Definition
The Fibonacci sequence is a special row of numbers where each new number is the sum of the two right before it. It starts with 0 and 1 and goes 0, 1, 1, 2, 3, 5, 8, 13, and so on. It shows up in nature surprisingly often, in the way leaves are arranged on a stem and in the seeds of a sunflower.
The simplest rule: add the last two
Imagine climbing a staircase where you decide your next step by adding the position one step back and the position two steps back. The Fibonacci sequence runs on exactly this simple idea. You start with 0 and 1, add them to get 1, then add 1 and 1 to get 2, and so on.
Write the numbers out in order and you get 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, going on forever. It looks so simple that even an elementary school kid could spot the rule in a minute. Yet this plain addition rule also shows up in the arrangement of leaves and seeds in plants.
Scholars in ancient India first worked with this sequence while counting poetic meters. It became widely known in Europe when the Italian mathematician Fibonacci introduced it in a book in 1202 through a rabbit-breeding puzzle. Imagine an ideal case where each pair of rabbits has a new pair every month, starting in their second month of life. Work out the numbers, and this sequence appears.
Why sunflowers and pinecones hold these numbers
Have you ever looked closely at a sunflower in a park or a pinecone on the sidewalk? The seeds of a sunflower are packed tightly along spiral curves that swirl both clockwise and counterclockwise. Count those spirals and, amazingly, you get consecutive Fibonacci numbers, like 34 and 55, or 55 and 89.
Of course, the plant isn't studying math to grow this way. At the center of the flower, each new seed often appears about 137.5 degrees around from the one before it. This angle is called the golden angle. With this spacing, the seeds fill the space evenly with no gaps, and as a result the spirals we can see come out as Fibonacci numbers.
You can often see the same rule in the order in which leaves sprout from a stem. Many studies explain why this angle takes hold through chemical signals at the growing tip, and whether it is the best possible adaptation is still being debated.
A bit more precisely: geometry linked to the golden ratio
Something strange happens when you divide each Fibonacci number by the one before it. 2/1=2, 3/2=1.5, 5/3≈1.666, 8/5=1.6, 13/8=1.625. As you go further out, the results get closer and closer to one special number, about 1.618. That number is the golden ratio. People often say we find this proportion the most beautiful, but experiments haven't clearly confirmed it.
Stack squares whose sides are Fibonacci numbers in a spiral, and you get a smooth swirling curve called the golden spiral. The nautilus shell, the arms of galaxies, and the swirl of a typhoon are often said to resemble this curve, but if you actually measure a nautilus shell, it grows at a different ratio from the golden ratio.
Not every spiral in nature is a perfect Fibonacci sequence or golden ratio, though. Nature has many kinds of curves, and the best-known places where Fibonacci numbers clearly appear are leaf arrangements and seed patterns in plants.
Common misconceptions
Every swirl and every flower petal count in nature must follow the Fibonacci sequence.
Plenty of living things don't follow Fibonacci numbers, such as plants in the mustard family, whose flowers have 4 petals. Fibonacci numbers are just a pattern that often appears in leaf arrangements and seed patterns, not a universal law of nature.
Where you meet it
The clockwise spirals (34) and counterclockwise spirals (55) in sunflower seeds
The number of spiral rows on pinecone scales (usually 8 and 13)
Petal counts of the lily (3 petals, 3 sepals), the moss rose (5), and the cosmos (usually 8)
In one sentence
A row of numbers built by adding the previous two, a simple rule that often shows up in how plants arrange their leaves and seeds.