Thank you for sharing this kind of grid. I've worked with that tiling before but didn't know the name of it. I like that it allows embedding a variety of shapes within the grid, but one drawback for use in a game is that the cells have more orientations and less symmetry than an equilateral tri-grid. So it may be most suitable for stylized art (see Hexels for a drawing tool that supports it).
I've prototyped the 3D equivalent, which is composed of tetrahedrons and 13 axes (I wonder if that 3D tiling has a name). The tetrahedrons aren't the same size, so it's a bit awkward to work with but might be suitable for 3D models/art.
I originally tried hexagons, but I ultimately chose triangles because they can compose into hexes and there were already so many hex-based games. First I tried a design using separate grids floating in continuous space but later switched to a complete grid-based world as it could make for a novel turn-based classic roguelike or strategy game. It's more limited than continuous space, although with tri- or hex-grids you get six rotations versus four with squares.
One difficulty with tri-grids is the poor spatial locality of triangles versus squares or hexes. For example, when calculating influence maps for AI on a triangle grid, each cell score is effectively stretched to the triangle vertices rather than representing a more consolidated or circular region (so for this purpose hexes would be ideal).
Like other grids, transforms (translation/rotation) on a tri-grid can be done with all integer math (and generally, if the game world is grid-based, everything but graphics/animations can be integer math). Rotations about a cell center are trickier (and are limited to three distinct rotations) but can be done by scaling up coordinates and then back down. I'll write up more details if anyone is interested.
I've prototyped the 3D equivalent, which is composed of tetrahedrons and 13 axes (I wonder if that 3D tiling has a name). The tetrahedrons aren't the same size, so it's a bit awkward to work with but might be suitable for 3D models/art.