Mathematics Chapter 1: Patterns in Nature

Study Notes

A pattern consists of repetition and arrangement according to a specific rule or sequence.
Types of natural patterns include symmetry, spirals, stripes, spots and dots, meanders, and cracks.

Symmetry: Objects exhibit symmetry when halves mirror each other along an imaginary line.
Spiral: Curved patterns often found in nature, centered around a focal point.
Stripes: Lines or bands of differing colors or tones seen in various animals.
Spots and Dots: Distinctive circular patterns on animals and plants that vary in size and color.
Meander: Regular curves and bends in the paths of rivers or streams.
Cracks: Linear openings in materials indicating stress relief, demonstrating elasticity.

Logical Patterns: Identified through rotating shapes, size changes, alternating colors/shapes, and mirror images.
Geometric Patterns: Repeating shapes create cohesive designs, including:
- Tessellations: Repeating polygons covering a plane without gaps.
- Fractals: Self-similar mathematical constructs, examples include Sierpinski Triangle and Koch Snowflake.
Word Patterns:
- Analogy: Comparisons of two different items by breaking them down into parts.
- Rhyme Scheme: Pattern of rhymes in poetry or songs.
- Haiku: A three-line poem with a 5-7-5 syllable structure.
Number Patterns: Sequences that adhere to a specific numerical rule:
- Even Numbers: Numbers divisible by 2 (e.g., …, -10, -8, 0, 2, 4, …).
- Odd Numbers: Numbers not divisible by 2 (e.g., …, -9, -7, 1, 3, …).
- Prime Numbers: Whole numbers greater than 1 with no divisors other than 1 and themselves (e.g., 2, 3, 5, …).
- Composite Numbers: Numbers with more than two factors (e.g., 4, 6, 8, …).
- Arithmetic Sequence: A sequence where each term is derived by adding a constant (common difference).
- Geometric Sequence: A sequence where each term is derived by multiplying by a constant (common ratio).
- Triangular Numbers: Represents a triangle's dot arrangement (e.g., 1, 3, 6, 10, …).
- Square Numbers: Squares of integers (e.g., 1, 4, 9, …).
- Cube Numbers: Cubes of integers (e.g., 1, 8, 27, …).

The Fibonacci Sequence is defined by each number being the sum of the two preceding numbers, starting typically with 0 and 1.
Named after Fibonacci, derived from the rabbit breeding problem, illustrating its emergence through natural growth patterns.
Notable properties include:
- Every nth Fibonacci number is divisible by F(n).
- The sum of squares of two consecutive Fibonacci numbers results in another Fibonacci number.