The third grid

Tessellation discourse 333, aka the triangular/triangle/3^6 grid/tiling/tessellation

Almost every tile-based game uses squares or hexes, while the 3rd regular tilling, the triangular/3^6 tiling, has been (almost) completely ignored; this is sometimes justified by redditors(& more recent AI that literally cites reddit🙄) by dismissing it as just “the worst of both worlds”, with inconveniently pointy tiles, no orthogonal lines, only 3 adjacencies, & horribly distorted distances.

The triangular(3^6) tiling is the dual of the hex(6^3) tiling, so playing on the vertices of a hex tiling(with the actual hexagons being meaningless) would be equivalent to playing on the tiles of 3^6

The diagonal question

While each triangle is only adjacent to 3 others thru its edges, the vertices touch 12! Of the other 9, 3 are straight diagonals & 6 are closer off-diagonals.

Even more remarkable are the lines, with only edge-adjacency you can move in 6 directions like on a hex grid, while diagonal moves would allow an impressive 12(not only including orthogonal directions, but having them work in 3 distinct orientations)

The downside to all this potential is that there isn’t a clear standard solution(or even 2) to how to deal with all these pesky diagonals. Tho a relatively simple reasonable-looking option would be making edge-adjacencies take 1 movement point, & both types of diagonals 2, while also requiring 1 edge-adjacency path around the vertex(so either going clockwise or anticlockwise) that isn’t blocked.

Other alternatives are just edge-adjacency, vertex-adjacency with no sanity checks(could look reasonable with the right tileset/level design/mostly having open game fields), or various arbitrary patterns for different abilities/units.

Parity

Tiles in the 3^6 tiling come in 2 orientations, which may be mildly inconvenient for the art(the very sharp corners don’t help) & programming, but hey, at least a triangle is a triforce of triangles! so unlike with hex grids, you can have chunks & subdivision.

(also because of how monitors are shaped, you probably want to orient the tiling to get horizontal lines in most cases where you’d have a fixed camera)

They do however also make parity very clear, & if keeping to the limited edge-adjacency, likely more relevant/fitting then even on square grids.

Δ Volvox Δ

Otherwise, just like on the square tiling you can break parity with null actions(waiting), allowing diagonal moves, or any sort of “tactical” style movement point system.

Quirkiness

3^6 has a lot of potential for funky movement rules/abilities to compensate for how limiting the basic adjacency movement can be with just 1 step/turn,

If you wanted bishop-like movement on a triangular grid, you could use only the straight diagonals, which splits the grid into 4 colors(interestingly enuf, each subset is connected like its own triangular tiling, but with hexagonal gaps)

Potential cone-shaped AoEs in 6 directions

Kriegesspiel being quirky in 1817(if you like genocide simulators, you should thank the Prussian army & their proto-nerd-fanboys‽‽)

It is also possible to have giant hexagonal or rhombic units on a triangular grid

Δ Colossal Citadels Δ may be trying this if it isn’t just abandoned

Visuals

As a hexagon can be split into 6 triangles, the triangular tiling has to be at least as visually flexible/smooth, & it has built in straight lines on top of that. Alternatively you could use the dual property to get a tileset that perfectly imitates hexes(but the transitions between tile types will be inside triangular tiles instead of on the hex edges).

Unfortunately there aren’t many examples or existing assets, tho there’s trixel art(like squixel(square…pixel) art but with triangles!)(note that while the isometric style is perfect for imitating 3d voxels, it might be confusing to play on if treated like a flat 3^6 grid)