Mr. Scandrett's ClassroomOS

Mathematics - Lesson 10

Fibonacci Sequence

Patterns Recursion Growth Golden Ratio Spirals

The Fibonacci sequence begins with two simple numbers, then grows by adding the previous two terms. That tiny rule creates a pattern that shows up in algorithms, population models, plant growth, art, architecture, and the golden ratio.

The Rule

A sequence is an ordered list of numbers. In the Fibonacci sequence, each new term is the sum of the two terms immediately before it.

F(n) = F(n - 1) + F(n - 2) starting with F(0) = 0 and F(1) = 1

The first terms are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34.... Every number carries the memory of the two before it.

Interactive Sequence Lab

Build the Pattern One Term at a Time

Move the slider to reveal more terms. Watch the ratios between neighboring terms settle near the golden ratio.

latest ratio 1.618

Ratio Trail target: φ ≈ 1.618

Fibonacci Spiral

Why It Grows So Fast

The Fibonacci sequence is recursive: the next value depends on earlier values. Recursive rules can create surprisingly large growth from very simple instructions.

Algorithm A repeatable rule: add the two previous terms to produce the next term.
Recursion A process that refers back to earlier results instead of starting over.
Ratio As terms grow, neighboring ratios approach about 1.618, the golden ratio.

Practice

Use the rule. Each answer should be a whole number.

1. What comes next? 1, 1, 2, 3, 5, __

2. What comes next? 8, 13, 21, 34, __

3. If F(7) = 13 and F(8) = 21, what is F(9)?

Where Fibonacci Appears

Fibonacci-style patterns arise whenever each new stage depends on previous stages. That simple rule shows up across nature, art, and computing.

Plants & Seeds Sunflower seeds pack in 34 and 55 spirals — consecutive Fibonacci numbers — to maximize density.
Pinecones & Shells Scales on a pinecone spiral in 8 one way and 13 the other. Nautilus chambers grow by the golden ratio.
Art & Architecture The golden rectangle (sides in ratio φ) appears in the Parthenon and in many compositional grids.
Music An octave has 13 notes; a major scale uses 8. Fibonacci proportions appear in sonata structure.
Computer Science Fibonacci heaps, search trees, and recursive memoization all rely on Fibonacci-style counting.
Population Models Fibonacci originally described rabbit breeding — a simple model of exponential-like growth.
Three Scales of Nature

The Same Rule, Everywhere

Fibonacci's recursive growth appears from the microscopic to the galactic — always driven by efficient packing and self-similar expansion.

👁 Human scale · mm to m

Phyllotaxis

Golden angle 137.508° · 233 seeds

Plants place each new leaf or seed at the golden angle (≈ 137.5°) from the last. Because this angle is irrational, no two seeds ever share a radial ray — the densest possible packing with no wasted space. The spiral count around the finished head is always two consecutive Fibonacci numbers: sunflowers show 34 going one way and 55 the other.

Sunflower Pinecone Romanesco Pineapple Artichoke
🔬 Microscale · 10–500 μm

Diatom Symmetry

Pore rings: 5 · 8 · 13 · 21

Diatoms are single-celled algae that build glassy silica shells called frustules. The pores and ribs are arranged in Fibonacci multiples — 5, 8, 13, or 21 per ring — the same packing logic as a sunflower head, scaled to microns. Plant stem cross-sections show vascular bundles in an identical golden-angle spiral.

Diatoms Radiolaria Pollen Vascular bundles
🌌 Galactic scale · 10 000 ly

Spiral Galaxies

Logarithmic spiral · pitch ≈ 14°

Spiral galaxy arms follow logarithmic spirals — the continuous version of the Fibonacci spiral — where the distance from the center multiplies by φ for each quarter-turn. Gravitational density waves and initial angular momentum produce the same self-similar curve as seeds packing in a flower head.

Milky Way M51 Whirlpool NGC 1300 Hurricanes

Lesson Guide

Goal: recognize Fibonacci as a recursive sequence and connect it to ratios, spirals, and growth models.

  • Start with the rule: add the previous two terms.
  • Use the lab to compare terms and ratios.
  • Ask why a simple rule can create complex-looking structure.

Teacher Prompts

  • What information do you need before you can calculate the next term?
  • How does the ratio change as the terms get larger?
  • Where else do you see repeated growth patterns?