The hat's family

The hat's edges come in segments of two lengths, eight short and six long (its one edge of length 2 counts as two short ones). Give the short segments length a and the long ones length b and you have Tile(a, b): the hat is Tile(1, √3), the turtle Tile(√3, 1). Drag a and b and the same patch of hats bends with them, never overlapping and never leaving a gap (scaled to the same perimeter, so the tiles stay about the same size); hats that are turned over are tinted ice blue. Either length may be zero, but not both. Whenever a and b are both above zero and different, the shape tiles only in patterns that never repeat. There are just three exceptions: Tile(1, 1), where a = b, and the two ends where a or b reaches zero, the chevron Tile(0, 1) and the comet Tile(1, 0). At these shapes a window opens at the top right, showing another tiling of theirs that does repeat; the two coral segments are the two shifts that carry it onto itself. For Tile(1, 1) that tiling needs a turned-over copy (ice blue). For a = b the two readings must match exactly; the arrow keys move one step at a time. The patch on screen is still the hats' patch, bent; it is not the one in the window.