Fibonacci Numbers in Nature

Count the spirals on a sunflower head or a pine cone and you will almost always land on numbers like 8, 13, 21, 34 or 55. These are Fibonacci numbers, and their appearance in plants is one of the most striking meeting points of mathematics and biology. In this article we explore where the sequence came from, why nature seems to love it, and how to calculate with it.

Where the Sequence Comes From

The Fibonacci sequence starts with 1 and 1, and each new term is the sum of the two before it:

$$ F_1 = 1, \quad F_2 = 1, \quad F_{n} = F_{n-1} + F_{n-2} $$

This gives

$$ 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, \dots $$

A Story From Pisa (and Earlier From India)

The sequence is named after Leonardo of Pisa, later nicknamed Fibonacci, an Italian merchant's son who learned Hindu-Arabic numerals while travelling in North Africa. In his book Liber Abaci (1202) he posed a famous puzzle: a pair of rabbits produces a new pair every month, starting from their second month, and no rabbits die. How many pairs are there after a year? The monthly totals form the sequence above.

However, the numbers were known much earlier in India. Scholars studying the rhythms of Sanskrit poetry asked how many ways a line of a given length can be built from short syllables (1 unit) and long syllables (2 units). The answer is a Fibonacci number. The idea is traced back to Pingala (around the 2nd or 3rd century BC), and the sequence was written out explicitly by Virahanka (around the 7th century AD) and Hemachandra (around 1150 AD).

Worked Example 1: Generating and Checking the Sequence

Find \( F_{13} \) and verify the identity \( F_1 + F_2 + \dots + F_n = F_{n+2} - 1 \) for \( n = 10 \).

Step 1: List the terms using the rule.

So \( F_{13} = 233 \).

Step 2: Add the first ten terms.

$$ 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 = 143 $$

Step 3: Compare with \( F_{12} - 1 \).

$$ F_{12} - 1 = 144 - 1 = 143 \ \checkmark $$

The identity holds. It works because each term can be written as a difference of two later terms, and the sum "telescopes".

Fibonacci Numbers in Plants

Spirals in Sunflowers, Pine Cones and Pineapples

Look closely at a sunflower head and you will see two families of spirals, one turning clockwise and one anticlockwise. In typical sunflowers the counts are pairs of neighbouring Fibonacci numbers such as 34 and 55, or 55 and 89; very large heads may show 89 and 144. Pine cones commonly show 8 and 13 spirals, and pineapples usually show 8, 13 and 21 rows of scales in three different directions.

Leaves Around a Stem

As a stem grows, new leaves often appear at a fixed fraction of a turn from the previous one. Botanists record these as fractions such as \( \tfrac{1}{2}, \tfrac{1}{3}, \tfrac{2}{5}, \tfrac{3}{8}, \tfrac{5}{13} \). Every numerator and denominator is a Fibonacci number. Arrangements like these reduce the shading of lower leaves by upper ones.

Petals

Many flowers have a Fibonacci number of petals: lilies and irises have 3, buttercups and wild roses have 5, many daisies have 13, 21 or 34. This is a tendency rather than a law; plenty of flowers have 4 or 6 petals.

Why Does Nature Choose Fibonacci?

The explanation lies in the golden angle. At the growing tip of a plant, each new seed or leaf bud forms in the largest available gap, about

$$ \frac{360^\circ}{\varphi^2} \approx 137.5^\circ $$

away from the last one, where \( \varphi = \frac{1 + \sqrt{5}}{2} \approx 1.618 \) is the golden ratio. Because \( \varphi \) is extremely hard to approximate with simple fractions, seeds placed at this angle never line up in straight rows. They fill the space evenly and tightly.

When seeds are packed this way, our eyes pick out spirals formed by seeds that are close together. The best fraction approximations to the golden angle come from ratios of Fibonacci numbers, so the visible spiral counts are Fibonacci numbers. In 1979 the mathematician Helmut Vogel proposed a simple model: place seed number \( n \) at angle \( n \times 137.5^\circ \) and distance \( c\sqrt{n} \) from the centre. A computer plot of this model looks remarkably like a real sunflower.

Worked Example 2: Ratios Approach the Golden Ratio

Show that the ratios of consecutive Fibonacci numbers approach \( \varphi \approx 1.618034 \).

Step 1: Compute some ratios \( F_{n+1}/F_n \).

$$ \frac{2}{1} = 2, \quad \frac{3}{2} = 1.5, \quad \frac{5}{3} \approx 1.6667, \quad \frac{8}{5} = 1.6 $$

$$ \frac{13}{8} = 1.625, \quad \frac{21}{13} \approx 1.6154, \quad \frac{34}{21} \approx 1.6190, \quad \frac{89}{55} \approx 1.6182 $$

Step 2: Observe the pattern. The ratios alternate above and below 1.618, each time getting closer.

Step 3: Explain the limit. If the ratio settles to a value \( x \), then dividing \( F_{n+1} = F_n + F_{n-1} \) by \( F_n \) gives, in the limit,

$$ x = 1 + \frac{1}{x} \quad\Rightarrow\quad x^2 - x - 1 = 0 \quad\Rightarrow\quad x = \frac{1 + \sqrt{5}}{2} = \varphi $$

This is why Fibonacci spirals and the golden angle are two sides of the same story.

Worked Example 3: Binet's Formula

There is a direct formula for the \( n \)th term, named after Jacques Binet (1843), though it was known earlier to Abraham de Moivre:

$$ F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}}, \qquad \psi = \frac{1 - \sqrt{5}}{2} \approx -0.618 $$

Use it to find \( F_{10} \).

Step 1: Compute \( \varphi^{10} \). Using a calculator, \( \varphi^{10} \approx 122.9919 \).

Step 2: Compute \( \psi^{10} \). Since \( |\psi| < 1 \), this is tiny: \( \psi^{10} \approx 0.0081 \).

Step 3: Substitute.

$$ F_{10} = \frac{122.9919 - 0.0081}{2.2361} = \frac{122.9838}{2.2361} \approx 55.0 $$

which matches the table. Because \( \psi^n \) shrinks so quickly, for large \( n \) we can simply round \( \varphi^n / \sqrt{5} \) to the nearest whole number.

Fibonacci in Animal Family Trees

Honeybees give another neat example. A male bee (drone) hatches from an unfertilised egg, so he has only a mother. A female bee has both a mother and a father. Tracing a drone's ancestry backwards, the number of ancestors in each generation is 1, 2, 3, 5, 8, 13, ... exactly the Fibonacci sequence.

Did You Know?

• The rabbit problem is biologically unrealistic, but the sequence it produced turned out to describe real plants with surprising accuracy.
• Every third Fibonacci number is even, and every fifth one is a multiple of 5.
• A journal called The Fibonacci Quarterly has been publishing research on the sequence since 1963.

Key Takeaways

• The Fibonacci sequence is defined by \( F_n = F_{n-1} + F_{n-2} \), and was known in India long before Leonardo of Pisa's Liber Abaci of 1202.
• Sunflowers, pine cones and pineapples show spiral counts that are neighbouring Fibonacci numbers.
• The cause is growth at the golden angle of about 137.5°, which packs seeds efficiently.
• Ratios of consecutive Fibonacci numbers approach \( \varphi \), and Binet's formula gives any term directly.
• Fibonacci patterns are a strong tendency in nature, not an unbreakable law.

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