ZenoRogue | Stretched spherical geometry @ZenoRogue | Uploaded May 2020 | Updated October 2024, 1 hour ago.
The first part of this video shows the space of rotations of the sphere. As well known in 3D graphics, this space is the same as unit quaternions, i.e., a 3-sphere (with opposite points identified). The small image in the corner shows the rotation corresponding to the current camera position. A point in the 3-dimensional space is filled iff the rotation it corresponds to is centered inside a wall (and the color corresponds to the color of that wall).
Today, we will modify this to get a new geometry!
The straight lines you see are called fibers. We change the geometry by stretching the metric along the fibers. As if the speed of light was slower/faster along the fiber, and light still took the fastest route, as usual.
We will call the stretching factor k. In the second part of the video, we stretch from k²=0.1 to 1.9.
In the third part we stretch the fibers to just k²=0.02. There is only one white block on every fiber -- but since it is so fast to travel along the fiber, we see many copies of every white block! In the limit we get the S2xE geometry.
By filling all the fibers, we missed one cool feature of this geometry so far! Without stretch, if light ray nearly misses a block, it travels π units around the sphere, also nearly misses it again. With stretch, it may hit another block.
In the fourth part we show only the blocks which were previously white, not the whole fibers. This way we can see "afterimages" obtained by hitting another point on the fiber after going around the sphere.
In the last part of the video we stretch the fibers to k²=10. The afterimage effect seems to be caused by errors in numerical solving of differential equations (it gets smaller when precision is increased), but it is still an interesting effect.
Thanks to Jeff Weeks for suggesting this geometry, and some hints!
See also: youtu.be/KBYPQaoBgz0
The first part of this video shows the space of rotations of the sphere. As well known in 3D graphics, this space is the same as unit quaternions, i.e., a 3-sphere (with opposite points identified). The small image in the corner shows the rotation corresponding to the current camera position. A point in the 3-dimensional space is filled iff the rotation it corresponds to is centered inside a wall (and the color corresponds to the color of that wall).
Today, we will modify this to get a new geometry!
The straight lines you see are called fibers. We change the geometry by stretching the metric along the fibers. As if the speed of light was slower/faster along the fiber, and light still took the fastest route, as usual.
We will call the stretching factor k. In the second part of the video, we stretch from k²=0.1 to 1.9.
In the third part we stretch the fibers to just k²=0.02. There is only one white block on every fiber -- but since it is so fast to travel along the fiber, we see many copies of every white block! In the limit we get the S2xE geometry.
By filling all the fibers, we missed one cool feature of this geometry so far! Without stretch, if light ray nearly misses a block, it travels π units around the sphere, also nearly misses it again. With stretch, it may hit another block.
In the fourth part we show only the blocks which were previously white, not the whole fibers. This way we can see "afterimages" obtained by hitting another point on the fiber after going around the sphere.
In the last part of the video we stretch the fibers to k²=10. The afterimage effect seems to be caused by errors in numerical solving of differential equations (it gets smaller when precision is increased), but it is still an interesting effect.
Thanks to Jeff Weeks for suggesting this geometry, and some hints!
See also: youtu.be/KBYPQaoBgz0
