How to make a Penrose pattern
- Choose Kites and darts (P2) or Rhombs (P3), and a starting patch: the sun, the star, or a single tile.
- Raise Subdivisions. Every step replaces each triangle by two or three smaller ones, so the tiles shrink by a factor of 1.618 and the count grows by about 2.6.
- Pick colours, a line width and, if you want the rules visible, the Matching-rule arcs. Drag to pan, scroll or pinch to zoom, and use the rotation slider to turn the whole pattern.
- Set the Tile edge in millimetres, decide what happens to tiles cut by the frame, and press Save SVG. The info line under the buttons gives the size of the file in millimetres before you save it.
Every setting, including the view, is kept in the address bar, so Copy link to this pattern gives you a link that reproduces the exact frame you exported.
Kites and darts, or rhombs?
Roger Penrose found these tilings in 1974. The two best-known sets are both here. P2 uses a kite (angles 72°, 72°, 72°, 144°) and a dart (36°, 72°, 36°, 216°), whose long and short edges are in the ratio φ = (1 + √5) / 2 ≈ 1.618. P3 uses a thick rhomb (72° and 108°) and a thin rhomb (36° and 144°) with all four edges equal — which is why it is the usual choice for floors and wooden puzzles: every tile has the same edge, so any edge can meet any other.
Both sets can be cut in half into the same two shapes: a triangle with a 36° apex and one with a 108° apex, the Robinson triangles. Tick Show the Robinson triangles to see the cut. A kite is two 36° triangles joined along its axis, a dart is two 108° triangles; a thin rhomb is two 36° triangles joined along their short base, a thick rhomb two 108° triangles joined along their long base. The generator works entirely with these triangles and joins mirror-image pairs into whole tiles at the end.
The rule that builds everything
A subdivision step (also called deflation) cuts each triangle into smaller Robinson triangles whose edges are 1/φ of the old ones:
| P3 rhombs | P2 kites and darts | |
|---|---|---|
| 36° triangle becomes | one 36° + one 108° | two 36° + one 108° |
| 108° triangle becomes | one 36° + two 108° | one 36° + one 108° |
The new corners sit at golden-ratio points of the old edges — in code, a point P = A + (B − A) / φ on an edge from A to B. Nothing else is needed: no search, no backtracking, no table of allowed neighbours. Starting from the sun (ten 36° triangles around a point), five rounds give 1,440 triangles for P2 and 890 for P3; ten rounds give 177,110 and 109,460.
Placing tiles freely is a different matter. A kite and a dart put together make a rhombus that tiles the plane periodically, so the shapes alone do not force a Penrose tiling. The matching rules do. Turn on Matching-rule arcs: every tile carries two arcs in two colours, and in a legal tiling every arc continues across every edge into the next tile, forming unbroken curves. Here the kite has an arc of radius 1/φ of its long edge around its head and one of 1/φ³ around its tail; the dart has arcs of 1/φ of its short edge around its head and its notch; the thick rhomb has arcs of 1/φ and 1/φ² around its two sharp corners and the thin rhomb two arcs of 1/φ² around its blunt corners. The verification script for this page checks that every arc meets its partner on every interior edge.
The rules also limit what can happen at a corner. In the patches this page generates, counted at eight subdivisions, the kites and darts meet in exactly seven different ways — Conway’s sun, star, ace, deuce, jack, queen and king — and the rhombs in exactly eight.
Why the counts close in on φ
Because one 36° triangle produces one of each and one 108° triangle produces one 36° and two 108° (in the rhomb tiling), the two counts obey a Fibonacci-type recurrence, and the ratio between them is pushed towards the golden ratio. Nothing in the rule aims at φ; it falls out. The Japanese article this tool comes from counted the rhomb tiling from the sun start:
| Subdivisions | 36° triangles | 108° triangles |
|---|---|---|
| 1 | 10 | 10 |
| 3 | 50 | 80 |
| 5 | 340 | 550 |
| 8 | 6,100 | 9,870 |
| 10 | 41,810 | 67,650 |
After ten subdivisions the ratio agreed with φ to eight significant digits. Choose Rhombs (P3), the sun start and ten subdivisions (the preset Ten subdivisions does it) and the table under the tool shows the same counts; the ratio is 67,650 / 41,810 = 1.6180339…, against φ = 1.6180339887. The counts are ten times consecutive Fibonacci numbers, and each subdivision moves two places along the sequence, so the gap to φ shrinks by a factor of φ⁴ ≈ 6.9 every time — the straight line the article drew on a log scale. Kites and darts behave the same way with the roles swapped: there the 36° triangles are the majority.
It never repeats — measured, not just stated
“Aperiodic” has a concrete meaning: no translation of the whole tiling maps it onto itself. The article tested that directly. It shifted the set of vertices by 1,500 difference vectors between nearby vertices and recorded how many vertices landed on vertices. A periodic triangular lattice reached 100.00%, as it must; the Penrose tiling’s best shift reached only 90.53%. Nine vertices in ten line up, and the last tenth never does.
The second card under the tool runs the same kind of test on 6,000 vertices around the centre of your view: the 1,500 translations are the vectors from the most central vertex to its 1,500 nearest neighbours. With the rhombs and the view at the centre it finds a best overlap of 90.68%, from a shift of about 13 edge lengths; the triangular lattice gets 100.00%. That is close to the article’s figure but not identical, because the window and the choice of candidate shifts differ. In the other windows the verification script tried, the rhombs’ best value stayed between 90.68% and 91.30% — never near 100%.
The other card computes the diffraction pattern, |Σ exp(−i k·r)|² summed over vertices, which is what an X-ray experiment sees. The article found the strongest angular order to be 10, with sharp spots. Sharp spots mean long-range order; ten-fold symmetry is impossible for a periodic crystal, which is allowed only two-, three-, four- and six-fold symmetry. Real materials with this combination — quasicrystals — were later found, and the discovery won the 2011 Nobel Prize in Chemistry. One caution: the sun patch is itself five-fold symmetric, so with the window at its centre the ten-fold answer is guaranteed by the patch. Pan away first. With the window off-centre this page still finds order 10 strongest, and a triangular lattice gives order 6.
Using it for quilts, floors, wallpaper and laser cutting
- Quilts. Use Fill tiles by: Tile direction — the rhombs point in only five directions, and colouring by direction gives the classic star-and-flower look. The tile counts tell you how many pieces of each shape to cut; switch on Whole tiles only and the SVG contains only complete pieces.
- Floors and wallpaper. Rhombs are easier to lay because all edges are equal. There is no repeat to print side by side, so zoom out until the frame covers the whole wall and export once.
- Laser cutting and plotters. The Laser cutting outline preset turns off fills, sets thin lines and chooses Each edge once, so a shared edge is not cut twice. The stroke width in the SVG is the line-width percentage of the tile edge, shown in millimetres on the info line.
- Puzzles. Print the One tile start with arcs shown: the arcs are the rule a player has to follow, and they are what makes the puzzle hard.
Limits
This is a finite patch, not an infinite tiling. Triangles whose partner would lie outside the patch are drawn as halves along the rim; the frame options decide whether they reach your file. Ten subdivisions is the maximum, about 177,000 triangles for kites and darts, which is already finer than a screen can show without zooming. And a pattern made by subdivision is always a correct Penrose tiling, but it is one particular one: the sun and star starts give patches with exact five-fold symmetry at their centre, which an arbitrary Penrose tiling does not have.