Science · Nature

Why Fibonacci Numbers Appear in Nature

Where the pattern is genuine, and where claims get stretched.

The question

Why do sunflower seeds so often form 34 and 55 spirals?

The short answer

When a plant adds each new seed or leaf at a fixed angle of about 137.5°, the result packs efficiently and the visible spirals usually come in Fibonacci numbers. It is common, not universal.

137.5° between each seed
Seeds placed one after another at the golden angle, 137.5°, pack evenly and form interlocking spirals.

What science knows

Established Science

In phyllotaxis, the arrangement of leaves and seeds, new organs form at the growing tip roughly 137.5° from the last, the “golden angle”. A simple 1979 model by Helmut Vogel reproduces sunflower heads this way.

Seeds placed at this angle leave few gaps, and the spirals your eye picks out usually number 21 and 34, 34 and 55, or 55 and 89.

The pattern

Established Mathematics

Each Fibonacci number is the sum of the two before it: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89. The golden angle is 360° divided by φ², which is why the two are linked.

What is still uncertain

Mixed Evidence

A 2016 citizen-science study of hundreds of sunflowers found Fibonacci patterns in most heads, but around one in five showed other or irregular counts. Other popular claims fare worse: measured nautilus shells are logarithmic spirals, but not golden ones.

Why humans find it interesting

Historical Fact

Leonardo of Pisa introduced the sequence to Europe in Liber Abaci (1202) through a puzzle about rabbits. Indian scholars, including Virahanka and Hemachandra, had described it centuries earlier while counting poetic rhythms.

CUE perspective

CUE Perspective

Fibonacci patterns show how simple rules repeated over time create order that looks designed. That is a real and lovely finding.

CUE is interested in repetition and growth too, and in being precise about how often a pattern truly holds.

Explore next

Sources

  • Vogel, “A better way to construct the sunflower head”, Mathematical Biosciences 44 (1979)
  • Swinton, Ochu et al., “Novel Fibonacci and non-Fibonacci structure in the sunflower”, Royal Society Open Science 3 (2016)
  • Sigler, Fibonacci’s Liber Abaci (Springer, 2002)

Written by CUE from the sources above. Some topics were suggested by Polymath; its claims are not reproduced.