There's one difference. On squares you have only 4 equal move directions (diagonals work differently and break the system a bit). On octagons you have 8 equal move directions, so it would be theoretically more variable than hexes and at the same time more logical and consistent than squares. But I think I must mention one very slight problem - you cannot build a 2D grid from octagons, it's mathematically not possible (There are octagons + squares on the image from Uberfrog, so it technically is NOT an octagon grid.)
(Btw, I understood that this thread is a joke, but I think it's a nice idea to discuss anyway )
Of course a triangle grid would always be an opportunity. Not sure how equal that would make movement, though I suppose that you could only move over the sides, not over the edges (or you'd be able to move to 12 triangles from any triangle).
Triangles have the problem that their orientations change between adjacent tiles. So you can move, say, north with your first movement point, but on the next tile, you'd have to move either northeast or northwest (or go back the way you came). This is quite unintuitive. Hexagons and squares feel much more natural, as unit paths wouldn't have to unnecessarily zig-zag like this.
Squares are good in that they offer nice straight paths for each compass direction. However they do not preserve distance, making diagonal travel more efficient in a way that seems to break movement rules. It also makes your landmasses look very blocky.
Hexagons are I think on balance the best possible choice. They are round enough that you can have more attractive map features, preserve equal distances in each available direction, and they offer intuitive straight paths in a good number of directions. The only thing that's slightly wonky is that you cannot directly travel east or west. But that's a reasonable trade-off for everything else, in my view. It also makes the adjacency bonuses that are key to Civ VI's design immediately clear in a way squares wouldn't (i.e. do diagonals count as adjacent?)
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