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by MoltenMan 221 days ago
Super Mario Galaxy is not non- euclidean; it's just a regular 3D projection onto a 2D screen. The spheres you travel on are technically non-euclidean if you consider the surface in a vacuum, but you aren't playing on the surface, you're playing in a full 3D space.

Additionally, 2D sphere surfaces are very understandable because they can actually exist in our 3D world and we're used to them; hyperrogue has a hyperbolic non-euclidean 2D geometry which is impossible in real life, much more confusing, and impossible to display with no confusion. I don't know what Smart Kobold is, but I presume it's also not just a regular 3D game like Super Mario Galaxy.
1 comments
thaumasiotes 221 days ago
> but you aren't playing on the surface, you're playing in a full 3D space.

That is not true in any meaningful sense. You can jump. But you can't move through the 3D space. You're stuck to the surface of the closest sphere. All of your pathfinding problems are non-Euclidean pathfinding problems.

If you were a sprite embedded within the surface, moving from sphere to sphere by using pipes, that would add nothing to the difficulty of pathing around the sphere.

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MoltenMan 221 days ago
Good point, the movement is more isomorphic with a fully spherical 2D geometry game than I was thinking. But I still wouldn't say it qualifies as non-euclidean in the same sense as hyperrogue at all, given that you aren't fully stuck on a sphere (and they don't give any way to change the view to a top down 2D spherical geometry view; because it's actually just a regular 3D game).

More relevantly to your original comment, it is always going to be far easier to understand 2D spherical geometry (possible to create in 3 dimensions, we see it every day) than 2D hyperbolic geometry (not possible to create in 3 dimensions, completely foreign to us), so I don't think it is a 'display' issue at all.

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thaumasiotes 221 days ago
> 2D hyperbolic geometry (not possible to create in 3 dimensions, completely foreign to us)

Foreign to us, yes. It's perfectly possible to represent a hyperbolic paraboloid in three dimensions.

https://mathworld.wolfram.com/HyperbolicParaboloid.html

> so I don't think it is a 'display' issue at all.

...you just complained that moving around a sphere should count as non-Euclidean if it's displayed inconveniently, but not if it's displayed conveniently. But you don't think that's a display issue?

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zenorogue 219 days ago
The hyperbolic paraboloid is not an isometric embedding of the hyperbolic plane H2 in Euclidean space E3, because it does not have constant negative Gaussian curvature.

In general it is impossible to have an isometric embedding of H2 in E3, although it is possible to have isometric embeddings of fragments of H2.

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