[some spoilers for a mechanical detail in Diaries of a Spaceport Janitor, specifically an answer to why it's so easy to get lost]

What's the shape of the map in Diaries of a Spaceport Janitor? Ok, I mean, there's a map in the game's manual (above) and it's clearly a rectangle. But among other things it's a game about getting lost, so this map is more of a suggestion of what the Spaceport is like than anything. In fact the actual Spaceport is laid out in a way that I don't think I've seen in any other game.

It becomes clear that there's some sort of looping going on: you can always just keep walking around obstacles, it never seems like there's some sort of boundary wall to the spaceport. There are a couple big buildings and one cliff where you can look out over a lower level you can never reach (these points are on the corners of this map), but that's it. Even the screenshots on the Steam store page loop.

It's obvious that something is up when you climb the "9-sided Ziggurat" and see... yourself! Just around the other sides of the building. And multiple copies of the city are visible below you too. Whatever's going on, this isn't a feature of typical video game torus-type warping, where you go off the edge of the map and come out the opposite side. The whole map has been rotated and placed multiple times around this one building... so it must be wedge shaped?

But that can't be right. If it was a big wedge, it would have a long exterior wall, and there's only a short one - the cliff - at the opposite corner of the map. What's going on is that the other corners of the map also hide points like this. If you look out across the cliff area in the top right (instead of down into the city below), you can just about make out identical cliffs, with identical copies of yourself looking across it. And the buildings in the top left and bottom right of the map are the same building. You can't see yourself as you walk around it, it's too large, but you can walk all the way around by only turning 180 degrees. You might not notice this because it's the only place in the game to feature magic travellators on the ground that make you walk faster, so it's easy to just think of how quick it is to get around as a result of those.

If you build out a diagram of this, you'll find the only way it works is if adjacent sides of the map warp to eachother, not opposite sides. The left and bottom sides are joined, and the top and right sides are joined. It's a clever design for how it makes the spaceport feel massive and easy to get lost in while also being surprisingly cramped. Take a shortcut through an alley and realise you're now somewhere you thought was super far away; make one wrong turn and feel immediately lost; walk in what you think is a straight line and the world itself turns you around by 90 degrees; wander aimlessly and somehow keep ending up at the Ziggurat when you're just trying to find your way home.

But what shape is it? If you join the opposite sides of a square you get a torus, but what if it's the adjacent sides? I only worked this out from the Golly manual of all things: it's a sphere! Which seems wrong, right? When you walk in a straight line on a sphere, you don't get turned around, you just end up back where you started. But if you think of the bottom left and top right of this map as the poles, it kind of seems to make sense.

It isn't possible to unflatten a sphere without somehow distorting it. In this case, the "poles" seem to have been distorted to be smaller: they're only 90 degrees around instead of 360. I wonder what it would look like if you projected the Spaceport onto a ball.

So what map projection is the Spaceport, then? It can't be original to this game because it's one of the available geometries in Golly, and the idea of joining adjacent sides of a square to make what is kind of technically a "sphere" is so simple it can't be that original, so somebody must have done the reverse. But it's not in the Wikipedia list of map projections. "Adams hemisphere-in-a-square" is close, but it's only one hemisphere, and the sides don't join up.

This projection has been described, though, and it was in 1925, by the same Adams! It's the Adams World in a Square II. (I think this has to be the right one, anyway. The question of why the corners of this map are displayed too large while the corners of the map in Diaries of a Spaceport Janitor seem too small is very confusing and fun to think about).

The original description of this projection, which you can read on google books, begins:

The problem of projecting the entire sphere conformally within a square, with the poles on one of the diagonals and symmetrically placed with respect to the center, has been in consideration from time to time for several years.

When you buy Diaries of a Spaceport Janitor on Itch you can pay a bit extra for some bonus content (which I'd say is worth it, there's some cool stuff there). One of the things included is a very early build of the game, a "very first proof-of-concept", with no Spaceport, no Janitor and no Diaries, just a top-down view of a map and a cube with an arrow on it that you move around. But the one thing it does have: if you go to the corners of the square map, you can see other copies of yourself, rotated by 90 degrees.

[and then if this was a video essay that'd be the bit where the synth line of No Output comes in and it would cut to the credits]