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Comment by wcoenen | Hacker News Reader

> rather than actually modeling non-Euclidean space natively

In mathematics such non-Euclidean spaces are called manifolds. A manifold is defined as a set of Euclidean-like regions that are connected by transition map functions. That formalism is basically the same thing as these portals.

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

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Except that that method wouldn't let you have, for example, a hyperbolic plane that could be traversed without obstruction.

I think you could actually approximate an unobstructed hyperbolic plane to arbitrary precision with portals. In this game, the hyperbolic plane is represented as a tiling of hexagons and heptagons: https://www.youtube.com/watch?v=lX5eCfRSCKY&t=742s

If you take this model but upgrade each cell to a little 3D euclidean region, then you could completely surround it by portals that connect it to the neighbouring cells. The edges of the portals wouldn't be visible. The smaller the euclidean cells, the better the approximation to the hyperbolic plane.

For easy computations assume octagonal cells. Their vertices will have 135 degrees each. The edges of the portal will be visible because the left and right images do not match (since 3*135 ≠ 360). And a object (as long as it is modelled as an Euclidean 3D object, not a point) will not be able to go through such a vertex.

There is a game in development that uses these ideas too, and is aptly called Manifold Garden:

https://store.steampowered.com/app/473950/Manifold_Garden/

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