Theory
An introduction to the mathematical principles underlying phyllotaxis — the fascinating science of spiral arrangements in plants.
1What Is Phyllotaxis?
Phyllotaxis, from the Greek words for "leaf" and "arrangement," describes the patterns in which leaves, seeds, florets, and other plant organs are arranged around a stem or within structures like flower heads.
Despite the diversity of plant forms, many species share the same geometric rules, which leads to remarkably consistent and efficient packing patterns.
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2The Golden Angle
Most spiral phyllotactic patterns are built on a specific angle of rotation between successive organs: the golden angle, approximately 137.5°.
It is derived from the golden ratio (φ = (1 + √5) / 2):
θ = 360° × (1 − 1/φ) ≈ 137.5°
Because φ is irrational, this angle ensures that no two organs ever align perfectly, distributing them evenly around the stem.
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3Fibonacci Numbers in Nature
Count the spirals on a sunflower or pine cone — you'll often find Fibonacci numbers: 3, 5, 8, 13, 21, 34, etc. These are the counts of parastichies, or visible spiral arms.
This is no coincidence. When each new organ is added at the golden angle, spiral patterns naturally emerge whose counts are consecutive Fibonacci numbers.
θn = n × 137.5°, rn = c × √n
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4How Do Plants Create These Patterns?
Plants don't do math — but they follow local rules. New primordia (young organs) appear on the shoot apex (meristem) where space is available.
Plant hormone auxin accumulates at specific points, and existing primordia inhibit auxin nearby. As a result, new organs form at the point farthest from existing ones — roughly the golden angle away.
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5Variations and Exceptions
Not all plants follow the golden angle. Some show whorled phyllotaxis (multiple organs at the same level), or distichous (alternating at 180°), common in grasses.
Occasionally, other sequences like Lucas numbers (2, 1, 3, 4, 7, …) appear. Developmental changes can cause pattern transitions as a plant grows.
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