Why do sunflower seeds form spirals?

Look closely at the center of a ripe sunflower. The seeds do not sit in rings or rows. They curve out in two sets of spirals, one clockwise and one counterclockwise. Count each set, and you will often get two neighboring Fibonacci numbers. In that sequence, each number is the sum of the two before it: 1, 2, 3, 5, 8, 13, 21, 34, 55. Plus Magazine: Citizen scientists count sunflower spirals.

How can one angle make spirals?

Picture a seed head built one seed at a time. Each new seed sits a little farther from the center than the last. It also turns by the same angle from the seed before. That single angle decides the whole pattern.

Pick a simple fraction of a turn, and the seeds line up. At 120 degrees, a third of a turn, every third seed lands on the same line from the center. The head grows three straight spokes, with big empty wedges between. At 135 degrees, three-eighths of a turn, it grows eight spokes.

Now pick about 137.5 degrees, the golden angle. It comes from the golden ratio, about 1.618, and it is not any exact fraction of a turn. So no seed ever lands exactly on an earlier seed's line. The seeds fill the gaps instead. Near misses still line up into curves, and those curves are the spirals you count. Wikipedia: Golden angle.

Botanists call the arrangement of seeds and leaves on a plant phyllotaxis. Wikipedia: Phyllotaxis.

▶ Play with this now in Golden Angle Garden: drag the angle from 135° to golden and watch spirals appear.

Why do the spiral counts change?

The spirals you see depend on the size of the head. In our model, a 400-seed head shows 34 spirals one way and 55 the other. Shrink it to 150 seeds, and the rim shows 21 and 34. Grow it to 1,200, and it shows 55 and 89.

Each pair is two neighbors in the Fibonacci sequence. The golden angle's closest near misses come at Fibonacci steps. Every 34th seed lands about 4.7 degrees off the first seed's line. Every 55th seed lands about 2.9 degrees off. The misses keep shrinking but never reach zero.

The model places seeds with a rule Helmut Vogel published in 1979. Each seed's distance from the center grows with the square root of its number. That gives every seed the same share of area. Wikipedia: Fermat's spiral.

What does the model leave out?

Golden Angle Garden is a mathematical placement model inspired by plants. Real plants vary and do not follow one perfect rule. The model has no growth physics, no florets and no petals, and every seed is the same size.

In a growing plant, each new bud tends to form in the least crowded spot near the tip. That jostling can produce turns close to the golden angle. It does not make the angle exact. Wikipedia: Phyllotaxis.

Real seed heads are messier, too. In the Turing's Sunflowers project, volunteers grew sunflowers and counted their spirals. In the most reliable data, 565 of 768 counts were Fibonacci numbers, about 74 percent. Another 67 fit related patterns, and the rest fit neither. Swinton, Ochu and colleagues (2016).

The golden angle is also not the only angle that packs seeds evenly. The model includes another irrational angle, about 99.5 degrees, that packs them about as well.

Can you count the spirals yourself?

A pinecone works well, because its scales are big. A sunflower head or a pineapple works too. These steps are a suggestion, so adapt them to the child beside you.

Predict. Turn the pinecone over and look at its base. The scales curve out in spirals, some turning left and some turning right. Ask your child: will the two counts be the same? Will they be Fibonacci numbers?

Try. Mark one spiral with a twist of thread or a dab of washable paint. Count the spirals that turn one way, all the way back to your mark. Then count the other way. Count each set twice to check.

Explain. Pinecones often show two neighboring Fibonacci numbers. Wikipedia: Fibonacci sequence. Yours might not, and that is a real result too. A plant grows by a process, not by a perfect rule. If you have a sunflower, count it next and compare.

Turing's Sunflowers was run by Jonathan Swinton and Erinma Ochu with Manchester's Museum of Science and Industry. Plus Magazine shows how to spot both families of spirals on a real head.

Golden Angle Garden is an ExplainerTools Original. Its seed rule is Helmut Vogel's 1979 construction.