Given a complete set of 2-D connectable square tile "primitives" comprising:
- a "fully immersed" inner
- a 4-crossing
- a singleton
- two parallel outer edges, one horizontal one vertical
- four straight outer edges
- four arrowhead outer edges
- four arrowhead outer edges with inner corner
- four single inner corners
- four double, neighboring inner corners
- four triple inner corners
- two with opposing inner corners each
- four with one outer edge and two corners each, T-shaped
- four with three outer edges each, U-shaped
How would I go about finding a packing that uses all of them, as few as possible each, properly coalesced, covering a minimal area, holes permitted?
I have already written an algorithm to validate any given combination, and measure both parameters "area covered" and "elements used", but cannot get my head around any sensible rule for composition apart from randomized brute-force.
My previous research efforts are limited to internet recherche based on keywords like "2-topology arrange connectable squares cross T U least area", and some wasted sheets of quad paper. I came to the conclusion that this is probably called a "packing problem with constraints" but this seems to be too abstract as a meaningful term to search literature.