustered around it. This meant that instead of the house having an entrance with rooms coming off it, the rooms led onto each other round the chimney.
The best thing about this when I was little was that my sister and I could chase each other round the house in circles, looping round and round the ﬁreplace, through the kitchen, dining room, sitting room, kitchen, dining room, sitting room.
Even though I don’t often chase my sister round in circles any more, I still like buildings with loops in them because it means I can go on an inﬁnitely long walk. Mathematicians – and no doubt others – often like walking to help them think. There’s something about the gentle movement that focuses my thoughts. Sometimes I think it’s that the act of walking keeps the logistical parts of my brain just busy enough that my dreamy mathematical brain can roam free without the other part constantly bothering it and saying things like ‘Don’t forget to buy some eggs’.
However, the trouble with going for an inﬁnitely long walk is that usually you would end up inﬁnitely far away from home. Also, if I go for a walk outdoors and think about maths at the same time, I am very likely to get lost and also be hit by a car. Having a building with a loop in it solves all these problems, because it means you can walk round and round in circles ‘forever’, without feeling like the circles are futile, because unlike when I just walked in a circle around my kitchen, you’re really going round something solid, not just empty space. As I have already mentioned, the maths department at the University of Nice is entirely based on a circular corridor. The inside of the circle is a beautiful circular inner courtyard, with ﬂoor-to-ceiling windows from the corridor looking down on it. All the ofﬁces are on the outside of the circle, and as you walk round and round the corridor you are always facing onto this beautiful courtyard. I’m glad to say I was not the only person who sometimes just walked round this in circles, thinking. There’s something about the possibility of inﬁnity that is liberating.
The problem with this department was it was so symmetrical it was impossible to remember where I was. There were four staircases, equally spaced around the circle, and whenever I emerged from a staircase I could never work out whether it would be quicker to go left or right to get where I wanted. The funny thing is the number of times I ended up going round the circle more than once to get where I wanted, because I missed it on the way.
Another satisfyingly inﬁnite structure is a paternoster. There is a quite famous one of these in the Arts Tower at the University of Shefﬁeld. It’s like a lift, except much more interesting, consisting of a circular loop of capsules, each of which carries two people (in Shefﬁeld’s case). At any given moment there’s one capsule at each ﬂoor going up, and one going down, and a couple in the process of looping over the top or under the bottom. The paternoster has no doors, and it never stops moving. You just have to jump into a capsule as it goes past where you’re standing, and jump off when it reaches the ﬂoor of your destination. It moves quite slowly in reality, but still the ﬁrst few times I took it I was terriﬁed, and my pulse shot up when I had to get on and off. It’s amazing that this thing hasn’t fallen foul of health and safety regulations yet. It is technically forbidden to ride over the top or under the bottom, but it is everyone’s ﬁrst thought, and I’d guess there aren’t many people at the university who haven’t tried it.
It’s called a paternoster because it’s supposed to remind us of a string of rosary beads, and the idea of rosary beads is to keep track of how many times you’ve said certain prayers. There might be ten small beads in a row to keep count of saying ten Hail Marys, and then a larger bead to indicate that it’s time to say a Lord’s Prayer before starting another ten Hail Marys. ‘Paternoster’ is how the Lord’s Prayer starts in Latin. The idea of the beads is that having a physical way of keeping track of the number of repetitions leaves your mind more free for thought.
I feel similarly about paternosters. I wish there were always paternosters in buildings instead of lifts. To me it makes the whole building feel connected in an inﬁnite loop, rather than being made up of separate, ﬁnite-sized ﬂoors that you have to travel in between. It means that my brain doesn’t feel divided up into ﬂoors when I’m moving between ﬂoors, and the predictable continuous motion means that my mind can feel more free for thought than when I’m waiting for the unpredictable arrival of a lift. This is a bit melodramatic, but when I’m thinking about maths this sort of thing makes a surprising amount of difference to my progress.
Besides, a paternoster is an amazing vertical version of being able to go round and round in circles (although you can’t really chase anyone round it) and it makes me feel that a building is inﬁnitely big. If you attach your Theseus-like string to the ground ﬂoor and then get in a paternoster and go round and round in it, you will again need an inﬁnitely long piece of string.
We previously claimed that there was nothing really inﬁnite in the world, but now we see that circles are wonderfully inﬁnite if you think about the paths that you can take on them. Circular (or oval) racetracks are clever because you can run any length of race on them. A little boy called Jacob recently wrote to me to tell me that maths is his favourite subject, and also, by the way, that he could do 84 metres on the monkey bars. I wondered if he had a very long series of monkey bars, or kept turning round, or if he had circular monkey bars so that he could keep going just as long as he wanted.
The Circle Line on the Tube used to be more fun because you could sit on the train forever and keep going round and round, which is much more like an inﬁnite journey than going back and forth between the end stations of a non-circular line. Some people are very excited by driving round and round the M25 around London repeatedly (though personally I’m always glad if I just managed to get part way round it without getting completely stuck in trafﬁc). I expect there are also such fanatics of other circular highways round cities, like the Périphérique around Paris, or the Capital Beltway around Washington, DC. An even more interesting version of a circle is a Möbius strip. This is made from a strip of paper, where you stick the ends together but instead of sticking them in the obvious fashion you twist the paper before sticking the ends together.
The intriguing thing about this, both physically and mathematically, is that you have now stuck the front of the paper to the back, and the back to the front, meaning that there is now only one side – the front and back have become the same.
There are many interesting things you can do with this, but one simple one is just to pick it up and go round it with your ﬁngers. It’s even easier to lose where you are on it than if you’re just going round a circle, as you can get confused about whether you’re on the front or the back – because there’s no difference. You can also try tracing your ﬁnger around the edge of the Möbius strip. The edge is a single circle, but it loops back on itself and then appears to ‘go round’ the Möbius strip again, although in reality it’s only gone round once. It’s a sort of ﬁgure of eight, another very satisfying shape to go round and round forever. How ﬁtting that these should be the symbols that begin and end our quest for inﬁnity:
Except now we know there is no end, because not only does inﬁnity go on forever, but the hierarchy of inﬁnities goes on forever, with bigger and bigger inﬁnities, even though we’re sitting quietly inside our large but ﬁnite world. Perhaps we should no longer be surprised that something inﬁnite can ﬁt inside something ﬁnite. Where inﬁnity is concerned, it seems that almost anything is possible. The world of mathematics ﬁts in our brain but is bigger than the universe.
Inﬁnity is liberating but sometimes too liberating. I can think much more freely and creatively if I feel I have an effectively inﬁnite amount of time at my disposal. I am more likely to prove a theorem if I have no ﬁxed engagements for the rest of the day, even if the proof only takes two hours. If I only had two hours available, I probably wouldn’t be able to do it.
And yet, we’ve seen that if we were immortal we could procrastinate forever. Knowing me, I probably actually would procrastinate forever. Having inﬁnite dimensions at our disposal gives us inﬁnite subtleties that we can’t deal with. But we can still dream about them even if we can’t explain all those dreams.
I truly believe that trying to explain everything isn’t the point. Rather, the point is to explain as much as we can, and, importantly, to be clear where the boundary is between what we can and can’t explain. In my mind’s eye, the sphere of things we can logically explain is at the centre of the universe of ideas, and the aim of mathematics is to move as much as possible into that sphere. So the sphere is always expanding, and as it does so, its surface keeps growing. The surface is where the explicable and the inexplicable meet.
The most beautiful things to me are the things just beyond that boundary of logic. It’s the things we can get quite a long way towards explaining, but then in the end they just elude us. I can get quite a long way towards explaining why a certain piece of music makes me cry, but after a certain point there’s something my analysis can’t explain. The same goes for why looking at the sea makes me so ecstatic. Or why love is so glorious. Or why inﬁnity is so fascinating. There are things we can’t even get close to explaining, in the realm far from the logical centre of our universe of ideas. But for me all the beauty is right there on that boundary. As we move more and more things into the realm of logic, the sphere of logic grows, and so its surface grows. That interface between the inside and the outside grows, and so we actually have access to more and more beauty. That, for me, is what this is all about.
In life and in mathematics there is often a trade-off between beauty and practicality, along with a contrast between dreams and reality, between the explicable and the inexplicable. Inﬁnity is a beautiful dream, inside the beautiful dream that is mathematics.
The winner of the 2017 Royal Society Insight Investment Science Book Prize will be revealed on 19 September.