Uploaded April 2021 | Updated September 2026, 1 week ago
Several recent games use this trick where a scene contains a smaller copy of itself. Sometimes called non-Euclidean, although it is more of an affine (similarity) manifold rather than non-Euclidean. So let's make it actually non-Euclidean!
0:00 To make the animation smooth, our speed is proportional to the distance from the center. We can assume that that's how the distances in our space work: distances are larger and larger as we
come close to the center. However, in this scene, the light rays travel still according to the rules of Euclidean / affine geometry.
0:15 Here we make the geometry work according to the metric outlined above! For example, when the blue person wants to reach the purple person, they won't go through the center, because the distances
are larger closer to the center. Because of this, if you want to reach the point in the same distance from the center, it is useless to come closer (or further), and the closest path would be a spherical
arc around the center.
The light rays also always take the shortest path, so for example, when we are above the head of the purple person, we see the blue torso in every direction orthogonal to the
central direction! This is because the light rays travel in spherical arcs and hit the blue person.
0:45 The Euclidean scene again.
Recent games based on this trick include:
Patrick's Parabox: store.steampowered.com/app/1260520/Patricks_Parabox
Maquette: store.steampowered.com/app/762840/Maquette
Spaceflux: store.steampowered.com/app/1344440/Spaceflux
Also e.g. glkitty.itch.io/spheres and this Minecraft video reddit.com/r/Minecraft/comments/mlkieu/since_the_trend_seems_to_be_noneuclidean
This geometry is called S2xR or S2xE. Discuss this stuff and non-Euclidean geometry in general in the #geometry channel in the HyperRogue discord: discord.gg/8G44XkR
Music: "Palace" by Will Savino, from the HyperRogue soundtrack (CC BY-SA 3.0)
Several recent games use this trick where a scene contains a smaller copy of itself. Sometimes called non-Euclidean, although it is more of an affine (similarity) manifold rather than non-Euclidean. So let's make it actually non-Euclidean!
0:00 To make the animation smooth, our speed is proportional to the distance from the center. We can assume that that's how the distances in our space work: distances are larger and larger as we
come close to the center. However, in this scene, the light rays travel still according to the rules of Euclidean / affine geometry.
0:15 Here we make the geometry work according to the metric outlined above! For example, when the blue person wants to reach the purple person, they won't go through the center, because the distances
are larger closer to the center. Because of this, if you want to reach the point in the same distance from the center, it is useless to come closer (or further), and the closest path would be a spherical
arc around the center.
The light rays also always take the shortest path, so for example, when we are above the head of the purple person, we see the blue torso in every direction orthogonal to the
central direction! This is because the light rays travel in spherical arcs and hit the blue person.
0:45 The Euclidean scene again.
Recent games based on this trick include:
Patrick's Parabox: store.steampowered.com/app/1260520/Patricks_Parabox
Maquette: store.steampowered.com/app/762840/Maquette
Spaceflux: store.steampowered.com/app/1344440/Spaceflux
Also e.g. glkitty.itch.io/spheres and this Minecraft video reddit.com/r/Minecraft/comments/mlkieu/since_the_trend_seems_to_be_noneuclidean
This geometry is called S2xR or S2xE. Discuss this stuff and non-Euclidean geometry in general in the #geometry channel in the HyperRogue discord: discord.gg/8G44XkR
Music: "Palace" by Will Savino, from the HyperRogue soundtrack (CC BY-SA 3.0)
