On commonly-used magic systems

So the basis for... probably most CRPG magic systems is the classical four elements. This can be seen in, for example, Final Fantasy X.

Final Fantasy X doesn't actually use the classical elements outright; rather, its four primary elements are fire, ice, water, and lightning. However, they're still opposed in exactly the same manner as the classics: where Paracelsus had fire/water and earth/air opposed, FFX has fire/ice and water/lightning. It's the same graph, just with different labels.

Of course, this can be extended to more than two opposing pairs, usually by adding light/darkness, or a close facsimile thereof. Chrono Cross used a similar, three-way opposing system: red/blue, green/yellow, and black/white. Exactly what physical concept was represented by each color was a little loose, but the strict opposition and noninteraction between them was not.

Intuitively, these map to `RR^2` and `RR^3` respectively — although the situation is actually a little more complex than that, since games using this system tend to find ways to break the axes at the origin: e.g., a player character wielding a Flame Sword but wearing an Antifreeze Charm, so that they can take on Ice-aspected enemies.

(TBC)