Sunflower Phyllotaxis Pattern Calculator
Explore the golden-angle spiral that guides sunflower seed placement. Enter a seed count, scale, and how many coordinates you would like to preview, then copy the results to design garden beds, art pieces, or fabrication layouts.
Phyllotaxis and the Beauty of Numbers
The mesmerizing spiral pattern seen in sunflower heads, pinecones, and cactus spines arises from phyllotaxis, the arrangement of leaves or florets on a plant stem. Botanists have long marveled at the efficiency of these patterns, which often reflect the golden angle and Fibonacci sequence. This calculator lets gardeners, artists, and mathematicians explore the geometry by generating radial coordinates for a sunflower-style seed layout. By adjusting the number of seeds and the scaling factor, you can design fractal artwork, optimize planting densities, or simply appreciate the mathematics of nature.
The Golden Angle Formula
Plants that display radial symmetry tend to place successive seeds or leaves at a constant divergence angle to avoid overlap and maximize light exposure. The most efficient packing occurs when this angle is irrational relative to a full rotation, preventing seeds from lining up in radial spokes. The ideal value is the golden angle , derived from the golden ratio . The golden angle itself satisfies
Once the divergence angle is fixed, the radial distance of the -th seed from the center is
where is a scaling constant. The angular position is . Converting to Cartesian coordinates yields and . These compact formulas fuel the coordinate generator below.
Example Coordinates
| n | x (cm) | y (cm) |
|---|---|---|
| 1 | 0.38 | 0.11 |
| 2 | 0.12 | 0.66 |
| 3 | -0.54 | 0.69 |
| 4 | -0.91 | 0.04 |
| 5 | -0.36 | -0.93 |
Fibonacci and Visible Spirals
As seeds accumulate, observers often count two sets of interleaving spirals winding in opposite directions. Remarkably, the number of visible spirals typically corresponds to consecutive Fibonacci numbers such as 34 and 55. This phenomenon arises because fractions formed by successive Fibonacci ratios provide the best rational approximations to the golden ratio. In phyllotaxis, these approximations create near alignments every few seeds, generating the illusion of spiral families. Our calculator uses the exact golden angle, so the Fibonacci structure appears naturally.
Table of Sample Radii
| Seed Count N | Scale c (cm) | Max Radius (cm) |
|---|---|---|
| 100 | 0.5 | 5.00 |
| 200 | 0.5 | 7.07 |
| 500 | 0.5 | 11.18 |
| 500 | 0.8 | 17.89 |
Because the radius grows with the square root of the seed index, doubling the number of seeds increases the pattern diameter by only about 41%. This insight helps gardeners estimate how much space a spiral bed will require and lets makers plan material usage for installations.
Continue exploring spirals and number patterns with the Fibonacci sequence calculator, the circle area calculator, and the Parker spiral magnetic field calculator to see how rotational symmetry appears across math, physics, and design.