Overview
This lecture introduces the Fibonacci sequence, its appearance in nature and art, and explores the concept and impact of the golden ratio.
Introduction to Fibonacci Sequence
- The Fibonacci sequence is a series of numbers where each term is the sum of the two preceding numbers (e.g., 1, 2, 3, 5, 8, 13).
- This sequence appears in various patterns in nature, such as the number of petals on flowers.
Historical Background
- Indian mathematicians like Pingala, Virahanka, and Hemachandra studied the sequence before it was popularized in Europe by Leonardo Fibonacci.
- Leonardo Fibonacci introduced the Hindu-Arabic numeral system to Europe in his book "Liber Abaci."
- Fibonacci formulated the famous rabbit population problem, which is solved by the Fibonacci sequence.
The Rabbit Problem
- Start with one pair of newborn rabbits; each pair produces one male and one female every month after reaching one month old.
- The number of rabbit pairs each month follows the Fibonacci sequence; after 12 months, there are 233 pairs (466 rabbits).
Fibonacci Sequence in Shapes and Nature
- Fibonacci numbers can be visualized with tiling squares to form the Fibonacci spiral.
- The spiral and sequence appear in natural forms, such as shells, plant patterns, and animal horns.
The Golden Ratio
- The golden ratio (about 1.618, symbolized by φ) is found by dividing consecutive Fibonacci numbers.
- A rectangle whose length divided by width equals 1.618 is in the golden ratio and considered aesthetically pleasing.
Applications of the Golden Ratio
- Used in art and architecture (e.g., works of Da Vinci, Michelangelo, Parthenon, Taj Mahal).
- Present in classical music compositions and various architectural structures worldwide.
- Cited in religious texts for design proportions (e.g., Ark of the Covenant).
- Observed in the proportions of the human body, facial symmetry, and DNA structure.
- Seen in honeybee colonies (male:female ratio 1:1.618), planetary patterns, and even smartphone photography.
Real-World Uses
- The Fibonacci sequence and golden ratio are used in finance for market analysis and prediction.
- These patterns unify mathematics, nature, art, and practical applications.
Key Terms & Definitions
- Fibonacci Sequence — a sequence where each number is the sum of the two preceding ones.
- Golden Ratio (φ) — the ratio of approximately 1.618 found by dividing consecutive Fibonacci numbers; symbolizes ideal proportion.
Action Items / Next Steps
- Review the Fibonacci sequence up to the 13th term.
- Observe examples of the golden ratio in art, nature, or architecture around you.