Naively, and with general interest public forum curiousity, these tiling problems seem to be about iterating the proportions of the sides of the tiles to get sy

Naively, and with general interest public forum curiousity, these tiling problems seem to be about iterating the proportions of the sides of the tiles to get symmetrical shapes, but given each tile is also necessarily a hamiltonian circuit between the angles/nodes of a shape and the tiles are aperiodic, the implied visual symmetry of the shapes doesn't seem meaningful. It seems like there would be infinite possible aperiodic tiles (with real valued side lengths), so long as the number of angles (or nodes/vertices) for a whole tile (like a triangle) has the same evenness or oddness as the number of vertices extending from the node as the number of sides of the shape it is a part of. So to completely tile a plane aperiodically, each node/angle of a triangle must have an odd number of "sides" from its adjacent tiles stemming from it to completely tile a plane, where each angle of a hexagon must be a node with an even number of sides connecting at its vertices. Once you are more than one "hop" away from another tile, you can have even or odd numbers of verticies. The perimeter of any plane with a complete aperiodic tiling must still be a hamiltonian path around its edge, therefore the graph of the verticies representing the angles the aperiodic tiles must also reduce to being made of other "shapes" with hamiltonian paths. It implies to me that for every tile that is odd-sided, it requires a complementary odd-sided shape somewhere in the tiling to form a hamilonian path of "hops." It's not a sufficient condition, but naively it looks like a necessary one. No math will get done here, but from a general interest reasoning perspective, I'd wonder if tiles and hamiltonian cycles are the same thing.