Penrose Tiling

Two triangle shapes, one deflation rule, and a tiling of the plane that never repeats.

Two triangles

Roger Penrose’s 1974 tiling is usually drawn as two rhombs — a fat one and a thin one. This construction uses the equivalent pair of Robinson triangles: each rhomb cut along its short diagonal, giving two isoceles triangles whose sides are in the golden ratio.

Working in triangles rather than rhombs makes the recursion easy to write, because each triangle’s replacement is a fixed arrangement of triangles rather than a partial rhomb.

Step 01

The seed: ten triangles arranged around a common point with five-fold symmetry. That starting arrangement is what makes this the “sun” rendering; other seeds give the star, and other patches again.

Deflation

for (var iter = 0; iter < iterations; iter++)
{
    var next = new List<Triangle>(triangles.Count * 2);
    foreach (var t in triangles) Subdivide(t, next);
    triangles = next;
}

Each triangle is replaced by two or three smaller triangles of the same two shapes. The new pieces are smaller by exactly φ = 1.618…, and that is not a stylistic choice — it is forced. The two triangles’ sides are in the golden ratio, so the only scale factor at which a subdivided triangle’s pieces have the same two shapes is φ.

Step 02

One deflation.

Step 03

Two. The tile count grows by about φ² each step, so a seven-deflation tiling has roughly φ¹⁴ ≈ 843 times as many tiles as the seed.

Why it cannot repeat

Step 04

Three deflations. Local five-fold symmetry is visible all over the patch, and that on its own is the proof sketch: the crystallographic restriction theorem says a periodic tiling of the plane can only have 2-, 3-, 4- or 6-fold symmetry. Five-fold is impossible for anything periodic.

The stronger statement is about ratios. In any Penrose tiling the number of fat rhombs to thin rhombs tends to exactly φ, which is irrational — and a periodic tiling would have to have a rational ratio, because it would repeat a finite cell containing whole numbers of each.

Step 05

Seven deflations.

Every patch appears infinitely often

A Penrose tiling never repeats, and yet every finite patch of it appears infinitely many times, somewhere. Take any region of this drawing and you will find a congruent copy of it arbitrarily far away — just never at a fixed offset.

That property has a name — the tiling is repetitive but not periodic — and it is why quasicrystals produce sharp diffraction patterns despite having no lattice at all. Dan Shechtman’s 1982 discovery of exactly that was rejected for two years and won the Nobel Prize in 2011.

Six renderings

Rendering	Seed	Iterations	What varies
Sun	Sun	7	Warm gold kites, deep blue darts
Star	Star	7	Terracotta kites, ocean darts
Coarse	Sun	5	Large bold tiles on a light ground
Mesh	Sun	7	Stroke as lattice; fills near-invisible
Frost	Star	7	Cool monochromatic blues
Museum	Sun	7	Sun tiling as a matted medallion

The seed decides which of the finitely many local arrangements sits at the centre; the deflation rule is identical in all six, and so is everything beyond the first few rings. The coarse rendering uses five deflation steps instead of seven, producing tiles roughly φ² ≈ 2.6× larger — individual kites and darts read as bold shapes rather than fine texture. The mesh rendering inverts the usual convention: kite tiles share the background colour and vanish, leaving the stroke network — the Penrose lattice itself — as the subject. The museum rendering presents the sun tiling as a circular medallion centred on a warm ivory mat with a thin gold border, as it might appear in a gallery.

Determinism

Nothing is seeded and nothing is random. Deflation is exact complex arithmetic on golden-ratio multiples of the previous vertices, applied in list order.

Progression 05

Variant 06