The Fibonacci Sequence

Combinatorics · Axiom Academy

Discover nature's most famous pattern, where each number tells a story of growth and recursion. In 1202, mathematician Leonardo Fibonacci posed a puzzle: Start with one pair of baby rabbits. Each month, every adult pair produces a new baby pair. Baby rabbits become adults after one month. Use the slider to advance through the months and watch the population grow! The Fibonacci sequence follows a simple rule: F(n) = F(n-1) + F(n-2) , starting with F(0) = 0 and F(1) = 1. Click the button below to build the sequence step by step! Here's a surprising connection: How many different ways can you climb a staircase if you can take 1 or 2 steps at a time? Try different staircase heights below! The Fibonacci sequence appears everywhere in nature! Click on each card to reveal where these magical numbers hide. When you arrange squares with Fibonacci-sized sides in a spiral, and draw quarter-circles through them, you create the famous Golden Spiral . Click the button to watch it draw! The Fibonacci sequence demonstrates how recurrence relations define sequences recursively. This fundamental concept in combinatorics allows us to count complex arrangements by breaking them into simpler subproblems. From rabbit populations to staircase paths, Fibonacci numbers solve counting problems where choices at each step build upon previous choices - a key technique in combinatorial enumeration.

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