ZenoRogue | Can we simulate spherical geometry in Euclidean space? @ZenoRogue | Uploaded April 2020 | Updated October 2024, 1 hour ago.
Three-dimensional spherical space can be created from 120 spherical dodecahedra. Four of these are filled. In the first part of the video, we see the effects mentioned in youtu.be/leuleS9SpiA
Can we simulate these effects using an Euclidean game engine and portals? The remaining two parts of the video show that this does not work, but they should still be fun!
In the second part, we construct the same scene from 120 Euclidean dodecahedra. Some of the spherical effects can be seen if you look close enough. This looks like some cool abstract art, but it does not work very well as a simulation of 𝕊³.
The edges look strange because we have only 349.695° of space around them.
In the third part, we follow the suggestion of Jos Leys [ josleys.com/article_show.php?id=83 ] we attempt to construct the scene from the stereographic images of spherical dodecahedra.
This concentrates the curvature on faces instead of edges [ geometrygames.org/HyperbolicBlanket ].
Again, this looks more like abstract art than 𝕊³.
Three-dimensional spherical space can be created from 120 spherical dodecahedra. Four of these are filled. In the first part of the video, we see the effects mentioned in youtu.be/leuleS9SpiA
Can we simulate these effects using an Euclidean game engine and portals? The remaining two parts of the video show that this does not work, but they should still be fun!
In the second part, we construct the same scene from 120 Euclidean dodecahedra. Some of the spherical effects can be seen if you look close enough. This looks like some cool abstract art, but it does not work very well as a simulation of 𝕊³.
The edges look strange because we have only 349.695° of space around them.
In the third part, we follow the suggestion of Jos Leys [ josleys.com/article_show.php?id=83 ] we attempt to construct the scene from the stereographic images of spherical dodecahedra.
This concentrates the curvature on faces instead of edges [ geometrygames.org/HyperbolicBlanket ].
Again, this looks more like abstract art than 𝕊³.
