Uploaded March 2021 | Updated September 2026, 2 weeks ago
Another non-Euclidean portal!
0:00 Through a great circle
This one is a great circle in the three-dimensional spherical space. It connects six worlds, each with different fog colors and a different color of the "lone brick".
These videos use 270° field of view (obtained with the stereographic projection) -- with a small field-of-view, there would be no way to see the whole ring at once. We are going straight roughly the center, so the ring would be actually seen around you.
0:10 Different perspective
0:30 Berger sphere
In the previous videos, we could only see two lone bricks at once (the other four others being hidden by their counterparts in closer worlds).Here, we make the geometry less symmetric by stretching it along the Hopf fibers (i.e., obtaining the Berger sphere).
1:00 Knot portal
Let's try a knot portal, based on a trefoil knot. The idea is similar to youtu.be/vFLZ2NGtuGw but the spherical geometry makes it quite hard to grasp. It is "self-hiding" -- in some worlds the portal is not there -- but it may still appear to be there, because the light will travel around the sphere and hit the copy of the portal in another world. The knot has been obtained by embedding it in a torus in the standard way ((3,2) torus knot) and then embedding that torus in the sphere in the standard way (Clifford torus).
A slightly different version on Twitter: twitter.com/ZenoRogue/status/1373298292520550404
Music: the "Laboratory" theme from the HyperRogue soundtrack, composed by Shawn Parrotte
Made with the HyperRogue engine aka RogueViz
Source code: github.com/zenorogue/hyperrogue/blob/master/rogueviz/notknot.cpp
A playable Windows exe at roguetemple.com/z/sims/notknot.zip (don't go into the walls; works in VR too!)
Another non-Euclidean portal!
0:00 Through a great circle
This one is a great circle in the three-dimensional spherical space. It connects six worlds, each with different fog colors and a different color of the "lone brick".
These videos use 270° field of view (obtained with the stereographic projection) -- with a small field-of-view, there would be no way to see the whole ring at once. We are going straight roughly the center, so the ring would be actually seen around you.
0:10 Different perspective
0:30 Berger sphere
In the previous videos, we could only see two lone bricks at once (the other four others being hidden by their counterparts in closer worlds).Here, we make the geometry less symmetric by stretching it along the Hopf fibers (i.e., obtaining the Berger sphere).
1:00 Knot portal
Let's try a knot portal, based on a trefoil knot. The idea is similar to youtu.be/vFLZ2NGtuGw but the spherical geometry makes it quite hard to grasp. It is "self-hiding" -- in some worlds the portal is not there -- but it may still appear to be there, because the light will travel around the sphere and hit the copy of the portal in another world. The knot has been obtained by embedding it in a torus in the standard way ((3,2) torus knot) and then embedding that torus in the sphere in the standard way (Clifford torus).
A slightly different version on Twitter: twitter.com/ZenoRogue/status/1373298292520550404
Music: the "Laboratory" theme from the HyperRogue soundtrack, composed by Shawn Parrotte
Made with the HyperRogue engine aka RogueViz
Source code: github.com/zenorogue/hyperrogue/blob/master/rogueviz/notknot.cpp
A playable Windows exe at roguetemple.com/z/sims/notknot.zip (don't go into the walls; works in VR too!)
