After Cory Poole posted some great Escher snowflakes, and Cerek Tunca had the great idea of using it as a base for a tetrahedron, well, I just had to give it a go. I will post a few more pictures and variants later (I think this was what Cerek was envisaging—if not let me know!)
Funny how the octahedron works best. In 2D, equilateral triangles tessellate in sixes and alternate, in 3D they alternate in fours. The icosahedron and tetrahedron has sides that touch similar sides, but there is no way around that.