Phyllotaxis
Draw a phyllotaxis seed pattern: place seed n at distance c√n from the centre and turn by the same angle between one seed and the next. With the golden angle, about 137.508°, the seeds pack evenly and form crossing spiral arms; change the angle a little and the arms break up, or pick a simple fraction of a turn and they line up into straight spokes. The tool draws 10 to 20,000 seeds, counts the two families of arms from the seeds themselves, can draw those arms, and can draw Fermat’s spiral through the first seeds. Number the seeds, colour them by arm, and print a seed template at real size. Download a PDF, a 300 DPI PNG or an SVG with every seed as an exact circle. Nothing is uploaded.
100% private — everything runs in your browser. Everything runs in your browser; nothing is uploaded.
Print at 100% (Actual size). Real size prints on one sheet when the drawing fits; the check bar in the footer lets you confirm the scale. The SVG holds the drawing at the printed size.
Seeds
Page
Runs in your browser with our own code: nothing you type is uploaded or saved. Every seed is placed by r = c√n, θ = n × angle, and the arm counts are worked out from the seeds by our own code. You may use what you make here, including commercially, with no credit needed.
How to draw a phyllotaxis seed pattern
- Pick a preset, or set the number of seeds and the divergence angle (the golden angle or another one).
- Set the spacing c and the seed size, and the colours.
- Turn on the arms, Fermat’s spiral or the seed numbers if you like.
- Check the arm counts under the preview, choose Fitted to one page or Real size and the paper.
- Download the PDF and print at 100% (Actual size), or the PNG or SVG.
What you can use it for
- Maths lessons: the golden angle, Fibonacci numbers and spirals
- Biology and art lessons on seed-head patterns
- Seed and dot templates for paper, clay or embroidery
- Posters and wall art
- Plotter, laser and vinyl art from the SVG
- Trying angles near the golden angle
The 15 presets
| Preset | Angle | Seeds | Arms found in the outer seeds | |
|---|---|---|---|---|
| Golden angle, 500 seeds | 137.50776° | 500 | 34 arms clockwise and 55 arms anticlockwise | fitted |
| Golden angle, 1,000 seeds | 137.50776° | 1,000 | 55 arms anticlockwise and 89 arms clockwise | fitted |
| Golden angle, 2,000 seeds, small seeds | 137.50776° | 2,000 | 89 arms clockwise and 144 arms anticlockwise | fitted |
| Golden angle, 60 seeds, numbered, Fermat’s spiral through them | 137.50776° | 60 | 13 arms clockwise and 21 arms anticlockwise | fitted |
| Golden angle, 800 seeds, the two arm families drawn | 137.50776° | 800 | 55 arms anticlockwise and 89 arms clockwise | fitted |
| Golden angle, 1,000 seeds coloured by arm | 137.50776° | 1,000 | 55 arms anticlockwise and 89 arms clockwise | fitted |
| Golden angle, 1,500 seeds, colour fades outward | 137.50776° | 1,500 | 55 arms anticlockwise and 89 arms clockwise | fitted |
| 137.3° (just under the golden angle), 1,000 seeds | 137.3° | 1,000 | 21 arms anticlockwise and 97 arms clockwise | fitted |
| 137.6° (just over the golden angle), 1,000 seeds | 137.6° | 1,000 | 34 arms clockwise, no second family crossing them | fitted |
| 90° (a quarter turn): 4 straight spokes | 90° | 400 | 4 straight spokes | fitted |
| 144° (two fifths of a turn): 5 straight spokes | 144° | 400 | 5 straight spokes | fitted |
| 137.1428…° (= 360° × 8/21): 21 straight spokes | 137.1429° | 800 | 21 straight spokes | fitted |
| 222.49° (360° minus the golden angle): the mirror image | 222.4922° | 500 | 34 arms anticlockwise and 55 arms clockwise | fitted |
| Real size template: 300 seeds, c = 4.8 mm, 8 mm seeds, numbered | 137.50776° | 300 | 34 arms clockwise and 55 arms anticlockwise | real size |
| Poster: 3,000 seeds | 137.50776° | 3,000 | 89 arms clockwise and 144 arms anticlockwise | fitted |
Every arm count is worked out by the tool from the seeds it draws.
Arm counts at the golden angle
| Seeds | First family | Second family | Neighbouring Fibonacci numbers |
|---|---|---|---|
| 50 | 13 clockwise | 21 anticlockwise | yes |
| 100 | 13 clockwise | 21 anticlockwise | yes |
| 200 | 21 anticlockwise | 34 clockwise | yes |
| 300 | 34 clockwise | 55 anticlockwise | yes |
| 500 | 34 clockwise | 55 anticlockwise | yes |
| 1,000 | 55 anticlockwise | 89 clockwise | yes |
| 2,000 | 89 clockwise | 144 anticlockwise | yes |
| 5,000 | 144 anticlockwise | 233 clockwise | yes |
| 10,000 | 233 clockwise | 377 anticlockwise | yes |
| 20,000 | 233 clockwise | 377 anticlockwise | yes |
Counted from the two nearest seeds of every seed in the outer fifth. The more seeds, the more arms you see at the edge.
Other angles, 1,000 seeds
| Angle | What the outer seeds form |
|---|---|
| 137.3° | 21 arms anticlockwise and 97 arms clockwise |
| 137.6° | 34 arms clockwise, no second family crossing them |
| 90° (1/4 turn) | 4 straight spokes |
| 144° (2/5 turn) | 5 straight spokes |
| 135° (3/8 turn) | 8 straight spokes |
| 137.1429° (8/21 turn) | 21 straight spokes |
| 149.1169° (360° × (√2 − 1)) | 70 arms clockwise and 99 arms anticlockwise |
| 222.4922° (360° − golden angle) | 55 arms clockwise and 89 arms anticlockwise |
A simple fraction of a turn gives straight spokes; angles just off the golden angle give arms that do not cross evenly.
Fibonacci fractions of a turn
| Fraction of a turn | Angle | Difference from the golden angle |
|---|---|---|
| 3/8 | 135° | −2.5078° |
| 5/13 | 138.4615° | 0.9538° |
| 8/21 | 137.1429° | −0.3649° |
| 13/34 | 137.6471° | 0.1393° |
| 21/55 | 137.4545° | −0.0532° |
| 34/89 | 137.5281° | 0.0203° |
| 55/144 | 137.5° | −0.0078° |
Each fraction F(k)/F(k + 2) of a full turn is closer to the golden angle than the one before, from alternate sides.
Seeds on Fermat’s spiral (c = 1, golden angle)
| Seed n | r = c√n | θ = n × golden angle (radians) | a√θ, a = c / √α |
|---|---|---|---|
| 1 | 1 | 2.399963 | 1 |
| 2 | 1.414214 | 4.799926 | 1.414214 |
| 3 | 1.732051 | 7.19989 | 1.732051 |
| 4 | 2 | 9.599853 | 2 |
| 5 | 2.236068 | 11.999816 | 2.236068 |
| 10 | 3.162278 | 23.999632 | 3.162278 |
| 20 | 4.472136 | 47.999265 | 4.472136 |
The second and fourth columns agree: every seed lies on Fermat’s spiral r = a√θ.
What prints
| Download | What is in it |
|---|---|
| The seeds fitted to one page or at real size, vector, with the arm counts and a check bar | |
| PNG 300 DPI | The same sheet as a picture |
| SVG | Every seed as an exact circle at the printed size, with the arms, Fermat’s spiral and numbers if shown |
For a cutter that needs DXF, use SVG to DXF.
Which tool to use
| You want | Use |
|---|---|
| Seeds placed by the golden angle, arm counts, Fermat’s spiral | Phyllotaxis (this page) |
| Numbers on a square spiral with the primes marked | Ulam Spiral |
| Curves drawn by rolling circles | Spirograph Generator |
| Circular patterns to colour | Mandala Maker |
| Golden ratio sizes | Golden Ratio Calculator |
Other tools: the Spirograph Generator draws rolling-circle curves, the Mandala Maker draws circular patterns and the Golden Ratio Calculator works out golden sizes.
More free tools
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Frequently asked questions
What is phyllotaxis?
In botany, phyllotaxis is the arrangement of leaves on a plant stem; the word is also used for spiral seed and floret patterns. This tool draws the seed pattern of Vogel’s model, the standard mathematical picture of a seed head.
What is Vogel’s model?
A model proposed by H. Vogel in 1979 (A better way to construct the sunflower head, Mathematical Biosciences 44, pages 179–189): seed n sits at distance r = c√n from the centre and angle θ = n × 137.508°, where c is a scale. The tool uses exactly those formulas, with any angle you choose.
What is the golden angle?
The smaller of the two angles you get by splitting a full turn in the golden ratio: 360° × (2 − φ) = 180° × (3 − √5), about 137.50776°, or about 2.39996 radians. The tool uses the exact formula, and our tests check it to 12 decimal places.
Where is Fermat’s spiral?
Fermat’s spiral is r = ±a√θ. Vogel’s seeds lie on it: with the angle α in radians, seed n is at angle nα and distance c√n = a√(nα) with a = c / √α. Turn on Fermat’s spiral to draw it through the first seeds; the dashed curve is the other branch, r = −a√θ. Our tests check that every seed lies on the curve.
What are the spiral arms (parastichies)?
The spirals your eye picks out, running one way and the other. The tool finds them from the seeds: for every seed in the outer fifth it finds the two nearest seeds and notes how far apart their numbers are. The two most common gaps are the arm counts: with a gap of d, seeds n, n + d, n + 2d … sit on one arm, so there are d arms. At the golden angle the counts come out as two neighbouring Fibonacci numbers (21 and 34, 34 and 55, 55 and 89 …) for every seed count in the table on this page, from 50 to 20,000.
Why do other angles make spokes?
If the angle is a simple fraction of a turn, such as 1/4 (90°) or 2/5 (144°), then seed n and seed n + 4 (or n + 5) point the same way, so the seeds line up on 4 (or 5) straight spokes. The golden angle is not a fraction of a turn, so the seeds never line up exactly. Fractions of Fibonacci numbers, such as 8/21 or 13/34 of a turn, come close to it; a table on this page shows how close.
Does every plant follow this?
No claim of that is made here. The Fermat’s spiral article on Wikipedia gives the sunflower and the daisy as examples of disc phyllotaxis and Vogel’s 1979 model as a model of the sunflower head; this tool draws the model, not any particular plant.
How big can it go?
10 to 20,000 seeds; every preset draws in under 2 seconds in our tests. Numbers on the seeds work up to 600 seeds, and Fermat’s spiral can be drawn through up to 300 seeds.
Can it print at real size?
Yes, when the drawing fits on one sheet: choose Real size and set the cell size (Ulam), the diameter (Sacks) or the spacing c and seed size (phyllotaxis). Every printed cell, dot and seed is within 0.2 mm of its true place and size in our tests, on US Letter and A4, as long as you print at 100% (Actual size); the check bar in the footer lets you confirm the scale. Bigger drawings print fitted to the page.
Is it free?
Yes. There is no sign-up, no watermark and no limit.
Is anything uploaded?
No. Every number, cell and seed is worked out by JavaScript in your browser. Nothing you type is sent anywhere or saved when you close the page.