Overview
This lecture introduces mathematics as the study of relationships among numbers, quantities, and shapes, explores patterns in nature, and explains the Fibonacci sequence and the golden ratio.
Definition and Importance of Mathematics
- Mathematics studies relationships among numbers, quantities, and shapes.
- It enhances critical thinking, reasoning, spatial thinking, and creativity.
- Mathematics helps us organize patterns and regularities in the world.
Patterns in Nature and the World
- Symmetrical patterns have identical halves when folded (e.g., butterfly wings).
- Spiral patterns are curved designs that revolve around a central point (e.g., plant growth).
- Fractal patterns consist of repeated shapes that shrink in size (e.g., romanesco broccoli, spider web).
- Tessellations use identical shapes that fit together with no gaps (e.g., pineapple, beehive).
The Fibonacci Sequence
- Discovered by Leonardo Pisano (Fibonacci), nickname means "son of Bonacci."
- Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, …
- Each number is the sum of the two preceding numbers.
- November 23 is Fibonacci Day because its digits (1, 1, 2, 3) match the sequence's first four non-zero terms.
The Golden Ratio and Its Relation to Fibonacci
- The golden ratio, denoted by φ (phi), is approximately 1.618034.
- The ratio of two successive Fibonacci numbers approaches the golden ratio.
- The larger the Fibonacci numbers, the closer their ratio is to the golden ratio, but it never equals it.
Mathematical Patterns and Regularities in the World
- The motion of a pendulum demonstrates mathematical regularity in time and movement.
- Reflection in a plane mirror follows mathematical laws, with image size and distance predictable by the law of reflection.
Key Terms & Definitions
- Symmetry — A pattern that is identical on both halves when divided.
- Spiral Pattern — A curved pattern revolving around a central point.
- Fractal Pattern — Design made of smaller repeated versions of itself.
- Tessellation — Repeating pattern of shapes with no gaps.
- Fibonacci Sequence — Sequence where each number is the sum of the two before it.
- Golden Ratio (φ) — Special number approximately equal to 1.618034.
Action Items / Next Steps
- Reflect on and write about the application of mathematics in your chosen course.