ZenoRogue | Impossible Triangle Possible in Nil Geometry @ZenoRogue | Uploaded February 2020 | Updated October 2024, 3 hours ago.
The Penrose triangle is a figure that is impossible in Euclidean space. You can try to build it, but you will end up over (or below) where you started.

Nil is a non-Euclidean geometry where when we walk one step to the north, one step to the west, one step to the south, one step to the east, we end up one step above where we started.

So, when you put the (correctly rotated) Penrose triangle in Nil geometry, it becomes possible...

3x3x3 quotient space of Nil

Links:
en.wikipedia.org/wiki/Penrose_triangle
sandlotscience.com/oscar-reutersvard

zenorogue.itch.io/hyperrogue/devlog/99827/hyperrogue-112-thurston-geometries-free-update (short explanation of Thurston geometries including Nil)
roguetemple.com/z/hyper/geoms.php (explanation of all geometries in HyperRogue)
twitter.com/ZenoRogue/status/1187378654470430720 (Penrose staircase in Nil)
twitter.com/stevejtrettel/status/1225102030928171008 (a Twitter thread on Nil, and a link to an explanatory paper)
twitter.com/stevejtrettel/status/1224460645644435457 (another Twitter thread on Nil)
http://www.cs.bme.hu/~pappd/pub/nilgeo_proc.pdf (older paper about geodesics in Nil)

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