Post #2923562

The deflation operation can be abelianised (in some sense) to give a 2 x 2 complex matrix with eigenvalues \(\phi\) and \(\phi^{-1}\), where \(\phi\) is the square of the golden mean. If we project the lifted centres of tiles with more frequent handedness onto the left eigenspace of \(\phi^{-1}\) (spanned by \((1, -\xi\phi)\)) and colour the projected points according to the orientation of the tile, we get a pattern which is the superposition of the two triangles shown, translated so their centres coincide. On the other hand, consider the projection onto the first coordinate. For a given tile centre, we can track its position as we continuously deform the projection from that eigenspace projection to the \((1,0)\) projection, and we can translate the region that the eigenspace projection of the point landed in as we do so, keeping the relative position of the projected point within the region fixed. This spreads out the regions, and we obtain the periodic pattern shown in the second image. I've overlain this with hexagons, which are the images of the tiles with more frequent handedness under this projection. A short line indicates the orientation of the lifted tile. In the notation of the previous post, these lines extend from \((0,0)\) towards \((\xi^2,0)\) (for a tile in that particular orientation) (3/n)