Post #2923563

We can now change the projection again, going from the \((1,0)\) projection to the \((1, -1)\) projection, dragging the regions along as before. The result is that the triangles spread out further, but the grid points in adjacent triangles no longer lie on a common rectangular grid (at least, not one with the same distance between points). This is why my construction here https://mathstodon.xyz/@pieter/110520611405361464 required two steps. But if we shift the control point to the position Peter chose, the grids become compatible, yet the regions (which must move along with the control points) fortunately don't overlap. So far I've ignored the tiles with less frequent handedness. From the second image, we can deduce regions that must correspond to their control points by considering their 'H8' neighbourhoods and looking at the intersections of the regions associated to these tiles (If this isn't clear, this post on empires of turtle tilings may help explain the general principle: https://mathstodon.xyz/@pieter/111696562690124891) (4/n)