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 (:
Somehow we have not yet visualized this, so it was a bit surprising! That "stalactite" is infinitely long, and also infinitely wide -- we cannot go around it.
The floor is a hyperbolic plane embedded in Solv geometry (as the set of points with x=0), while the roof is equidistant to it.
And here is the (z=0) plane, which has Euclidean intrinsic geometry. We have had visualizations of this earlier (e.g., it appeared in https://youtube.com/watch?v=yqUv2JO2BCs). It looks more like a horotorus than like a horosphere because hyperbolic planes (x=0), (y=0) expand in opposite z directions.
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
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