Last week I went to see Radiohead play at Victoria Park, London, and travelled by train up from Cornwall. It’s been three years since I used this train, and the carriages have been renewed in the intervening period, replaced by what seem to be more comfortable, better thought out spaces.

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The next thing I noticed was the fold-down tray in the back of each seat. Even if you have no need of it, you are drawn to play with this kind of thing, and normally they are flimsy, rattling affairs, which have the air of having been drawn up by someone really pissed off by having been given the low-grade job of ‘tray designer’. They do the job, but reluctantly.

This tray, however, had purpose, it knew what it was there for and it wanted to do it well. The base of it was metal, a good few millimetres thick, cast as a complete unit together with the hinges, had a nicely textured finish. The hinges had perfect friction. This is vitally important, and someone had given very careful consideration to it. Imagine you’re sat in your seat, a drink in one hand, and you have a moment of minor stupidity; you release the tray from its upright position with your free hand, forgetting about the scalding hot pretend coffee in the other. Normally, the crappy tray will crash down to its horizontal position, knocking the ‘beverage’ from your grasp, with any luck into the shoe of the snoring bore next to you, but more likely, into your lap.

This tray won’t do that. It has just enough friction in the hinges to stay at any angle between the upright and horizontal. Not so much friction that it’s annoying to move, just the perfect amount that allows you to notice it. There’s also a neat slide-out frame in the front edge of the tray, ideal for giving you that extra bit of resting space for a laptop or large book. Again, just the right amount of friction in it so that it will actually stay in the position that most suits the user.

Aside from useful power points within easy reach of all seats, and a couple of other minor things I won’t bore you with, the other detail that jumped out was a small silver knob stuck at an angle into the edge of the seat opposite the pink ear. Exactly the right thing for hanging a bag, jacket, camera, etc from. Something so right once it’s there.

Anyway, there I was, hurtling backwards towards the city and heaving crowds, thinking great, I’m going to write about this as an example of good usability, detail that makes you happy to have encountered it, that sort of thing. Slowly, though, as the cows zipped past the window, and the excitement of the things wore off, another nagging sense started to weave its way into the back of my head, a sense that the space of the seat that was filling my forward view was somehow right, over and above its use.

Normally, when that sense comes into play, there’s a reason. In a church, for instance, it doesn’t take long to realise that all aspects of the architecture work, because they are in proportion to each other, they are related by the mathematics of their dimensions.

In this case, a seat on a crowded train, nothing stood out at first. But then slowly pieces started to fit together, and I found myself holding up the flat edge of the Safety Information pamphlet against certain sections to see if the hunches were correct.

First, the silver knob is positioned so that its bottom edge aligns exactly with the bottom of the First Bus logo in the middle of the seat.

Also, the point at which the base of the pink ear joins the seat aligns with the center of the First Bus logo.

The angle of the right edge of the First Bus logo aligns with the point at which the pink ear joins the top of the seat. This line flows seamlessly into the rounded edge of the ear. It also intersects the previously drawn line at exactly the center of the First Bus logo.

The distance from the top of the seat to the center of the logo is the same as the distance from the center of the logo to the outer edge of the seat.

I badly wanted the top right edge of the seat to form a perfect triangle with the line linking the logo to the ear, but it wasn’t to be. Not far out though.

However, when moving that line to join the other diagonal at the top edge of the square, I noticed that it intersected the bottom diagonal exactly at the edge of the seat.

Astonishingly, if you then trace the circle that creates the top right section of the pink ear, that circle aligns perfectly with the moved diagonal, and also intersects the right edge of the square at exactly half its length.

How many of these relationships ( and more that I’ll leave you to find ) are intentional, and how many are coincidental ?