This will seem like a nit at first but it's really a key driver to why you'd look at other shapes: a hexagon is more comparable to a quadrangle, a square is more comparable to what is called "a regular hexagon". Regular meaning the sides and angles which make up the shape are all equal. In the 2D world, such as on board games, equilateral triangles, squares, and regular hexagons can all tile a plane perfectly. This is not the case for the surface of a sphere, there is no tiling regular polygon in that case.
From there you're just trying to optimize uniformity in distance to neighbors, how big the adjustments to the irregular polygons need to be to get them to tile on the surface are, how easy the polygon is to split up into smaller similarly shaped variants of itself as sub tiles, and trying to be somewhat close to a circle in shape as that means the average distance to the center of the area defined by the index is as close to as it can be.
If you chunk through those you'll find quadrangles aren't attractively simple anymore and hexagons tend to optimize the parameters very well. H3 actually uses both hexagons and the occasional pentagon (12 total, no matter the zoom level). It all comes down to "tiling isn't going to be perfect - what is the most optimal answer for the purpose of the tiling".
From there you're just trying to optimize uniformity in distance to neighbors, how big the adjustments to the irregular polygons need to be to get them to tile on the surface are, how easy the polygon is to split up into smaller similarly shaped variants of itself as sub tiles, and trying to be somewhat close to a circle in shape as that means the average distance to the center of the area defined by the index is as close to as it can be.
If you chunk through those you'll find quadrangles aren't attractively simple anymore and hexagons tend to optimize the parameters very well. H3 actually uses both hexagons and the occasional pentagon (12 total, no matter the zoom level). It all comes down to "tiling isn't going to be perfect - what is the most optimal answer for the purpose of the tiling".