I had a free twenty-four hours and a hangover. I hadn’t slept well in a while. I needed to be in the North on Sunday morning and ended up in Manchester, largely because I could, and because I knew nobody in the city.

Mist was beginning to pool in the road when we drove home from a party the night before, and the morning was shrouded in it. Inclement weather carries a broader variety of affects than I had noticed before: low cloud and low pressure is flattening, a downer, while strong wind and heavy rain beat refrains of the nested comforts of home. Mist is mysterious, as though the two are etymologically linked, and a sense of adventure creeps out of it. I got fernweh.

It rolled like half-formed hay bales across the fields on the way to Hereford station. I was certain of the rightness of leaving home now; I had a sense that something was up, that the train ride would sort me out. I had a deal with myself that I work on trains, and it is a deal I never break. The hills of Shropshire rose and fell, and I wrote, half-heartedly at first, peering over the laptop at the Long Mynd, and by the time the train passed the plump suburbs of Cheshire the sun was out and I had made sense of it all and the trip felt vindicated.

And the deal, then, was that I got to do fun stuff. I wandered through cafes and tried on too much lipstick and went dancing, alone, which is the best way to do it. I wandered through the city, half lost. The lostness was a key bit of the attraction, for the city was a stranger in that way that cities, like people, can be handsome or ugly, friendly or menacing, and vary in their knowability.

The not-knowing, the way that streets and buildings carried no embedded meaning for me, and simply presented themselves as paths and vistas, made it possible to see the shapes and patterns that make cities beautiful when they are. I took terrible blurry pictures of their geometries on my phone, and thought about the nature of geometry, or tried to. The tiredness made my mind soft, and it softened the shapes around me too so that streetlights had an aura and cars left a trail.

The natural world has its own geometries, some of them obvious: a pinecone’s perfect spiral in Fibonacci sequence, the six-pronged star of a snowflake. And then there are the subtler ones, the fractal repetitions of shapes that you get upon magnification, the patterning of the secondary structure of proteins where the chain of amino acids forms bonds at regular intervals to create a helix or sheet.

Proteins are manufactured in the endoplasmic reticulum and packaged in the Golgi apparatus – the industrial organelles, the Manchesters of the cell, built out of long factory production lines. I fear at this point that the overextended metaphor will be taken for anthropomorphism, which is on balance fine: production, like geometry, replicates itself across all scales of existence, and if you want to tag it as mistakenly human-centred the human part of that assumption is all yours. Back to geometry.

There was a whole row about geometry that happened in the nineteenth century, at the same time that much of Manchester’s architecture came into being. Old-school Euclidean geometry was being ditched and dismissed as “planimetry” – a system that worked only self-referentially in the unreal confines of the flat plane. The new geometers were getting their teeth into new forms with new methods that better described the complexities of spacetime, or so the story went.

The new geometers thought that Euclidean geometry with its neat forms created a conceptual trap in which the world was thought to correspond to geometry, and when the world, or at least spacetime, appeared to correspond to any of the new geometries that better described its curvatures and folds the assumption that the world must be shaped like a geometry somehow persisted. It would simply be too strange to see the geometries as shapes that only ever existed in the human imagination, and from which we sought out mirrors in external data, or created external things in their image.

But what fantastic objects we created from them. The repetitions of shadowed railings on steps, so that the lines of the railings fell diagonally across them; the diminishing lines of steel struts beneath a bridge, backlit from the road; an alleyway between the hooped part-moons of railway arches and the striplit planes of a multi-storey carpark, bridged by a vertical fence that opened precisely halfway along.

And, strangest of all, in the last cheap hotel in town, which looked like it might have been repurposed out of student halls that gave the kids strange dreams, a vast pendulum hung and swung in perpetuity in an octagonal space in the lobby. Maybe the human capacity for folly sets us aside from the rest of nature; I’m not at all sure about that, though.