Yeah, Jeff calls it "non-Cartesian". He changes the topology but the geometry is Euclidean. There are also lots of roguelikes doing other experiments with topology, tilings, dimensions (Verdagon 7DRLs, geometry episode of Roguelike Radio, etc.)

Wouldn't any geometry that violates any of Euclid's five postulates be non-Euclidean? Yes, the parallel lines one is the most famous, but surely a geometry that violates any of the other four would be too.

How would you violate one of the other four?

Postulate 1 defines what a line is.

Postulate 2 defines what a direction is.

Postulate 3 defines what a circle is.

Postulate 4, in modern terms, doesn't say anything at all. (We aren't even sure why the Greeks felt it was necessary to state it!)

Well, for 1, consider spherical geometry. In Euclidean geometry, there can be only one line between any two points. But consider the lines on a sphere between the antipodes (like lines of longitude on Earth) -- there are infinite lines of longitude that connect the North and South Poles, not just one.

That's postulate 5, parallel lines meeting. Postulate 1 says that a line can be drawn between any two points. Drawing more than one line between a certain pair of points doesn't contradict that. If you can draw two lines, you can draw one line.

No. Postulate 1 says that for any two points A and B, there is a line AB connecting them. While you could argue that "infinite lines" are a superset of "a line" it is pretty obvious that Euclid meant one and only one line (which is true in Euclidean geometry)

You don't seem comfortable with math.

You might observe that the way Euclid uses postulate 1 is to provide geometric constructions of shapes. If you want to argue about what he meant, what he meant was "you have a straightedge". Similarly, postulate 3 says "you have a compass", and postulate 2 says... "you have a straightedge".

You've described drawing several parallel lines that meet at two opposite points of a sphere. Postulate 1 has no problem with that - all of those lines are straight. Postulate 5 has a problem with it, because they meet.

If your geometry has portals, the parallel line postulate will no longer hold. If your portals can have different angles between the entrance and the exit you can straight up cross "parallel" lines, and even if you require them to be oriented at the same angle you'll lose "when given a line and a point exactly one parallel line can be drawn".

Euclidean geometry is an extremely precise geometry, and there is nothing wrong with calling any deviation from it "non-Euclidean". Since Euclidean geometry is a fairly specific point in geometry-space, the only real objection is that calling something "non-Euclidean" doesn't tell you much about what it is, but Euclidean geometry isn't a bad choice for a "default" geometry and at least it tells you it's not that. There is no requirement for it to be continuously curved in some spherical or hyperbolic manner to be "non-Euclidean". Even just permitting a single portal into an otherwise Euclidean geometry would utterly shred Euclid's Elements from top to bottom.

By my examination of the original postulates, not only would portals wreck the parallel line postulate, it in fact ruins 4 out of 5 of them, leaving only that a line can be continuously extended (and that line can still self-intersect, which may in fact violate the definition of a "line" and mean all 5 axioms are busted if you really dug in). It means it is no longer possible to draw a line between any two points (portals can shadow portions of the space such that some points are not connectible), cicles lose a lot of their properties that we rely on (imagine the unit circle, and a portal starting x = .5 and going straight up... they're no longer necessarily continuous), right angles may not be equal to each other any more (portals can change the apparent angles, or give two interpretations rather than one), and of course the parallel line doesn't hold. Putting portals into a geometry fairly comprehensively makes it non-Euclidean.

> It means it is no longer possible to draw a line between any two points (portals can shadow portions of the space such that some points are not connectible

This feels a bit like saying that putting a wall in a room makes it non-euclidean since there's now a barrier in the way.

I know where you're coming from, but this is confusing geometry and topology. (Curvature vs. how the space is connected.)

> This feels a bit like saying that putting a wall in a room makes it non-euclidean since there's now a barrier in the way.

Maybe that's what H. P. Lovecraft meant about non-Euclidean architecture in R'lyeh. ;D

(It probably isn't - the quote from The Call of Cthulhu says "the geometry of the dream-place he saw was abnormal, non-Euclidean, and loathsomely redolent of spheres [...]".)

If you define "a line" as "blocked by another line (standing in for a wall)", it would be non-Euclidean, by virtue of breaking the axiom that says a line can be infinite extended, so that's not a particularly compelling argument.

I don't need to "confuse" geometry and topology. Euclid's axioms require both flat geometry and no "connections" in the space. It's right there in the axioms, if you can read them properly. They have no accommodations for "topology", and it is certainly not just a handwave "oh, whatever, it's no big deal" to extend them to handle it.

By that logic, the game Portal is non-Euclidean. Which might be true in a certain interpretation. But nobody looks at portal and thinks it’s anything but a normal 3D world with portals.

Here the geometry is very unusual. But using the existence of portals to define that seems insincere.

It is non-Euclidean. 4 out of 5, if not 5 out of 5, of the postulates that define Euclidean space fail. It can't fail any harder.

This isn't a matter of emotion or feeling like "oh, gee, I don't want to, like, demote the space or make it feel bad for calling it 'non-Euclidean'." This is math. The postulates won't work. The proofs won't hold. Even a single portal in the space make it non-Euclidean. It doesn't fall down a little bit. It doesn't "mostly hold, you know, except for a few exceptions". It breaks the axioms from top to bottom.

I think we both love Math. But I respectfully disagree. These are games. In a game like Portal, standard intuition about geometry mostly applies, as long as there isn’t a portal in the portion of space you’re considering. The boundaries between reality and magic are what make it interesting. And the fact that it’s mostly normal but sometimes not is exactly what makes it interesting. Because for the player, it doesn’t matter if the proofs are invalid, their intuition mostly holds.

In HyperRogue, normal Earth intuition is mostly inapplicable because of the hyperbolic geometry. The fact that it has portals and therefore proofs fail shouldn’t IMHO be the defining features of the world. Unless you think the key audience of the game is topologists or something, but if you care about non-mathematicians getting into the game, I’d say focus on intuition not axioms.

Considered in three dimensions, I would guess that most Rogue-likes (in a narrow, classical sense) are non-Euclidean (in that broad sense): the down stairs on one level don't necessarily line up with the up stairs on the next.

At least in Angband, the game explicitly tells you that you pass through a maze of staircases. The level you just left still notionally exists, but you can never find it again.

(This is also true of the recall effect, for no reason stated by the ingame text. But then again, the ingame text never claims anything other than that you'll recall to a particular depth, as opposed to a particular place.)

No, you are confusing geometry and topology here. Topology does not change when you stretch the space but changes when you cut/glue it. Geometry does not change when you cut/glue but changes when you stretch. So portals change the topology but the geometry is Euclidean (you get a Euclidean manifold). This are the meanings used by mathematicians working in these areas.

(The original meaning is of course about breaking 5th postulate while all the others hold, showing that it was possible was a celebrated result in mathematics, while it is trivial to break the postulates in some arbitrary way.)

The modern meaning of "geometry" may not change, but cutting and gluing space definitely break Euclidean geometry, as in, specifically the one defined by Euclid. You can't break a mathematical system much harder than to kill it at the axiomatic level. 4 out of 5, if not 5 out of 5, axioms do not hold if you include portals. That's pretty dead.

For an analogy: an Abelian group is a structure in which four axioms hold. A non-Abelian group is a structure in which three of these axioms hold, and the fourth does not. It is not a structure in which some random proper subset of axioms holds, because such a notion would be useless. While structures where specific subsets of axioms hold (non-Abelian groups, semigroups, monoids, etc.) are useful.