The aperiodic Tile(a, b) can be constructed as a 4D skew nonplanar polygon then projected to 2D. Changing the projection direction, we can get the whole continuum including hat, turtle, and the non-curvy spectre. For some directions, we get a self crossing polygon in 2D. 1/2

In the tiling mixing hats and turtles, all 4D tiles are the same skew polygon but in different orientations. At the same time, some tiles are projected to hats, some turtles. The coordinates of a skew polygon and the projection matrix are here. They are easy to construct. 2/2

In the continuum of the tiles, some edges are expanding and some are shrinking at the same time. I put the expanding edges on the first two dimensions in 4D, and the shrinking ones on the last two dimensions. The 4D rotation mixes the two pairs of dimensions together.

In my animation, I use triangles to fill the skew polygon. @ArnaudCheritat proposed a design to use parallelograms to fill it the same skew polygon. Here's how it looks. I think this new design looks better and more consistent.

This is so cool!!! 😭

can these 'sheets' be stacked vertically without gaps and if so, does that then constitute a volumetric tiling? pardon my amateur language. and if so....reminded me of amplituhedrons ... so this shape MIGHT be the answer to life the universe and everything afterall?

If we pause at the Tile(1,1) frame (spectre tiling), we can extrude the shape “up”. Things are a bit complicated because in 4D there are two dimensions we can extrude into. We have the freedom to choose one of them as “up”.

Just marvelous! I was wondering if anyone could possibly extend/generalize it into a universal hidden/behind-the-scene all-(a-)periodic monotiles generator/regulator, please.

2d compactification of 3d spacetime just dropped. (Which suggests the 3d-4d exists?)

This animation uses the Spectre tiling pattern, it seems. Can the Hat tiling pattern be animated in this way as well?

Here you go. This is based on the original Hat tiling using "mirrored" hats, from the paper in March. In 4D, the 4D skew polygons are rotated but not mirrored.