Overview
This lecture explores the history, mathematical properties, and significance of the Fibonacci sequence and the golden section, highlighting their surprising appearances in nature, art, architecture, music, and finance.
Fibonacci and His Sequence
- Leonardo of Pisa (Fibonacci) lived from 1170 to 1240 and introduced Arabic mathematics to Europe.
- His major work, Liber Abaci (1202), helped popularize the Arabic numeral system.
- Fibonacci posed a famous rabbit population problem, leading to the Fibonacci sequence: each number is the sum of the two preceding numbers.
- The sequence models certain growth processes, even though actual conditions differ in nature.
Fibonacci Sequence in Nature and Geometry
- The Fibonacci sequence appears in phyllotaxis (leaf arrangement) and seed spirals in plants like sunflowers.
- The angle between consecutive leaves (the golden angle) is about 137.5°, maximizing light and rain absorption.
- Geometrically, Fibonacci numbers are used to build squares and a spiral (Fibonacci spiral), which appears in shells like the Nautilus.
The Golden Section and Its Properties
- The golden section (or golden ratio, φ ≈ 1.618) divides a line so the ratio of the whole to the longer part equals that of the longer to the shorter.
- Noted by Euclid and called “Divine Proportion” by Luca Pacioli.
- Johannes Kepler observed that the ratio between consecutive Fibonacci numbers approaches the golden ratio.
Golden Section in Art and Architecture
- The golden ratio is found in buildings (Pyramids of Giza, Parthenon), artworks, and design.
- Rectangles and triangles with golden proportions are considered especially harmonious.
- Modern architects like Le Corbusier designed proportion systems based on the golden section.
The Pentagram and Symbolism
- The pentagram, based on the golden section, is a recurring symbol in art, religion, and culture.
- It forms through intersecting diagonals of a regular pentagon and is made from golden triangles.
- The pentagram is found naturally (e.g., in an apple core).
Fibonacci and the Golden Ratio in Music
- The piano keyboard and musical intervals reflect Fibonacci numbers.
- Composers like Béla Bartók and possibly Bach, Mozart, and Schubert used golden section proportions.
- The golden ratio is also used in the design of musical instruments by makers like Stradivari.
Fractals and Fibonacci Numbers
- Fractals are mathematical structures with self-similarity, exemplified by Romanesco cauliflower and the Mandelbrot set.
- The structure of the Mandelbrot set features sizes following Fibonacci periods.
Fibonacci in Finance: Elliott Wave Principle
- Ralph Nelson Elliott developed the Elliott Wave Principle, linking market price cycles to Fibonacci numbers.
- Five waves up, three waves down patterns mirror Fibonacci sequences and show self-similarity across time scales.
Key Terms & Definitions
- Fibonacci Sequence — A series where each number is the sum of the two preceding numbers.
- Golden Section (Golden Ratio, φ) — A proportion (≈1.618) where a line is divided so the larger part to the smaller equals the whole to the larger.
- Golden Angle — The smaller angle (about 137.5°) dividing 360° into golden ratio proportions.
- Pentagram — A five-pointed star with proportions based on the golden ratio.
- Fractal — A complex geometric structure with self-similarity at different scales.
- Elliott Wave Principle — A theory proposing that market trends move in repetitive cycles related to Fibonacci numbers.
Action Items / Next Steps
- Review the properties and calculation of Fibonacci numbers and the golden ratio.
- Identify examples of the golden section in art, nature, and architecture for deeper understanding.
- Read more about fractals and their appearance in mathematics and natural forms.