Fibonacci Bloom

A field of seeds, placed one at a time at the golden angle (137.507 degrees) from the previous seed, just like a sunflower head builds itself. The whole bloom breathes in and out gently, contracting toward the center on the inhale and radiating outward on the exhale. The slider lets you change how many seeds the bloom contains; when you land on a Fibonacci number, the two sets of curving arms running through the pattern snap into perfect alignment and glow.

Those are called parastichies. They are not drawn by the plant; they are an emergent visual pattern that your eye traces through the seeds. There are always two sets of them, one curving clockwise and one counterclockwise, and the number of arms in each set is always two consecutive Fibonacci numbers: 5 and 8, 8 and 13, 13 and 21, 21 and 34, and so on. Slide through the Fibonacci snap points to watch the arms reorganize.

Because plants do, in a sense. The seed primordia of a real sunflower expand outward over weeks of growth, with the newest seeds nearest the center. The breathing animation compresses that lifetime growth into a six-second cycle so you can see the structure assemble and disperse continuously. The cycle length is also tuned to match slow, conscious diaphragmatic breathing, the rate associated with parasympathetic activation. Synchronize your breath with the bloom and notice how it feels.

It is a turn of 137.507 degrees, which comes from dividing a full circle (360 degrees) by the golden ratio (about 1.618, written with the Greek letter phi). Because the golden ratio cannot be written as any simple fraction, no two seeds placed at this angle ever line up, even after thousands of placements. That makes the golden angle the most efficient possible way to fill a flat space evenly.

It is the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987... where each number is the sum of the two before it. As you move further along, the ratio between each pair of consecutive Fibonacci numbers gets closer and closer to phi (the golden ratio). That is why Fibonacci numbers and the golden angle are inseparable in nature.

The visible parastichy arms are determined by which Fibonacci numbers best approximate the rate at which seeds wrap around the center. When the total number of seeds happens to be a Fibonacci number, the alignment between the radial structure (spiral arms) and the count of seeds becomes exact instead of approximate, so the arms become crisper and more visually defined. It is a moment where the geometry and the arithmetic agree.

Sunflower heads (typically 34 and 55 arms, or 55 and 89 in larger varieties), pinecones (8 and 13 arms), pineapples (8 and 13, sometimes 13 and 21), artichokes, succulents like Aloe polyphylla, the seed heads of daisies and chamomile, and the leaf arrangements of many plants from cacti to ferns. The same packing problem is solved by the same answer wherever it appears.

The pattern this tool draws is called Vogel's model, after mathematician Helmut Vogel who described it in 1979 with the elegant formula r = c√n, theta = n × 137.5°. The deeper biological explanation came in 1992 when physicists Stephane Douady and Yves Couder showed experimentally that any system where new units repel existing ones in turn produces the golden angle automatically. The plant does not have to know the math; the math is what efficient packing looks like.

Yes, closely. The golden spiral is a logarithmic spiral that grows by phi every quarter turn, the smooth curve you see in nautilus shells and galaxy arms. The parastichy arms in this bloom are discrete approximations of that same logarithmic curve, traced out by the seeds. If you smoothed the arm of dots into a continuous line, you would have a golden spiral.

Several likely reasons. The breathing rhythm matches the slow pace of conscious diaphragmatic breathing (about six seconds per cycle, the rate associated with parasympathetic nervous system activation). The pattern itself is highly regular without being repetitive, which the brain finds easier to process than either pure noise or pure repetition. And we evolved surrounded by phi-based growth patterns in plants and shells, so the proportions feel familiar and natural at a level deeper than thought.