Overview
This lecture explores the Fibonacci sequence and the golden ratio, highlighting their presence in nature, art, architecture, and more.
Fibonacci Sequence in Nature
- Fibonacci sequence: each number is the sum of the two preceding numbers (e.g., 1, 2, 3, 5, 8, 13).
- Common in nature, such as the number of flower petals.
- Discovered by Indian mathematicians and popularized in Europe by Leonardo Fibonacci.
Fibonacci’s Rabbit Problem
- Problem: Calculate rabbit population growth with idealized assumptions (pair reproduces monthly, matures after 1 month).
- At the start of the 13th month, there will be 233 pairs or 466 rabbits.
- Illustrates how the Fibonacci sequence models real-world phenomena.
Visualizing the Sequence
- Squares with side lengths corresponding to Fibonacci numbers can be arranged into a spiral (Fibonacci spiral).
- Fibonacci spiral is observed in natural patterns (shells, flowers), art, and architecture.
The Golden Ratio
- Golden ratio (approximately 1.618, symbolized as φ or phi) relates to dividing two consecutive Fibonacci numbers.
- A golden ratio occurs when the ratio of two numbers approaches 1.618.
- Used to create aesthetically pleasing proportions in design.
Applications of the Golden Ratio
- Present in famous art and architecture (e.g., Da Vinci’s works, Parthenon, Taj Mahal).
- Found in music compositions by composers like Mozart and Beethoven.
- Human body and DNA exhibit golden ratio proportions.
- Occurs in natural phenomena (bee colonies, tree branches, animal horns).
- Used in photography and app design for visually pleasing layouts.
Practical Importance
- Utilized in finance for market prediction and analysis.
- Seen as a fundamental pattern underlying natural and human-made systems.
Key Terms & Definitions
- Fibonacci sequence — a series where each number is the sum of its two preceding numbers.
- Golden ratio (φ or phi) — a special number (~1.618) seen when the ratio of two quantities is the same as the ratio of their sum to the larger quantity.
- Fibonacci spiral — a spiral shape created by drawing arcs connecting the opposite corners of squares in a Fibonacci number-based layout.
Action Items / Next Steps
- Observe and list examples of Fibonacci sequence or golden ratio in your environment.
- Review class notes on the Fibonacci sequence and solve related practice problems.