Uploaded February 2020 | Updated September 2026, 1 week 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)
Follow twitter.com/ZenoRogue and play HyperRogue!
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)
Follow twitter.com/ZenoRogue and play HyperRogue!
![[360° VR] Portals to Non-Euclidean Geometries](https://i.ytimg.com/vi/bQSfzDugH7s/mqdefault.jpg "[360° VR] Portals to Non-Euclidean Geometries
This is a 360° VR version of our earlier video:https://www.youtube.com/watch?v=yqUv2JO2BCs
On this tour, portals will take us to various non-Euclidean geometries. This is not Minecraft!
A cool holonomy effect happened during this tour, but it was not explained by our guide Tehora! Have you found it? Please tell us in the comments!
Tehora Rogues channel: https://www.youtube.com/channel/UCHRkB4HgLLReBxWJwblHsjQ
Visited geometries:
* Three-dimensional Euclidean space 𝔼³ (cyan floors, music: Icy Land by Shawn Parrotee)
* Product space ℍ²×ℝ (green floors, music: Living Caves by Shawn Parrotee)
* Three-dimensional hyperbolic space ℍ³ (yellow floors, music: RLyeh by Shawn Parrotte)
* Product space 𝕊²×ℝ (blue floors, music: Ocean by Will Savino)
* Three-dimensional spherical space 𝕊³ (purple floors, music: Land of Eternal Motion by Shawn Parrotte)
* Solv (brownish floors, music: Lost Mountain by Lincoln Domina)
HyperRogue soundtrack under the Creative Commons BY-SA 3.0 license
Link to RogueViz: https://zenorogue.itch.io/rogueviz
To learn more about non-Euclidean geometry, play HyperRogue or visit our discord: https://discord.gg/8G44XkR")
