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Comment by DaiPlusPlus | Hacker News Reader

What happens is the graphics and movement get glitchy and eventually unplayable - I wouldn’t call this “non-Euclidean Doom” - just “consequences of messing with the constants of the universe” - whereas I’d expect an actual non-Euclidean Doom to be like this: https://youtu.be/kEB11PQ9Eo8?si=0HNlpGFBii2AIK1n

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To be a bit pedantic, that video is incorrect in its use of the term non-euclidean. The folks behind hyperrogue made few videos that show actual non-euclidean geometry: https://youtu.be/yqUv2JO2BCs?si=AutaqS5unvT7cDjw

Comparing some of the weird geometry effects in the doom video, like things sliding sideways when you move forward, I feel like it's reasonable to call this doom non-euclidean!

Part of the problem is that Doom isn't truly 3D, it's a 2D universe with hacks to make it look 3D.

So in the final presentation, you're seeing a facsimile of a universe where the value of Pi has only changed in two out of three spatial dimensions.

This is not really true. It is true that this particular 3D environment has some limitations, but it is 3D. A video here discussing this exact topic: https://www.youtube.com/watch?v=ZYGJQqhMN1U

For a more modern take (that's kid friendly) have a look at https://store.steampowered.com/app/1256230/Hyperbolica/

This it's also not what non-euclidean means. It's just regular 3d world with portals. Games of that era often used portals in the engine, they just didn't use them to implement impossible rooms. I'm not trying to be nitpicky, but it's a common misunderstanding.

For actual non-Euclidean stuff I recommend ZenoRogue stuff. For example a simple game with Nil geometry[1], giantic bossfight in non euclidean world rogue game[2], or some general geometry weirdness [3]. Or just check out any of their stuff.

[1] https://m.youtube.com/watch?v=gejRg_q70EA&pp=ygUJemVub3JvZ3V...

[2] https://m.youtube.com/watch?v=jcnXI8IArRI&pp=ygUJemVub3JvZ3V...

[3] https://m.youtube.com/watch?v=yqUv2JO2BCs&pp=ygUJemVub3JvZ3V...

"Non-Euclidean" is probably just fine meaning "literally anything other than straight Euclidean". Euclidean geometry doesn't have portals in it. Being truly hyperbolic or spherical is an interesting case but I don't think we have to insist that's the only viable meaning. Adjoining portals to Euclidean geometry by my reckoning breaks 3 of the 5 core axioms of Euclidean geometry... which means by sheer axiom break count, Euclidean geometry plus portals is "more" non-Euclidean than spherical or hyperbolic geometry. (Which may not seem like a terribly meaningful count, but I think would be reflected in what proofs in a portalized space would end up looking like. You'd grow a forest of case analyses and conditions in even the simplest proofs.)

(And it's possibly 4 axioms broken; I'm reckoning "all right angles are equal" as still holding, though I'm not 100% sure a "normal" right angle and a right angle generated where a portal bisects the angle is necessarily "equal" in all relevant ways. It's been a while since I did geometry proofs but the more I ponder it the less sure I am. If not that would leave only "a straight line segment can be prolonged indefinitely" as holding, and unlike in Euclidean geometry a straight line segment may intersect itself arbitrarily often.)

Officially it cannot be called a "geometry" at all if it does not satisfy the other axioms. So it's a "non-Euclidean non-geometry."

Fair point.

What if it satisfies an as-of-yet undiscovered set of axioms?

The math term "geometry" has a certain definition. Things that fit it are geometries, things that don't, aren't.

The gee-whiz "what if" sort of question your asking doesn't really mean anything anymore. In about the mid-20th century math finally came to terms with basically anything you can define being a valid subject of investigation. Arguments about whether things like "imaginary numbers" are "real" and therefore worthy of study are largely gone now. (Whether they are "real" for some definition is now a philosophy question.)

Since axioms define the system you are building, if you want "Euclidean except plus portals" you just need to write them down and start studying them. There are multiple possible sets that describe different systems; for a trivial example, you can assert in an axiom that area is conserved on passing through a portal, or you can write axioms that don't conserve that, and follow the implications from there. Video games have had both kinds of portals. Do portals need to be straight or can they be curved? What do curved portals do to other curves if they pass through them? Does "passing through them" even mean anything (note Euclidean geometry doesn't have "time" in it)? How crazy can I be with the portals? What happens if I define a portal as having a boundary corresponding to a Cantor set [1] and pass a line through it in an axiom system that has "time"? There is no one answer to that question, it depends on the axioms you write down, which may or may not even permit such portals.

Mathematicians will accept any and all of these systems if you write them down. Some may prove to be "uninteresting". Some may prove to have contradictions in them. But it's been a long time since a mathematician would even consider berating you about any of those choices not being "realistic" or something.

And I expect it is very likely some topologist somewhere could hear all these idle musings of mine and say "Oh, yes, you want to go to $TOPOLOGY_SUBDISCIPLINE for that." I don't know what the subdiscipline is or I'd name it, but I'm sure there's something already out there for all this. Since the mathematicians stopped worrying about "realness" they've really spread their wings as as discipline.

[1]: https://en.wikipedia.org/wiki/Cantor_set

Rather than including “portal” as a kind of thing in your space, I would think to just say, “around each point, there is an open ball such that the metric (not the metric tensor, just the d(x,y) thing) is the same as a ball in Euclidean space”.

Oh, and then I guess you could add requirements that for any continuous path along with a, continuous choice of an orthonormal basis of the tangent space, along that path, that uh, transporting small volumes along it, not change the volume?

It can be given an as-of-yet unchosen new name :)

Axioms are not discovered but decided. A nice set of axioms gives birth to interesting anduseful interactions which you are the one that you discover, although in practice there's is a bit of back-and-forth, as you decides the axiom based on what you are trying to achieve and they get changed or tuned as time pass.

Bonus point if the new axioms play well with existing ones, and are useful to work on our reality.

> If not that would leave only "a straight line segment can be prolonged indefinitely" as holding, and unlike in Euclidean geometry a straight line segment may intersect itself arbitrarily often.

This one might not hold either, depending on your point of view. If you take the point of view of someone traveling along a line, that can still go on forever.

But if you're measuring the line, portals can easily prevent it from extending indefinitely, by wrapping it around to an earlier part of itself. It would then fail to be the case that, for any distance, there are two points on the line separated by at least that distance.

I'm surprised nobody has brought up Antichamber yet. It was an excellent game using a similar concept to create actual puzzles.

Come to think of it, long enough has probably passed that I might enjoy replaying it.

That's a great game, but it only uses portals to locally break the rules of geometry. The phrase "non-euclidean geometry" normally means either hyperbolic of elliptic (spherical) geometry.

If you like that, take a look at Viewfinder too: https://store.steampowered.com/app/1382070/Viewfinder/

You take pictures to create portals to other parts of the level, but in any orientation/angle that you want.

Superliminal is also good. (https://store.steampowered.com/app/1049410/Superliminal/)

If you play it, though, expect something that's not a challenge but more just an interesting presentation of some ideas.

It's made to make you feel smart when you solve the puzzles while not actually being really difficult, which is fine if you only play it once.

The writing is too wordy though, which is an issue when translated - I've watched some Japanese streamers play it and the subtitles try to keep every single idea in there, making them so long they're a headache to read.

Also consider Manifold Garden.

Man I spent so many happily stoned hours digging into that game. Sometimes I’d get stuck and just spend some time throwing myself off the edge of a cliff face and falling into eternity, watching the fractal platforms just unfold around me, until I felt like nudge my course back over to solid ground and resume puzzle solving.

This looks like portals. Essentially how FPS games of that era worked is that they had rooms stitched together by doorways (or 'portals') as a way of occlusion culling. While perfectly capable of representing sensible geometry, it was actually not a requirement that the geometry should make sense in any shape or form and allowed arbitrary connections between rooms.

The Marathon 1+2 engine back in the 90s allowed for "5D space", ie. "Spatial" Rooms that were capable of overlapping each other, because the engine only defined the floor polygon with a given height and these floor polys were connected to each other at the vertices to create the actual level spaces.

So even though you were in the same absolute space coordinate as another player, you were not necessarily in the same room.

Good times. <:o)

Huh I never knew that. I just knew you couldn’t have two floors directly above each other while also having both floors visible to the player at the same time. All of my Mara 1 maps I built would crash when I tried this. I wonder if what you just explained is why.

I don't know how they actually did it but conceptually I thought about it as traversing the map's segments depth first in 2D starting at one edge of the viewport & the currently occupied polygon. Simple, and allows freedom like overlapping rooms, but would go sideways in cases like the one you describe.

Doom actually didn't permit overlapping rooms like Duke3d and some other games did, although some source ports of Doom do allow it (notably used by the trendy MyHouse.wad)

Some vanilla Doom levels get exceptionally close to simulating it using portals and other tricks.

And bridges and such, but I'd put the line on Boom compatible engines.

FPS games of that era were usually either in the portals camp or the BSP camp, BSP being the "OG" way to do it precisely because Doom did it that way.

That is exactly what I expected as well. Here's another non-Euclidian implementation with a couple of other novel ideas (at least they were to me): https://www.youtube.com/watch?v=tl40xidKF-4

I expected MyHouse.WAD https://www.youtube.com/watch?v=5wAo54DHDY0

Here's another example of a "non-euclidean" engine/game, which in my opinion is more authentically non-euclidian: https://youtu.be/tl40xidKF-4