[ykierhyf] tilings of triangles and squares

tile the plane with squares and equilateral triangles all of the same side length.  this could be fun with physical tiles.

well investigated are the regular, semiregular, and uniform tilings, but random tilings also seem possible.

possible vertex configurations are (4 squares), (6 triangles), and (2 squares and 3 triangles).  2+3 has 2 possibilities: whether or not the squares are adjacent.

how careful do you need to be to avoid creating a gap into which nothing fits?

how difficult are such tilings to 4-color?  I think the following works: assign squares 2 colors.  2-color in a checkerboard pattern every connected component of just squares.  similarly, triangles with the other 2 colors.

this pair of tiles feels vaguely similar to Penrose tiles.  squares and triangles can tile periodically but they don't have to.

the edges form a penny graph.

these graphs feels aesthetically dense like the square and triangular grids.  the graph formed by the edges of the regular hexagon tiling does not feel aesthetically dense.