The tessellation of triangles can easily be mapped onto the square lattice, see the figures at the left and right. Upright triangles (colored green) and upside-down triangles (white) go into separate rows. The lines of interaction for the Von Neumann neighborhood are regular hexagons in the triangle lattice. These hexagons are distorted in the square lattice.
With this mapping it is easy to make up a cellular automaton on the lattice of triangles. Using the same methods as for the hexagonal and square lattice I get some new ornaments. For the result below I have used eight colors and the Von Neumann neighborhood augmented by the cell itself. Because triangles are related to fire I preferred a red color scheme.