Penrose P3

Two tiles. A fat rhombus with 72° corners and a thin one with 36°. Between them they cover the plane completely, and no matter how far you slide the result it never lands on itself. It is aperiodic — not disordered, but never repeating.

Drag the level up and the same figure refines. That is not a coincidence of the drawing: refinement is the rule.

Grown, not assembled

The tiling is not laid down tile by tile. It is grown from a string, by an L-system — a rewriting rule applied over and over, the way you might grow a plant in software.

Start with five copies of a symbol N, one every 72°:

[N]++[N]++[N]++[N]++[N]

That is where the five-fold symmetry comes from, and it is the only place it comes from. Then rewrite, every symbol at once, forever:

M → OF++PF----NF[-OF----MF]++
N → +OF--PF[---MF--NF]+
O → -MF++NF[+++OF++PF]-
P → --OF++++MF[+PF++++NF]--NF
F → (nothing)

M, N, O and P are the four half-tiles. + and - turn by 36°, brackets save and restore position, and F steps forward and draws. F rewrites to nothing, so the only Fs that survive to be drawn are the ones the last pass introduced — which is why each level draws a finer figure rather than adding to a coarser one.

At level 6 the string would be about 900,000 characters. This page never builds it: it walks the rules recursively and emits the geometry as it goes.

The count gives away the golden ratio

Every level, count the two kinds of tile and divide.

It is climbing to $\varphi = \tfrac{1}{2}(1+\sqrt5) \approx 1.6180$.

And $\varphi$ is irrational, which is the whole argument. If the tiling repeated — if some translation carried it onto itself — you could cut it into identical finite blocks, and the ratio of thick to thin would have to be the ratio of two whole numbers. It cannot be. The tiling cannot repeat.

Those counts are computed by this page as it draws, not typed in from a book.

What the sketch had to get right

The Processing sketch this came from drew the figure straight to PDF, in outline. Two things changed on the way here.

It drew on the tile symbols as well as on F, which splits every rhombus edge into two collinear halves — and then spent a whole pass finding those halves and gluing them back together. That pass kept only the edges it managed to pair, so any it missed vanished from the drawing. Not drawing on the tile symbols removes the doubling, the repair and the losses in one go.

The other was arithmetic. The sketch compared vertices with a tolerance while hashing them through a rounding function, so two points it called equal could land in different buckets — its own comment says as much, and shrugs. But a turtle taking unit steps at multiples of 36° never lands anywhere fuzzy: every vertex is an exact whole-number combination of five fixed directions, which collapse to four because $x^4-x^3+x^2-x+1 = 0$. Store those four integers and vertices compare exactly, with no tolerance anywhere. The five-fold symmetry can then be checked rather than eyeballed: rotating every vertex by 72° gives back the same set, exactly.

See also

The Spectre monotile does this with one tile instead of two — a question left open from Penrose’s tilings in the 1970s until it was finally settled in 2023.

Source

The rule set is from Przemysław Prusinkiewicz and Aristid Lindenmayer, The Algorithmic Beauty of Plants (Springer, 1990), and ships as one of Processing’s own examples. The tiling is Roger Penrose’s, from 1974.