Non-Euclidean Doom: what happens to a game when pi is not 3.14159 (2022) [video] | Hacker News Reader

Non-Euclidean Doom: what happens to a game when pi is not 3.14159 (2022) [video]

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607 pointsrobin_reala170 comments

There was an example of this in the classic 'Duke Nukem 3d'. It had a level by Richard "Levelord" Gray, 'Lunatic Fringe'.

https://dukenukem.fandom.com/wiki/Lunatic_Fringe

This level had a circular hallway ring around the outside that had two full rotations around without intersecting, using the 'build' engine's ability to separate areas by their room connections that also drove the 'room over room' technology which was groundbreaking at the time.

It made for fun multiplayer, and the illusion held well there. The central chamber has 4 entrances/exits if I recall, and you would only encounter two of them in each loop around the outside.

I recall building a toy level while experimenting with the engine that "solves" the "3 houses with 3 utilities without crossing" puzzle using this trick as well.

Room over room (as done in later build games) usually does not require the sectors to overlap in this manner as it is a different thing (it is extension of how swimable water works, where the engine renders the other sector instead of floor/ceiling).

While Lunatic Fringe is pretty in-your-face example of impossible geometry in build, the duke3d maps contain many more cases of intersecting geometry. Obviously such things are impossible in Doom because there is no way to build BSP tree out of that and because doom only tracks X/Y coordinates of player(s)/monsters.

> only tracks X/Y coordinates of player(s)/monsters.

It does, otherwise Lost Souls and Cacodemons wouldn't able to fly.

I meant that as in contrast to build tracking the sector the thing is in. Due to that you can have two sectors that intersect not only in 2D, but even in 3D and the engine still does the right thing (ignoring the fact that the renderer gets slightly confused when there are overlapping sectors in the view)

Is there no way of building a BSP out of it? I don't see why it has to be Euclidean to be partitioned, and loops are also possible (e.g. Deimos Lab). (The coordinates are definitely a blocker, so this question is academic.)

The “sectors” in the resulting level data have to be convex for the doom engine. The “asset pipeline” handles that by breaking up non-convex geometry into smaller convex sectors. So, there are no loops or holes in the actual level data, also from the cursory glance both of the large loops in Deimos Lab are actually not complete loops, but they have a place where the loop is broken. But that does not matter that much, as almost any level contains geometry that is either concave or has a hole (courtyard in E1M1, both large rooms in MAP01…)

The Doom design starts with X,Y,Z and then finds your BSP node. You can't warp to a different X,Y,Z without some type of warp portal which doesn't exist in Doom. Modern Doom forks have this feature and also have graphics portals so you can implement this (as MyHouse.pk3 does).

Portals allow weird stuff (non linear geometry) in a BSP level. I thought Doom had petals.

The primary thing that BSP does is that it maps coordinates onto a BSP node/sector. Thus you cannot have overlapping geometry, as this mapping would not be unique. Quake has some idea of portals (I'm not sure about Doom), but it is used only as an additional layer of optimization, the engine is not fundamentally portal-based.

There is a very interesting Doom level named “myhouse.wad” that does a lot of clever things to seemingly allow room-over-room.

I don't think that mod works on a normal Doom engine.

No mention of the real Prey (2006) in a discussion of weird geometry in games?

I was truly amazed by this game! You don’t see anything like that elsewhere

The portals in prey were cool but, definitely the games Portal and Portal 2 are places you could see it elsewhere. Also in portal you can actually choose where the portals are, whereas Prey's portals were fixed in place

Sure you can do this in doom, or at least it can be done in modern source ports.

https://www.youtube.com/watch?v=Iq1-TZXz9xo (myhouse.wad)

There is nothing about the data structures ID choose to use in doom that prevent portal shenanigans. ID(John Carmack) just did not implement them.

Modern source ports are a totally different thing. If any, try to do what the former comment states with a Boom compatible engine as Crispy Doom or Chocolate Doom.

You're confusing things a bit here, Boom is already an enhanced port and it got the kind of line-to-line teleport that allows you to do this.

Crispy and Chocolate Doom are what the community calls "vanilla" Doom, with no enhancements at all.

That said, if you are willing to hex-edit the map data, you can have non-euclidean geometry in vanilla Doom: https://doomwiki.org/wiki/Linguortal

Crispy Doom will play most Boom compatible levels just fine. I tried it with some Megawads, and I had no issues, even with FreeDoom as the IWAD.

No, I'm not confused. To me, Boom it's pretty close to vanilla. Zdoom and the rest are pretty much another league, closer to Quake/HL than Doom.

> the 'room over room' technology which was groundbreaking at the time.

Bungie's Marathon from 1994 could also do this and demonstrated it in the deathmatch map 5-D Space[1]. It was really only Doom that bugged out on overlapping sectors.

[1] https://www.lhowon.org/level/marathon/30

> also drove the 'room over room' technology which was groundbreaking at the time.

Descent did this in 1994.

Descent was a polygonal game though, Duke3D was a raycasting engine. It was the first four a raycasting engine, and quite impressive.

Duke3D isn't a raycaster; it's a portal-based quad rasterizer. Wall quads are forced vertical, and floor quads are rasterized using a neat trick. The renderer computes successive lines of constant distance along slooped floor surfaces, then draws textures using affine scaling, which is correct along those lines. The portal walls clip the draw of successive sectors, which solves the hidden surface problem.

https://fabiensanglard.net/duke3d/build_engine_internals.php

I remember a Descent PVP level that was a big room with a corridor that came off of one end and went to the other by looping back through the main room but was not visible inside of the main room. Was a bit of a mindbender in a game that already stretched the players spatial awareness.

In what sense is the Duke Nukem thing "an example of this"? The duke thing is an internally consistent programmed behaviour, this is... just random errors caused by a random change in the source code. Duke is maybe non-euclidean geometry, or something. This doom pi thing is... nothing to do with geometry. More an example of "garbage in, garbage out" maybe.

It's an example of "non-euclidean" space, and yes, it is a bit different than the article.

The DN example would be an instance of non-Euclidean topology whereas the other one is presumably non-Euclidean geometry.

This video does a good job explaining it, yes it really is quite mind-bending.

https://www.youtube.com/watch?v=UitzmhJe578

I wrote a little engine with this capability a long time ago. I didn't know about the Build engine at the time. I divided the world into convex sectors and allowed arbitrary links (portals) instead of following a BSP tree. Rendering is front-to-back and clips at the portal boundary.

If you can rasterize the inside of a convex shape, you can rasterize a sector-portal-world by marking certain faces as portals, and when rendering one, setting the clipping region (or stencil buffer) to where the face would be, then rendering the sector behind it with an appropriate transformation (which is part of the portal's data, alongside the ID of the other sector).

Collisions through portals are much harder than rendering through portals.

> the "3 houses with 3 utilities without crossing" puzzle

In graph theory's terminology, this is "K3,3", one of two irreducible nonplanar graphs. The other one is K5.

You can also make all the connections without crossing any edges if you embed the graph in a torus, which is equivalent to building a bridge over some set of edges that other edges are allowed to take.

There is an interesting VR game called "Tea For God" that puts you in your play area, and cleverly creates new corridors and rooms as you make it around the corner or lift so there is an illusion that you are in a very vast place, despite never leaving the same room, all without using the joystick or teleporting.

I spent so much time back in the day working with build. In many ways it was super easy (and fun!) to use.

Thanks for the throwback!

Funny I was just reading this Poul Anderson classic

https://en.wikipedia.org/wiki/Operation_Chaos_(novel)

which is set in an parallel world where magic is real and advancing rapidly along with science. (e.g. Edwin Land invents a device to transform a werewolf without moonlight using polarized light)

The protagonists have their child kidnapped and brought to Hell. They are told that the Army had tried to investigate Hell 20 years ago but everyone was driven insane. An adversary drops the hint that the space-time geometry of Hell is different from ours so it would be possible to go back in time and retrieve the child at the moment it arrives. This is enough of a hint for scientists to determine that the geometry of Hell is non-Euclidean and figure out spells for getting there safely, keeping safe, and getting back. They pray to get the assistance of two 19th century geometers, one a saint, to be able to navigate.

See also Jack Vance's wonderful story collection Tales of the Dying Earth, which also includes magic/science and daemon dimensions.

This might come off as trite (and it probably is) but why don't we get novels this creative nowadays?

I liked this one

https://www.amazon.com/Terraformers-Annalee-Newitz/dp/125022...

Honestly sounds like you're just disconnected from the culture. There are plenty of creative novels coming out these days. For a specific example check out Adrian Tchaikovsky's work.

Also you may want to subscribe to a short story publication like Clarkesworld. You'll have a drip feed of aspiring creative writers and can follow the ones you find especially good for when their forthcoming novels come out.

One that especially blew me away from Clarkesworld recently was Axiom of Dreams by Arula Ratnakar.

https://clarkesworldmagazine.com/ratnakar_09_23/

Have you actually read recent SF novels? Think what you will about awards, but I've never been exceedingly disappointed when I read a novel that won a couple out of Hugo, Locus, and Nebula. I'd suggest you start there.

Laundry Files just recently had some novels that involves cool weird dimensions.

The last Bobiverse book was about infiltrating a spaceship that is an artificial world.

Heck Bobiverse in general has some cool ideas.

The Andrea Vernon books are a hilarious take on the superhero genre.

The Bobiverse books are great IMHO.

I've read them and somewhat disagree. The protaganist(s) in the book almost never choose a sensible course of action for their goals, even given elapsed decades in which to make them. If I can come up with much better plans on the fly while reading, the main character surely could in decades.

The main character is not perfect, he has several major personality flaws that break through and screw up his actions again and again.

I think taking every thing into account, his actions are consistent with his personality.

That said, the first book's exploration of combat in space was, IMHO, some of the best Sci-Fi work in the series. Although I get why the books gave up on even trying to be hard sci-fi at all early on, science was pushed out of the way to make writing the story easier.

Stories with creative magic systems are still being produced. Check out Unsong, by Scott Alexander. It's about a world that's programmable by Hebrew and Kabbalah.

https://en.wikipedia.org/wiki/Diaspora_(novel) and of course other Greg Egan works

In particular Dicronaughts is quite mind-bending, with its two time dimensions.

I wonder that as well. And the same for movies, games, etc.

I think in order to "play it safe" against the 1% of people who cry over the dumbest shit, but who happen to screech the loudest, creators make everything about someone's feelings, edge case stuff you hardly see represented in day to day. You weave that web enough and you end up with the "creative" output of something like Gemini lol.

Huh? I would imagine that content that is about people’s feelings and the things those feelings are about, would be more likely to offend? Who gets offended by content about like, “How navigation would be different if space were like X?” or, “How would sheet music for music in a world with two time dimension look? Might we have music where along one direction the frequency is increasing in magnitude, but decreasing along another, played to depict a scene in which a pursuer gets closer to a target along one direction of time but further along the other?” ?

(Though, I’m not sure if two time dimensions exactly makes sense, because like, R\{0} has two connected components, which seems connected to the distinction between the past and the future, while R^2\{0} is connected (though not simply-connected) , so I’m not sure if the distinction between past and future, can still be made? Maybe if instead of a 0-dimensional point we have a 1-dimensional curve? )

As I imagine it, for the most part, the further something is from human experience, the less it would tend to remind people of whatever things in life they find objectionable.

Because in trying to tell that kind of interesting story today, the creators would force the current day politics and other bullshit into it and make that the front and center instead of the cool math or science stuff.

If even simple children's stories can't escape it with the remakes, forget it when it comes to something interesting.

I'm always happy when there are exceptions to this of course, but it is quite rare. One of my goals if I ever make it to the 9 figure club is to start a media company based outside the US where we can make great content about interesting concepts like this, without injecting politics into it or having to work with people who inject nonsense into everything. (on a side note, this is why I enjoy working with E. Euro and S. American engineers - they just focus on working and problem solving, not feelings and shit based on half truths they made up from taking drugs or whatever)

Anyway I was just responding to why stuff like that isn't made anymore. That's my opinion.

> Because in trying to tell that kind of interesting story today, the creators would force the current day politics and other bullshit into it and make that the front and center instead of the cool math or science stuff.

That's always been the case. If you look at Star Trek, the TOS Klingons were a very thinly veiled reference to the Soviet Union, the Federation was the USA (and in some cases not even thinly veiled, they had an episode where Kirk was "yo, sounds like you're quoting the constitution"); the whole of The Undiscovered Country was chock full of similarly themed memes with Praxis being Chernobyl, "Shakespeare is best in the original Klingon" was the kind of thing that the Soviet Union was famous for doing[0] with all the culture they liked, and with Rura Penthe being a very obvious reference to a Siberian Gulag.

TNG comes around, over a thousand people on the ship, statistically speaking 50 or so are probably gay… none are mentioned because that was just before "don't ask, don't tell" was introduced, which itself was a mild improvement on "we will discharge you for this". Even the attempt at an LGBT episode (the planet which had eliminated gender and brainwashed anyone who still had one) only got the T, and not the LGB parts.

DS9's handling of the Bajorans feels to me like it was riffing on then-current US attitudes towards the Mujahideen (who were still broadly seen as a positive because they had kicked the Soviets out of Afghanistan and 9/11 had not yet happened) and the Provisional IRA (who were, when the show began, still engaged in armed conflict with the government of the UK).

Enterprise's Archer felt like Bush Jr, and the Xindi arc felt like it was done as a reference to the 9/11 attacks (I don't know how much of a delay there is between scriptwriters experiencing real life and the result hitting the airwaves).

Every show you've ever seen has something like this, just to get distributed. But these days, we've got more options for distribution, so you're not limited to your own country's Overton Window for "what counts as art and not filth that corrupts the minds of our youth", so you can much more easily access things that those around you would revile you for — be that smoking tobacco or marijuana, or depicting casual consumption of pork or beef like those aren't حَرَام or अघ्न्या (in Islamic and Hindu cultures respectively), or which you are upset more about out of nipples and guns.

[0] https://tvtropes.org/pmwiki/pmwiki.php/Main/InTheOriginalKli...

Read early Philip K Dick. Lots of female homemakers and male breadwinners, and other 50s mores. Flying robot taxis, but the lady of the house wears a pinafore. It hasn't aged well.

Culture warriors are people with a dogmatic attitude to the social realm (and who shout about it - but that's redundant; all dogmatics are hegemonists).

SF is fundamentally iconoclastic.

What is a culture warrior doing reading SF, if you don't mind explaining?

Are you complaining about gay characters existing or something? Cause if you are then aaa lmao

You do, but then you get games like Sunless Sea which are praised by critics but don't sell millions

I loved Operation Chaos, but I found the much later sequel to be disappointing.

The space-time geometry of our world is not Euclidian, not even Riemannian. The story of "Operation Chaos" looks more chaotic than scientific. That fits with the books of Poul Anderson I've read, like the excellent High Crusade: he likes going wild and doesn't care much for the scientific pretense.

For another take on non-Euclidian geometries, I have good memories of Inverted World by Christopher Priest.

I read a lot of sci-fi as a kid but really never got into Anderson despite there being a lot of Anderson books at the library but I do remember enjoying Tau Zero (a better story about Bussard Ramjets than any of Niven's... And funny Niven was my favorite author when I was 10 and I've got much less regard for him now, the Pak really spoil Known Space for me)

I wound up re-reading Tau Zero and was really blown away which got me reading the Ensign Flandry books, Corridors of Time, Star Fox, etc. I don't agree with his politics any more than I agree with Niven or Heinlein but enjoy his books anyway. I'm planning to read all the Anderson books at my library.

Heinlein wrote a few crossover books like "Glory Road" and "Magic Inc" but those were the exception for Heinlein and closer to the rule for Anderson because Anderson was just as important as a fantasy writer as a sci-fi writer.

I do have a copy of The Inverted World that I ought to read.

> the Pak really spoil Known Space for me

Out of curiosity: why? They are rather silly, but I'm mostly thinking about their implausible evolutionary history when I say that.

That sounds awesome.

John Carmack may have misremembered the 10th digit of pi, but you should all search your codebases for 84,600 and see who typo'd the number of seconds in a day. It is surprisingly common and there is a lesson about whether or not you should type constants into your program when they likely exist in the standard library of your programming language.

If you can't do multiplication and you rely on a library for that. I feel sorry for you.

Magic numbers something something. If your language provides it as a constant, why not use that? You’ll always get the correct value AND a descriptive name for free.

It's really a matter of who's going to catch your typos. If you use the constant built into your language, millions of people are looking at it. If you type them into your throwaway bash script, nobody is going to notice you typed 60 * 60 * 24 * 356 to get a year's worth of seconds.

So meta that you typed 356 instead of 365

That was their point.

If I have it I’d use it, but I’m not going to go out of my way to add another dependency for it lol.

*the number of seconds in most days

Depends what planet you're on too, I suppose.

earth has leap seconds

But those don't increase the number of seconds in the day. Instead, the final second happens twice.

This is different from what happens in leap year, where February is actually lengthened and February 29 is considered a different day from February 28. It escapes me why leap seconds aren't treated that way. (I know, people say the stupid treatment of leap seconds is required by the POSIX definition of time. So what? Change the POSIX definition.)

> But those don't increase the number of seconds in the day. Instead, the final second happens twice.

i agree that this is a rather confusing way of putting it

> But those don't increase the number of seconds in the day. Instead, the final second happens twice

No, it adds a 61th second to the last minute.

UNIX timestamps happen twice, because it's broken. NTP too.

I spent an embarrassing amount of time debugging why some token refresh code that was supposed to trigger every 15 minutes didn't. Eventually I had to just print out the value for each variable and found that I defined 15 minutes as 15 * 60 * 60...

I like defining constants with math.

I’ve encountered similar issues for long lived tokens in cookies on Safari.

Turned out my Maths was okay, but Safari under many circumstances removes first party cookies after 7 days.

https://snowplow.io/blog/tracking-cookies-length/

I find it easier to do something like:

SECONDS_IN_MINUTE = 60

REFRESH_TIME_MINUTES = 15

REFRESH_TIME_SECONDS = SECONDS_IN_MINUTE * REFRESH_TIME_MINUTES

e.g. I define every constant separately, and ideally any constant defined by an operation will only have one such operation. This makes mistakes _much_ easier to spot.

    new TimeSpan(minutes = 15)

Depending on the language, this is definitely preferable!

Oh the places you'll go

When you roll your own date

You'll think you are great

'Til you're 6 minutes slow!

Oh wow, you weren't kidding. Even big projects like MySQL and Textmate look like they might have that error in their codebase.

https://github.com/search?type=code&auto_enroll=true&q=84600

https://github.com/mysql/mysql-server/blob/824e2b4064053f7da...

https://github.com/textmate/textmate/blob/346b52b108b387462d...

That is so smart to try it on Github.

None of those look fatal but I think they all differ from the intent of the author. That's why code review is so good. You just need someone to ask "what's 874600?". Turns out: not a thing.

MySQL's issue is a timeout on SSL sessions which is normally 300s. Textmate's is a "friendly" display of the time. Not a showstopper, but probably not what they had in mind. It's so easy to transpose digits. It's great to have someone ask "are you sure?".

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

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

Woah! Thanks for sharing, that video is amazing, I'd love to see an actual game using that effect.

https://en.wikipedia.org/wiki/HyperRogue

(not exactly the effect in the video you responded to, but similar category)

That'd pretty much be Portal; the technique he's using to render there is basically the same as the one Portal uses (the Portal engine can do "invisible" portals like this too, but there's only one point across both games where they're used in the finished game).

I suppose something around non-Euclidean levels without user-created portals could be interesting, but I think it'd be hard to flesh that out to a point where there's enough there for a whole game without coming up with some other gimmick.

I know it was left as an exercise for the reader, but I never worked out where that was used. (The turret live fire exercise comes to mind, but I thought I'd ruled that out.) Do you happen to know?

Remembered the entity name, and this information is on the wiki page: https://developer.valvesoftware.com/wiki/Linked_portal_door

> They are also used twice in the campaign (contrary to the commentary's claim that they are only used once): in the GLaDOS wakeup sequence where they are used to connect the incinerator shaft to GLaDOS' chamber, and in Finale 2 where they are used to connect the "trap" chamber to the main map. These are the only uses of this entity in the final game.

Antichamber: https://store.steampowered.com/app/219890/Antichamber/

Also look at myhouse.wad for actual Doom in actual non-Euclidean spaces.

Hyperbolica exists too, and is pretty fun even if I haven't finished it yet. it has hyperbolic and spherical geometry in it at least.

The only game to give me motion sickness by watching a streamer playing it, and no motion sickness at all playing the VR version. Very odd.

If you like this, try Superliminal (https://store.steampowered.com/app/1049410/Superliminal/)

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.

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.

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/

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.

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