Moiré

Two deflation levels, overlaid — and what deflation does off the Penrose class

A generalized Penrose tiling is labeled by its LI class, Σγ mod 1, with Penrose at 0. Deflation sends γj to −(γj+2 + γj+3), and every row of that matrix sums to −2, so on the class it is the doubling map with a sign: s → −2s (mod 1). A tiling off the Penrose class does not fail to deflate — it deflates into a tiling of a different class. What it stops being is self-similar, and that is why the two levels make a moiré.

A
B

Drag to pan, scroll to zoom — the two stay in register, converting between their scales, so B shows the same region of the plane as A however many times it has been deflated. With link on, B is A deflated, and anything you do to A's γ lands on B — including anything you do to B, which snaps straight back, reset included. Turn link off for two free reticula. merge stacks the two canvases into one and multiplies them together, which is where the moiré is.

Every parent vertex is either exactly on a child vertex or exactly φ−2 away, never anything between. On Penrose it is always 0 — every vertex of the parent is a vertex of the child, which is what the substitution says. Off it, the share that coincides is a property of the class rather than of the offsets: 81% at 1/8, 69% at 1/4, and a minimum of 59.6% at 1/2, symmetric about there. Which is worth a second look: 1/2 resembles its own deflation least, and it is the class that deflates into Penrose in one step. Landing in Penrose and looking like your parent are different questions.