Guide · Form No 176 · Aperiodic tiling P3

Making Penrose

One page, one job: make aperiodicity visible. A Penrose rhomb field grows by golden-ratio deflation while the camera eases in, so the pattern deepens forever — and the matching-rule arcs surface on a cycle to show why it can never repeat.

The concept

Penrose is a study page for the P3 rhombus tiling: two shapes — a fat 72° rhomb and a thin 36° one — that tile the whole plane yet never fall into a period. The audience is anyone who has seen the pattern on a bathroom floor or a Nobel citation and wants to feel the mechanism, not read a proof. The single job of the page is the hero: a live substitution tiling, four to five generations deep at rest, deflating in real time under an infinite eased zoom.

Palette — the Darb-i Imam lineage

The colour argument comes from the girih tilework of the Darb-i Imam shrine (Isfahan, 1453), whose strapwork shares the Penrose deep structure: gilt line on deep lapis ground, with dusty rose reserved for the fat rhombs. The bright gilt is a line and mark token; body copy sits in parchment ink so small text never rides the accent.

Gilt on lapis measures ≈ 8.3:1 and parchment ≈ 13:1 — both clear the 4.5:1 body floor, but gilt still stays reserved for rules, counters and italic accents.

Type

Cormorant Garamond carries the display voice — a Renaissance-blooded serif whose italic has the calligraphic drift of girih strapwork; the hero wordmark is set italic at up to 164 px. IBM Plex Mono runs the instrument panel: generation counter, rhomb tally and the fat:thin ratio ticking toward φ. Two faces, two registers — geometry gets the serif's romance, measurement gets the mono's neutrality.

Technique

Substitution tiling · canvas 2D

The field is built from Robinson half-rhomb triangles. Each generation, a fat half splits into three children and a thin half into two, every edge shrinking by 1/φ — the classic P3 deflation. Triangles are batched into six Path2D fill buckets (three shades per rhomb type), one gilt edge path and one arc path, so a frame stays at roughly a dozen draw calls no matter how many thousand tiles are alive.

The infinite zoom — a φ rebase

The camera scale grows exponentially, one factor of φ every seven seconds. When it crosses the threshold, the engine culls tiles outside a 2× viewport margin, subdivides the survivors once, then rebases the world: every coordinate is re-scaled by φ about the camera and the zoom divided by φ in the same frame. Nothing on screen moves, but coordinates and scale both return to O(1) — so the zoom can run for hours without floating-point drift. The camera then eases toward the centroid of whichever new triangle contains it, so the descent path is itself aperiodic.

Matching-rule arcs

Each half-rhomb draws one circular arc about its apex at radius 1/φ² or 1/φ of the edge — small at one corner of the rhomb, large at the opposite — the decoration Penrose used to state the matching rules. The arc layer breathes on a thirteen-second cycle, surfacing to gilt and sinking back, so the rule reads as a recurring revelation rather than permanent ornament.

Second read

One rhomb near the centre carries a faint gilt glow. It is a lineage: at every deflation the glow passes to one of its children, so the highlighted tile survives the zoom forever — the same rhomb, endlessly subdividing. On a fine pointer, hovering any tile names it in the HUD (fat 72°/108° or thin 36°/144°) and lights its mirror half.

Iteration log