Patterns and Numbers in Nature

Study Notes

A pattern is defined as a series of regular arrangements characterized by repetition according to specific rules.
Types of natural patterns include:
- Symmetry: When an object can be divided into two mirrored halves along an imaginary line.
- Spiral: A curved pattern circling around a center point, commonly seen in plants and animals.
- Stripes: Lines or bands of different colors or tones found in various living organisms.
- Spots and Dots: Circular patterns occurring on various surfaces in nature, exhibiting unique designs in size and color.
- Meander: Sinuous curves in water bodies, produced by natural erosion and the shifting channels of rivers.
- Cracks: Linear openings in materials indicating stress relief, which assess material elasticity.

Logical Patterns: To analyze, observe:
- Rotating shapes.
- Increases or decreases in numbers or sizes.
- Alternating colors and shapes.
- Mirror images or reflections.
Geometric Patterns: Combinations of shapes creating cohesive designs, including:
- Tessellations: Repeating polygon patterns covering a surface without gaps.
- Fractals: Self-similar objects, examples include the Sierpinski Triangle and Koch Snowflake.
Word Patterns:
- Analogy: Comparison of two different things based on related parts.
- Rhyme Scheme: The pattern of rhymes in poetry or songs.
- Haiku: A 17-syllable Japanese poem structured into three lines of 5, 7, and 5 syllables.
Number Patterns: Sequences of numbers following specific orders, including:
- Even Numbers: All integers divisible by 2.
- Odd Numbers: Integers that cannot be evenly divided by 2.
- Prime Numbers: Whole numbers greater than 1 with no divisors other than 1 and themselves.
- Composite Numbers: Numbers with more than two factors.
- Arithmetic Sequence: Sequence with a constant difference between consecutive terms.
- Geometric Sequence: Sequence obtained by multiplying by a constant (common ratio).
- Triangular Numbers: Numbers that form an equilateral triangle.
- Square Numbers: Numbers that are squares of integers.
- Cube Numbers: Numbers that are cubes of integers.

The Fibonacci Sequence is generated by adding the two preceding numbers, usually starting from 0 and 1 or 1 and 1.
Named after Leonardo Pisano Bogollo, also known as Fibonacci, his observation of rabbit breeding led to this sequence discovery.
Notable properties of the Fibonacci Sequence include:
- Every nth Fibonacci number is divisible by n.
- The squared sums of two consecutive Fibonacci numbers form another Fibonacci number.