Post by Zeno Rogue in AAAAXY comments

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Seems you are confusing geometry and topology here and in the description on GitHub... (geometry is a local property that is changed when you stretch the space and topology is changed when you cut/glue the space -- portals change the topology, not the geometry; in the "sphere" level you have the topology of the sphere but the geometry is Euclidean, not spherical. The geometry is Euclidean everywhere, at least from what I have seen so far)

Also not sure why you say it is "the universal cover of a massively twisted space"? This does not seem to be accurate either. (From this description I would expect a world in the shape of e.g. an annulus that always looks like an annulus, but the contents are different when you go around the hole -- this would be the cover of the annulus, and if you never go back no matter how many times you go around, that would be the universal cover (it is more fun with two holes as you get the full binary tell then); here, the walls also sometimes all change when you go around a hole)

A bit similar 2D portals were used in the 7DRLs by Jeff Lait (Jacob's Matrix was the first one IIRC and Vicious Orcs was another one), although he did not do that much with this concept.

divVerent2 years ago (1 edit)

To be clear, the universal cover property is mostly true; there are a few places where portals are enabled/disabled at runtime (in particular the places where one goes back and something changes in the previous place), and thus the topology is actually changed. In most cases the property is actually true, and things actually do change when you go around the hole (e.g. if the annulus contains a platform, you would be able to hold on to this platform and get a second one when you get to the “same” place again, and the existence of both objects together proves that the old and new place is not fully identical).

But you are right, the proper word, to be exact, is topology, although “non-euclidean topology” isn’t quite the right term either. https://hyperbolica.fandom.com/wiki/Fake_Non-Euclidean_Geometry is a quite good description.

HOWEVER, it isn’t Euclidean geometry either - e.g.

A list of things that have been mistakenly claimed as violating Euclid’s 5th postulate “for any line A and point B, there is exactly one unique line that crosses B but not A” is here:

is actually violated a few times accidentally, although not by every use of portals. Furthermore, from Euclid’s axioms, it follows that a line segment is the shortest path between two points, which is absolutely untrue in all “R^n with portals” worlds due to a trivial counterexample.

Feedback taken though, I will look for a better way to describe this. Thanks!

divVerent2 years ago (1 edit)

One issue here is the lack of a generally accepted and mathematically correct term.

It appears that the common term in game development has been “non-euclidean geometry”, even though it is often just “accidentally” violating Euclid’s axioms, and only in few places. The term “impossible spaces” appears to be used too and would be more accurate, but is not as commonly used for these games. I think I will try adjusting to that, as impossible spaces are more the focus than some accidental violations of the parallel axiom.

https://zenorogue.medium.com/non-euclidean-geometry-and-games-fb46989320d4 (which, I know, is your site)

Zeno Rogue2 years ago (3 edits)

I see the point about the platform returning to the same location after going around the hole, it is indeed a bit similar to would happen in the universal cover. However, IIRC if you move the platform A, go around the hole to find a yet unmoved platform, and then go back, the platform A also becomes unmoved, so it is still not really the same thing...

I would say that the parallel axiom is not violated (the spirit of this axiom is that parallel lines behave weirdly, they converge/diverge, or in 3D they could also twist -- since the gravity is preserved in most of the game, the gravity lines act like normal parallel lines, same with the lines orthogonal to them.). The only place I have found so far where gravity lines cross (in some sense) is the "sphere" level.

> Furthermore, from Euclid’s axioms, it follows that a line segment is the shortest path between two points, which is absolutely untrue in all “R^n with portals” worlds due to a trivial counterexample.

I would not agree with this, because by this logic, any L-shaped level is non-Euclidean because the shortest path is not straight.

It is still piecewise straight, so we should allow for piecewise straight lines here, and in portal spaces shortest paths are again piecewise straight lines (well, unless the obstacles are curved).

Also this is not really a property of non-Euclidean geometry -- shortest lines are straight lines in all classic non-Euclidean geometries.

(again, in both cases the weirdness happens due to topological rather than geometrical effects)

I do not think "non-Euclidean geometry" is that common, mostly they say just "non-Euclidean" without geometry (one thing contributing to the confusion is CodeParade's "non-Euclidean worlds" viral video, he thinks that "non-Euclidean geometry" should not be used for any weird space but just "non-Euclidean" is fine -- this makes some sense but it still confuses people so I do not agree with him). "Impossible space" is what most people who care say. Thanks for considering this!

divVerent2 years ago (5 edits)

I would not agree with this, because by this logic, any L-shaped level is non-Euclidean because the shortest path is not straight.

In fact, any bounded “space” already violates Euclid’s axioms, as you can e.g. find two lines that are not parallel in the “infinite extension”, yet nowhere cross within the space. So just reducing the plane to a rectangle’s contents already is breaking Euclid’s axioms.

I think the issue is different ways of interpreting “non-Euclidean” - a mathematician typically means by non-Euclidean geometry not anything that violates some of Euclid’s axioms, but something that violates them in a rather specific way: only the parallel axiom, and in a rather “deterministic” way (i.e. either there are no parallels, or infinitely many, no matter which point and line one starts from). However, it is quite clear that most game worlds violate Euclid’s axioms in some ways or another, thereby it is technically not wrong to call them non-Euclidean. Yes, even Quake… with its floating point inaccuracies making Euclid rotate in his grave :)

BTW, as for gravity lines crossing - this is even already the case in the very first room of the game.

However, IIRC if you move the platform A, go around the hole to find a yet unmoved platform, and then go back, the platform A also becomes unmoved, so it is still not really the same thing…

This comes from the rule of the game “if you can’t see it, it does not exist”, which is derived from the same rule in Super Mario games. I.e. the moment you take the platform A out of view, it gets destroyed, and if you come back to its spawn location, it reappears there. Just like a Goomba which will reappear even if you allowed it to drop down into a pit once you leave and re-enter the section.

Implementation wise, doing the portals in the tile loader means the game code already “sees” only the universal cover, meaning everything in game is a standard 2D tile map based game, and e.g. distance functions work normally. So it could be seen as an Euclidean game with above visibility rule working on the universal cover and thus an infinite world - or maybe a tiny world that is built and destroyed around you as you go. To the player, and the way the levels are designed, it behaves more like a typical portals based game, except that you will never see yourself or duplicates of the same object at once.

Zeno Rogue2 years ago

Yeah, usually "negative" qualifiers in mathematics mean some specific things. "Irrational number" does not mean numbers like i or Aleph-Zero even though they are not rational numbers. It means a *real* number that is not rational. Discovery of such numbers was quite important in math. Same as with non-Euclidean geometry, the discovery that all the axioms could be satisfied but not the fifth one was important again. So it originally meant just that, not satisfy the 5th but satisfy all the others. Some people extend this but they usually try to keep the spirit (hard to explain Nil geometry using axioms, but the spirit of parallel axiom is no longer there...). Sometimes "non-X Y" is meant to include "X Y" as a special case :)

Anyway, thanks for the nice game! I have added a mention to my article.

divVerent2 years ago (1 edit)

Thanks!

BTW, as for the teleportation trick - both Antichamber and AAAAXY could not really have been done with the teleportation trick alone, as objects (e.g. Antichamber’s matter gun blocks) remain available when going through a portal. In a few places a player can even clearly notice that the game isn’t just teleporting those blocks, too.

However, the first part of the game would entirely have been possible with just seamless teleportation.

BTW, Super Mario Bros. 1 is probably one of the first games with impossible spaces, in two ways, so it may too be worth mentioning:

Pipes are not connected in a way consistent with layout on both sides (but arguably this could be understood as a long twisted pipe one just doesn’t see, so this may not really count - after all, maybe pipes are just teleporters).
Levels 4-4 and 7-4 are “puzzles” where one has to pick an upper/lower path, and picking the wrong path seamlessly transports one to the start of the “puzzle”. This appears to be done by seamless teleportation and even in a bad way, as Firebar objects suddenly stop working when scrolling past the “portal”.

divVerent2 years ago (1 edit)

Another game in impossible spaces maybe worth mentioning is Unreal Tournament ’99 (the GOTY one). A few included maps use portals to shorten/lengthen pathways to balance out gameplay on the map. This is hardly noticeable in-game though, so it differs from the use in other games - the player is not intended to ever notice.

Of course, some third-party maps totally exploit that feature :)

itch.io

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