People tell me that maths is beautiful. I’ve heard it said that Euler’s identity is elegant. There’s a touching moment in a Horizon documentary when Andrew Wiles describes a eureka moment when he finally completed his proof of Fermat’s last theorem. ‘It was so indescribably beautiful’, he says and then almost breaks down in an affecting, slightly embarrassed manner.
I don’t doubt my friends who tell me about Euler’s elegance, or the sincerity of Andrew Wiles’ numinous moment but I can’t say that I feel it personally. I’m sure the proof of Fermat’s last theorem is very remarkable, but to fully appreciate this beauty one requires knowledge of modular forms, elliptic curves and the functioning of complex numbers. I still can’t quite divide two numbers without employing a pencil and paper.
I’m thinking about this after watching Manchester City play football. Pep’s Man City are often described as scoring beautiful goals when their cast of rotating cast of technical midfielders pinball through a defence with 15 short passes. Their pressing is relentless without ever being quite as exciting as a team like Liverpool. Their players are wonderfully gifted and perform some memorable feats of skill—I still enjoy watching Bernardo Silva’s epic dribble at Anfield – but none of them quite has the wow factor of an Erling Håland or the stand-up excitement of, say, Jadon Sancho at his best.
All of this is to say that watching Man City leaves me a bit cold. Browsing twitter and football blogs, I can see that I’m far from the only one who feels like this. Even those paid to be studiously impartial and kind often resort to revealing descriptors. Darren Fletcher, commentating on Man City’s game against Chelsea this afternoon, describing the contest as ‘fascinating’: a word only ever employed when a game has failed to produce a goal or amazing save.
But what if the fault is not in our stars but in ourselves, that we don’t appreciate them. Part of this comes from never seeing a Guardiola side in the flesh. Watching on TV, it’s close to impossible to understand how high-presses work to unbalance an opposition team. You never understand how fast a player like Sterling is until you’re in the stands and you have to physically turn your head to keep up with his runs.
It’s not just this though; what if I’m just not smart enough to understand them? Increasingly I think I need to understand the complex-number based forms that govern Man City’s precision attack in order to properly appreciate them. When they are set up in one of their complex attacking or pressing structures, Man city’s players look more like vertexes describing a polytope of immense and hidden symmetry than they do a conventional polytope. Not all great passing teams have this quality; the current Barcelona Femení have a claim to be the best such side of this century, but they have a more relatable and human quality. Maybe their players look like they enjoy it a bit more?
In the big games, when Pep’s city meet another highly technical team I sometimes feel like I need a live data analyst or a friggin geometry professor to understand how and why the two teams look like they do. The polytopes enter higher dimensions then; Pep Guardiola’s fifth-dimensional hypercube matching up against Thomas Tuchel’s Five-Cell Simplex. It has all the fascination of watching two mathematical diagrams meet up for a fistfight.
I don’t want to sound like I hate modern football or Pep’s huge influence on it. The sport after 2009 looks different because of Pep’s Barcelona men’s team. I like that there’s less physical violence involved in the game, and more room for technical players. But to really understand the beauty of the purest expressions of this style, like Pep’s Barcelona, I might have to go back to my fifth form Maths teacher and ask him to explain the unit circle again.
Paper Sail 