Uploaded December 2017 | Updated September 2026, 2 weeks ago
In the hyperbolic plane there is more area than in the Euclidean one -- in fact, the area of a circle of radius r is exponential in r. However, if you fold a small fragment of the hyperbolic plane, you can fit it in three-dimensional Euclidean space -- this can be done in the real world (see hyperbolic crocheting or beads [1]) or simulated [2].
This was negatively curved surface embedded in a flat space -- but what about embedding a flat surface into a positively curved space? Again, there is not enough space to do this without folding, In this video, we try to fit a fragment of a Euclidean plane into a three-dimensional sphere, S³ (imagine a two-dimensional sphere, e.g., the surface of Earth -- but add one extra dimension, and imagine light rays sticking to the surface).
It might appear that something strange is happening -- things that seem to be close are obscured by ones that seem to be distant... this is a weirdness of S³ we are in. All the rays coming out of your eyes will meet again at the antipodal point, causing things close to that antipodal point appear just as if they were next to you. Furthermore, all these rays will hit the back of your head (head not shown); this means that things you have walked past will appear in front of you again. These effects are visible in the video.
Want to explore S³ yourself (as in this video)? This will be possible in HyperRogue 10.3.
HyperRogue: roguetemple.com/z/hyper
[1] youtube.com/watch?v=7mqnd5aGWpw
[2] youtube.com/watch?v=GegO9ysaaio
See also: archive.bridgesmathart.org/2018/bridges2018-551.pdf
In the hyperbolic plane there is more area than in the Euclidean one -- in fact, the area of a circle of radius r is exponential in r. However, if you fold a small fragment of the hyperbolic plane, you can fit it in three-dimensional Euclidean space -- this can be done in the real world (see hyperbolic crocheting or beads [1]) or simulated [2].
This was negatively curved surface embedded in a flat space -- but what about embedding a flat surface into a positively curved space? Again, there is not enough space to do this without folding, In this video, we try to fit a fragment of a Euclidean plane into a three-dimensional sphere, S³ (imagine a two-dimensional sphere, e.g., the surface of Earth -- but add one extra dimension, and imagine light rays sticking to the surface).
It might appear that something strange is happening -- things that seem to be close are obscured by ones that seem to be distant... this is a weirdness of S³ we are in. All the rays coming out of your eyes will meet again at the antipodal point, causing things close to that antipodal point appear just as if they were next to you. Furthermore, all these rays will hit the back of your head (head not shown); this means that things you have walked past will appear in front of you again. These effects are visible in the video.
Want to explore S³ yourself (as in this video)? This will be possible in HyperRogue 10.3.
HyperRogue: roguetemple.com/z/hyper
[1] youtube.com/watch?v=7mqnd5aGWpw
[2] youtube.com/watch?v=GegO9ysaaio
See also: archive.bridgesmathart.org/2018/bridges2018-551.pdf
