Comment by penteract
20 days ago
No idea if they are doing this, but you can use Gosper islands (https://en.wikipedia.org/wiki/Gosper_curve) which are close to hexagons, but can be exactly decomposed into 7 smaller copies.
20 days ago
No idea if they are doing this, but you can use Gosper islands (https://en.wikipedia.org/wiki/Gosper_curve) which are close to hexagons, but can be exactly decomposed into 7 smaller copies.
Can Gosper islands tile the sphere though?
Yes! A Gosper Island in H3 is just the outline of all the descendants of a cell at a some resolution. The H3 cells at that resolution tile the sphere, and the Gosper Islands are just non-overlapping subsets of those cells, which means they tile the sphere.
Not quite - you need 12 pentagons in a mostly hexagonal tiling of the sphere (and if you're keeping them similar sizes, Gosper-islands force hexagon-like adjacency). I don't think it's possible to tile the sphere using more than 20 exactly identical pieces.
You could get a Gosper-island like tiling starting from H3 by saying that each "Hex" is defined recursively to be the union of its 6/7 parts (stopping at some small enough hexagons/pentagons if you really want). Away from the pentagons, these tiles would be very close to Gosper islands.
> I don't think it's possible to tile the sphere using more than 20 exactly identical pieces.
I was wrong about this (e.g. https://en.wikipedia.org/wiki/Rhombic_triacontahedron). It still seems possible to me that there's a limit to the smallest tile that can tile a unit sphere on its own. (Smallest by diameter as a set of points in R^3).