Patterns and Sequences
Hook
Look carefully at a sunflower. In the center, the seeds spiral outward. If you count the spirals going clockwise, you might find 34. Count the ones going anticlockwise: you will find 55. Look at a pinecone: 8 spirals one way, 13 the other. A pineapple: 8 and 13, or 13 and 21. These numbers are not random. They are consecutive terms in one of the most famous sequences in all of mathematics. And the sequence was described by a medieval Italian mathematician who was writing about rabbits.
Core Concept
A pattern is a regularity — something that repeats or changes in a predictable way. Mathematics is, in many ways, the science of patterns. Sequences are one of the most fundamental kinds of mathematical patterns: ordered lists of numbers where each term is generated by a rule. The simplest sequence is 1, 2, 3, 4, 5 — each term is one more than the last. Or 2, 4, 6, 8 — each term increases by two. Or 1, 3, 9, 27 — each term is multiplied by three.
Sequences become interesting when the rules become interesting. The Fibonacci sequence is built on a surprisingly simple rule: each number is the sum of the two numbers before it. Start with 1 and 1. Then 1+1=2. Then 1+2=3. Then 2+3=5. Then 3+5=8. Then 5+8=13. Then 8+13=21. And so on: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...
What makes this sequence astonishing is not the rule — it is what happens when you find it everywhere in nature. The number of petals on most flowers is a Fibonacci number. The spirals of seeds in a sunflower, scales on a pinecone, and bumps on a pineapple are always consecutive Fibonacci numbers. This is not coincidence. It is mathematics built into the growth patterns of living things — an optimisation that evolution discovered long before any human wrote it down.
Everyday Examples
Patterns are the first mathematical thing most children notice, long before they can count. Stripes on a zebra. Spots on a ladybird. The regular rhythm of a heartbeat. The repeating design of tiles on a floor. Once you start seeing patterns, you cannot stop. Computer programs are built on patterns — loops that repeat, sequences that generate. Music is made of patterns — rhythms, melodies, harmonies. The stock market, weather systems, the branching of trees, the spiralling of galaxies — all have patterns that mathematicians and scientists try to identify and describe. Pattern recognition is, in some ways, the deepest function of the human brain.
Foundational Discovery or Proof
In 1202, the Italian mathematician Leonardo of Pisa — later nicknamed Fibonacci, from filius Bonacci, "son of Bonacci" — published his Liber Abaci (Book of Calculation). It was primarily a book about arithmetic and the advantages of the Hindu-Arabic numeral system (the 0–9 digits we use today) over Roman numerals. But it included a puzzle: suppose you start with one pair of rabbits. Every month, each mature pair produces a new pair. Pairs take one month to mature. How many pairs will you have after twelve months? The answer generates the sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144.
Meanwhile, in China, the mathematician Liu Hui (around 263 CE) was exploring a different but related beauty: patterns within geometric shapes. He calculated pi (the ratio of a circle's circumference to its diameter) to extraordinary accuracy by inscribing polygons inside circles and counting their sides — a 96-sided polygon gave him 3.14159, accurate to five decimal places. He also proved the Pythagorean theorem using a beautiful visual argument — rearranging colored squares. Liu Hui's work shows that pattern recognition in mathematics transcends any single culture: the search for regularity in numbers and shapes was happening simultaneously across the ancient world.
The Scientist(s) Behind It
Fibonacci was born around 1170 in Pisa, Italy. His father was a customs official who traded in North Africa, and the young Leonardo traveled with him, learning mathematics from Arab scholars who had preserved and extended the knowledge of the ancient Greek and Indian mathematical traditions. He came home convinced that Europe needed the Hindu-Arabic numeral system — with its positional notation and its concept of zero — and wrote the Liber Abaci to demonstrate its power. His rabbit puzzle was a minor illustration in a long book. He had no idea it would become the most famous sequence in mathematics.
Liu Hui wrote his commentary on the Jiuzhang Suanshu (Nine Chapters on the Mathematical Art), a Chinese mathematical classic, in 263 CE. He was methodical, rigorous, and remarkably modern in his approach — proving results rather than just asserting them. He calculated areas and volumes with precision that would not be equalled in Europe for another thousand years. His work exemplifies the quiet, cumulative nature of mathematical progress: building carefully on what came before.
Connection to Your World
Patterns are something you may already be brilliant at noticing. The Fibonacci sequence is a good place to test that: can you find the next term of the sequence 1, 1, 2, 3, 5, 8, 13... ? And then the one after? Can you make your own sequence with a different rule — say, each number is twice the previous one? What does it look like after ten steps? Next time you see a flower, count its petals. Lily: 3. Buttercup: 5. Delphinium: 8. Black-eyed Susan: 13. The Fibonacci sequence is hiding in the garden.
Wonder Question
The Fibonacci sequence appears in sunflowers, pinecones, and pineapples because it is connected to the most efficient way to pack objects in a circular space — and plants discovered this through millions of years of evolution. But there is another number lurking in the Fibonacci sequence. If you take any two consecutive Fibonacci numbers and divide the larger by the smaller — 5÷3, or 8÷5, or 13÷8 — you get closer and closer to a special number: approximately 1.618. Mathematicians call it phi, or the golden ratio. Artists and architects from ancient Greece to the Renaissance believed it was the most beautiful proportion. Is there really a most beautiful number? Or do we just find beauty in what we are biologically programmed to see?
