Title: Fibonacci Sequence
Description: Discover the fascinating Fibonacci sequence, its mathematical properties, historical origins, and surprising appearances in nature. Learn how this simple number pattern connects to the golden ratio, art, and even financial markets.
Author: Alloya Huckfield
tags:
  - Fibonacci-sequence
  - Golden-Ratio
  - Leonardo-Fibonacci
  - spiral-patterns
icon: LiAsterisk

fibonacci-sequence

fibonacci-sequence 1.svg

The Fibonacci Sequence: Nature's Hidden Code

The Fibonacci sequence is one of the most famous and intriguing number sequences in mathematics. Named after Italian mathematician Leonardo of Pisa (Fibonacci), it appears throughout nature, art, and mathematics.

Definition
The sequence is defined recursively as:

The first terms are:

Mathematical Properties

Golden Ratio Connection
The ratio of consecutive Fibonacci numbers approaches the golden ratio :

Binet's Formula
An explicit formula for the Fibonacci number:

Cassini's Identity

In Nature
Fibonacci numbers appear in various natural phenomena:

Phyllotaxis: Leaf arrangement in plants
Sunflowers: Typically have clockwise spirals and counter-clockwise spirals
Pinecones: Often have spirals in one direction and in the other
Pineapples: Usually show , , and spirals

Applications
Computer Science

Dynamic programming examples
Efficient algorithms for Fibonacci heaps

Finance

Fibonacci retracements in technical analysis

Art & Architecture

Golden rectangle constructions
Aesthetic proportions

sacred-geometry