Lecture Notes: Fibonacci Sequence and Its Applications

Introduction to the Fibonacci Sequence

• Definition: A sequence of numbers where each number is the sum of the two preceding ones.
• Sequence Example: Starts as 0, 1, 1, 2, 3, 5, 8, 13, 21, etc., forming a swirling visual pattern.
• Origin: Named after Leonardo of Pisa, also known as Fibonacci.

Historical Context

• Fibonacci's Problem: Illustrated using a rabbit breeding problem:
  • Start with a single pair of rabbits.
  • Each pair of rabbits produces a new pair every month, beginning in their second month.
  • Questions how many pairs are produced in a year.

Fibonacci in Nature

• Natural Occurrences:
  • Branching of trees and leaves.
  • Distribution of seeds, such as raspberry seeds.
  • Patterns of flower petals.
  • Parts of the human body, including fingers and facial proportions.
• Relation to Golden Ratio:
  • Many natural structures follow the Fibonacci numbers, typically in a ratio of approximately 1:1.6, known as the golden ratio.

Applications in Design and Architecture

• Use of Fibonacci and Golden Ratio:
  • Guides design in architecture.
  • Influences the layout of websites and user interfaces.
  • Ensures aesthetic and functional harmony.

Summary

• The Fibonacci sequence is not only a fascinating mathematical concept but also a fundamental pattern found throughout nature and human design. Understanding its principles can provide insights into both natural phenomena and practical design applications.