Quasicrystal Pattern Generator
Make a quasicrystal pattern: a picture that has rotational symmetry no repeating tile can have, but never repeats. The tool adds up N plane waves, f(x, y) = Σ cos(K(x cos θₖ + y sin θₖ) + φ), with the N directions θₖ = πk/N spread evenly over half a turn, and colours the page by the value of f: bands of shade, two-tone (f above or below 0) or contour lines, all read from the same f. With phase 0 the pattern looks the same after a turn of 180°/N about the centre, so 5 waves give 10-fold symmetry. The pattern prints at real size, filling the page or at the width and height you type. Download a PDF, a 300 DPI PNG or an SVG. It is a picture made from waves, not a model of a real crystal. Nothing is uploaded.
100% private — everything runs in your browser. Everything runs in your browser; nothing is uploaded.
Print at 100% (Actual size): the pattern prints at real size. The pattern never repeats, so there is no repeat tile; the SVG holds the same bands or contour paths.
Waves
Look
Size
Page
Runs in your browser with our own code: nothing you type is uploaded or saved. Every band and line is worked out from the sum of waves by our own code. You may use what you make here, including commercially, with no credit needed.
How to make and print a quasicrystal pattern
- Pick a preset, or choose the number of waves N and the wavelength.
- Set the phase (0 keeps the full symmetry).
- Choose bands, two-tone, contour lines or two-tone with an outline, and the colours.
- Choose fill the page or a width and height.
- Download the PDF and print at 100% (Actual size), or the PNG or SVG.
What you can use it for
- Posters, wall art and prints
- Backgrounds for covers, cards and packaging (one panel; it does not tile)
- Maths lessons on symmetry and waves
- Laser and plotter art from the contour SVG
- Textile and screen-print panels
- Coasters and tiles cut at real size
The 14 presets
| Preset | Waves N | Wavelength | Phase | Look | Rotational symmetry | Printed size |
|---|---|---|---|---|---|---|
| 5 waves, grey bands (10-fold) | 5 | 12 mm | 0° | bands (4) | 10-fold | 184.5 mm × 236 mm |
| 7 waves, two-tone (14-fold) | 7 | 12 mm | 0° | two-tone | 14-fold | 184.5 mm × 236 mm |
| 5 waves, contour line f = 0 | 5 | 14 mm | 0° | contour lines (1 level) | 10-fold | 184.5 mm × 236 mm |
| 8 waves, teal bands (16-fold) | 8 | 12 mm | 0° | bands (8) | 16-fold | 184.5 mm × 236 mm |
| 9 waves, grey bands | 9 | 10 mm | 0° | bands (5) | 18-fold | 184.5 mm × 236 mm |
| 11 waves, plum two-tone | 11 | 10 mm | 0° | two-tone | 22-fold | 184.5 mm × 236 mm |
| 13 waves, three contour levels | 13 | 12 mm | 0° | contour lines (3 levels) | 26-fold | 184.5 mm × 236 mm |
| 6 waves, navy bands (12-fold) | 6 | 12 mm | 0° | bands (6) | 12-fold | 184.5 mm × 236 mm |
| 5 waves, phase 90° (shifted, not symmetric) | 5 | 12 mm | 90° | bands (4) | none (phase) | 184.5 mm × 236 mm |
| 7 waves, two-tone with outline | 7 | 14 mm | 0° | two-tone + outline | 14-fold | 184.5 mm × 236 mm |
| 5 waves, 30 mm wavelength (large) | 5 | 30 mm | 0° | bands (6) | 10-fold | 184.5 mm × 236 mm |
| 7 waves, 6 mm wavelength (fine texture) | 7 | 6 mm | 0° | two-tone | 14-fold | 184.5 mm × 236 mm |
| Real size 150 mm square, 8 waves | 8 | 10 mm | 0° | bands (4) | 16-fold | 150 mm × 150 mm |
| Real size 6 x 6 in, 5 waves, contour lines | 5 | 0.5 in | 0° | contour lines (2 levels) | 10-fold | 6 in × 6 in |
Sizes are worked out by the tool on US Letter; fill the page uses the printable area.
Symmetry and range, checked
| Waves N | Symmetry | Turn | f after the turn | f at the centre | f seen (4,000 points) | Possible range |
|---|---|---|---|---|---|---|
| 5 | 10-fold | 36° | same to 10^-9 | 5 | -3.96 to 4.90 | −5 to 5 |
| 6 | 12-fold | 30° | same to 10^-9 | 6 | -2.93 to 5.78 | −6 to 6 |
| 7 | 14-fold | 25.71° | same to 10^-9 | 7 | -5.74 to 6.32 | −7 to 7 |
| 8 | 16-fold | 22.5° | same to 10^-9 | 8 | -6.11 to 7.22 | −8 to 8 |
| 9 | 18-fold | 20° | same to 10^-9 | 9 | -4.02 to 8.12 | −9 to 9 |
| 10 | 20-fold | 18° | same to 10^-9 | 10 | -6.78 to 7.16 | −10 to 10 |
| 11 | 22-fold | 16.36° | same to 10^-9 | 11 | -7.01 to 9.93 | −11 to 11 |
| 12 | 24-fold | 15° | same to 10^-9 | 12 | -5.29 to 10.83 | −12 to 12 |
| 13 | 26-fold | 13.85° | same to 10^-9 | 13 | -8.71 to 11.73 | −13 to 13 |
Wavelength 12 mm, phase 0, random points within 150 mm of the centre.
What prints
| Download | What is in it |
|---|---|
| The pattern at real size, vector bands or contour lines, with N, the wavelength, the symmetry and a check bar | |
| PNG 300 DPI | The same sheet as a picture |
| SVG | The same bands (rectangles) or contour paths, at real size |
For a cutter that needs DXF, use SVG to DXF on the contour SVG.
Which tool to use
| You want | Use |
|---|---|
| A smooth never-repeating pattern from waves | Quasicrystal Pattern Generator (this page) |
| Kite and dart or rhombus tiles | Penrose Tiling Generator |
| Repeating star and girih tilings | Islamic Geometric Pattern Generator |
| Overlaid line screens | Moiré Pattern Generator |
Other tools: the Penrose Tiling Generator draws aperiodic tiles, the Islamic Geometric Pattern Generator repeating star patterns, and the Moiré Pattern Generator overlaid screens.
More free tools
Maths art and printables?
Drawing and print tools that run privately in your browser, with no sign-up and no watermark.
Frequently asked questions
What is a quasicrystal pattern?
A pattern that is ordered but never repeats. Here it is made by adding up N plane waves (stripes of cos) pointing in N directions spread evenly over half a turn. For N = 5 and above the directions do not fit any repeating grid, so no shift of the pattern lands it back on itself, and with phase 0 it has 2N-fold rotational symmetry about the centre: 10-fold for 5 waves, 14-fold for 7.
Is it a real quasicrystal?
No. It is a picture made from waves. Real quasicrystals are solids whose atoms are ordered in a pattern that never repeats; Dan Shechtman first saw one on 8 April 1982 and was awarded the 2011 Nobel Prize in Chemistry for the discovery (nobelprize.org press release, 5 October 2011). This page does not model any material.
Why 2N-fold, not N-fold?
Turning the plane by 180°/N moves each wave on to the next one, and the last on to the first pointing the opposite way, which is the same wave when the phase is 0 or 180°. So the picture repeats every 180°/N, which is 2N times in a full turn. The table below shows our test for N = 5 to 13: the value of f after the turn is the same to 10^-9.
What does the phase do?
It shifts every wave by the same amount. At 0 or 180° the full symmetry holds; other phases shift the pattern and break the exact symmetry, and the footer says so.
What values can f take?
Each wave is between −1 and 1, so f is between −N and N, and f = N at the centre, where all the waves peak together. Bands split that range into equal steps; two-tone shows f ≥ 0 in the light colour; contour lines are drawn at f = 0, or at 2 to 5 levels evenly from −N/2 to N/2.
Does it tile seamlessly?
No. The pattern never repeats, so there is no tile to repeat. Print it as one panel at the size you need; our tests check that no shift of up to 60 mm maps it on to itself.
How is it different from a Penrose tiling?
A Penrose tiling is made of two kinds of tile (kites and darts, or two rhombi) with sharp edges. This page draws a smooth function, the sum of waves, as bands or contours. Both have 5-fold style order that never repeats.
Is it free?
Yes. There is no sign-up, no watermark and no limit.
Is anything uploaded?
No. Every arrow, line and band is worked out by JavaScript in your browser. Nothing you type is sent anywhere or saved when you close the page.