Edge options on Hat make mirror image a separate tile
This page shows the same tiling as the Hat page, and with Line edges it looks the same. Here you can also curve the edges, as on the Spectre page. Curved edges are how the Spectre gets rid of its mirror images. On the Hat they can't do that.
No edge curve, of any shape, lets a hat and its exact mirror image tile in the hat pattern. Some edges meet in a loop of three: a hat's edge 1 meets another hat's edge 2, edge 2 meets a mirrored hat's edge 12, and edge 12 meets edge 1. Follow a curve round that loop and it comes back reflected in the edge's own line. Only a straight edge equals its own reflection.
Two tiles do work. Mirrored hats never touch each other, so each one can take whatever shape its ring of hats leaves. That shape is the mirror image of the curved hat with every bump turned inside out. With Double, every edge in the tiling carries the same S, and the mirrored hats keep the S instead of turning it into a Z. So once the edges curve, this pattern is made of two different tiles, not one tile and its mirror image. In the key, an m marks a mirrored hat, so 0° m is a mirrored hat at 0°.
Single puts one bump on each edge; on seven of a hat's fourteen edges it is turned half a turn, so it bulges the other way. Double puts an S on every edge. The a-edges and b-edges never meet each other, so each has its own height. At the hat's own shape the a-edges are the short ones. Push a height past the marked limit and the tiles collide: the red stripes show where they overlap. The tile shape slider works as on the Hat page, and the curves reshape with the tiles.
Drag to move. Scroll, pinch or double-click to zoom. Turn with the turn buttons, a two-finger twist, a right-drag, Ctrl and drag, or Shift and the arrow keys. A trackpad twist works in Safari. The address bar keeps your place, colours, shape and edges, so you can share a view.
This demo needs WebGL 2, which this browser does not offer.