Conway's pinwheel tiling

Love this tiling ! So beautiful

I made a few pinwheel tilings visualisation on shadertoy https://www.shadertoy.com/results?query=tag%3Dpinwheel including a few original use of pinwheel to simulate a "snake" walking inside the tiling https://www.shadertoy.com/view/4sSBRy . and this one gets really trippy if you wait a few minutes there are interferences patterns in the colors that become really crazy https://www.shadertoy.com/view/XsSfWm

The bottom of my home page has this as a PostScript signature:

    %!PS                           % -John Tromp http://tromp.github.io/
    /t{dup 1 sub gsave dup 0 gt{[.4 .2 -.2 .4 .4 .2]concat t currentgray
    .8 mul .2 add setgray -1 1 scale t -1 2 translate t 1 -1 scale t[0 1
    1 0 0 2]concat t pop}{0 moveto 1 0 lineto 0 2 lineto closepath clip
    fill}ifelse grestore}def 10 10 translate 600 600 scale 5 t showpage

which outputs this pinwheel tiling https://tromp.github.io/img/pinwheel.pdf

Super interesting! So if I understand correctly, all you need to do to have this in your home is gather a bunch of 1:2 tiles, cut them along the diagonal, and assemble them as shown? Awesome

How could one prove this is aperiodic? I'm guessing maybe you can prove that some or most of the triangles have globally unique rotations regardless of N?