Two-point equidistant projection of the hyperbolic plane, but one point is in the center, and the other point is in the infinity, and changes its direction during this animation. The frame where horocycles are mapped to straight lines is insighftul. (Basically, a circle of radius 𝑟 around the center of ℍ² is mapped to a cirlce of radius 𝑟 around the center of 𝔼², and concentric horocycles are similarly mapped to straight lines; these two conditions determine where every point is mapped.) Based on an idea by bengineer8u.
By the way, our video "Portals to Non-Euclidean Geometries" https://youtu.be/yqUv2JO2BCs has just passed 1M views!
You cannot tile a hollow sphere with hexagons. But you can pretend.
Some games and animations pretend they can tile a sphere from the outside [ https://twitter.com/ZenoRogue/status/1439246553877729286 ] but I have not seen this done with the inside. This is the WIP HyperRogue feature of embedding 2D geometries into 3D geometries; in this case, the Euclidean world map is embedded as a (hollow) horosphere in 3D hyperbolic space. Expect more weird visualizations based on this (:
If they expand in the same 'z' direction, but with different rates instead, it looks more like a horosphere (compare the first visualization in this thread); this is why hexes in the visualization below are squished when we look at them orthogonally from far away. #mathart#noneuclidean#hyperrogue#rogueviz#mathviz#visualization