LESSON #2: Patterns and Fibonacci Sequence

1. Introduction to Patterns

Definition & Behavior

  • A pattern may have a list of numbers in which a constant number is added to get the succeeding terms.

  • For other sequences, the terms increase because the number to be added increases in a predictable way.

  • There are number patterns whose terms decrease because the number to be added decreases in an expected way.

Why Patterns Provide a "Sense of Order"

Nature and the universe can seem chaotic. Patterns are the underlying rules that organize this chaos into predictable systems:


  • Predictability: Recognizing a pattern allows us to anticipate what comes next (e.g., weather cycles, celestial movements).

  • Simplification: Instead of memorizing infinite data points, a single mathematical formula or rule can explain a whole system.

  • Problem Solving: When scientists or engineers find a pattern in a problem, they can find a relationship, build a model, and create a solution.

"A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas."


— G.H. Hardy

2. Taxonomy Matrix of Patterns

All patterns stem from the same drive to find relationships, but operate through four distinct mediums:


Pattern Type

Medium

Object Characteristics

Rule Type

Logic Patterns

Abstract

Order, shared attributes, or attributes of objects.

Sequential changes in physical attributes.

Geometric Patterns

Shapes

Abstract shapes like lines, polygons, and circles.

Spatial repetition and topological growth.

Word Patterns

Language & Syntax

Metrical poetic patterns, grammar/plurals/past tense rules, encryption/ciphers.

Structural shifts in alphabet or meaning.

Number Patterns

Quantities

Sequences predicting subsequent terms; leads to functions.

Mathematical functions predicting sequential values.

3. Pattern Types & Examples

A. Logic Patterns

  • Definition: Deals with the characteristics of various objects, order, or sequences where items possess similar attributes.

B. Geometric Patterns

  • Definition: Deals with a motif or design that depicts abstract shapes (lines, polygons, circles) and typically repeats like wallpaper.

C. Word Patterns

  • Definition: Deals with the metrical patterns of poems and syntactic patterns (how we make nouns plural or verbs past tense). Supports both mathematical and natural language understanding.

  • Example (Analogy):

Doctor: Nurse: Leader: Follower

  • Example (Encryption):
    Encrypting MODERN to OQFGTP skips every letter by two in the alphabet (M→N→O). Applying this to WORLD yields YQTNF.


D. Number patterns

Definition: Deals with predicting the next term in a sequence. Forms the foundation for the concept of functions in mathematics (a formal description of relationships among quantities).

Examples:

Prime Numbers: 11,13,17,19,23,29

Adding Consecutive Prime Numbers: 5(+2)7(+3)10(+5)15(+7)22(+11)33

Adding Consecutive Integers: 12(+1)13(+2)15(+3)18(+4)22(+5)27

Alternating Operations (×4,−2):1×4=4;4−2=2;2×4=8;8−2=6;6×4=24;24−2=22;22×4=88

4. The Fibonacci Sequence

Who Was Fibonacci?

  • European Mathematician (c. 1175–1250), real name Leonardo of Pisa (or Pizano).

  • Discovered the Fibonacci sequence while investigating how fast rabbits could breed under ideal circumstances.

What is the Fibonacci Sequence?

  • An infinite series of integers where each number is the sum of the two preceding ones.

  • Traditionally starts with 0 and 1 (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,...).

  • Generates mathematical patterns found in computer programming, data sorting, random number generation, and nature.

Fibonacci in Nature & Science

  • Petals on Flowers: White flower (5 petals), Clematis (8 petals), Sunflower (13petals).

  • Fibonacci Spiral Objects:


  • Nautilus Shell: Forms a Fibonacci spiral.

  • Pinecone: Exhibits clockwise and counter-clockwise Fibonacci spirals.

  • Spiral Aloe: Leaves form a distinct Fibonacci spiral.

5. The Golden Ratio (ϕ)
Definition

  • A special irrational mathematical constant approximately equal to 1.61803398..., denoted by the Greek letter Phi (ϕ).

  • Two quantities are in the Golden Ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities.

The Golden Ratio in Architecture

  • Ancient Greeks / The Parthenon: Built extensively on the golden section; its front elevation is roughly 0.618 times as wide as it is tall.

  • Great Pyramid of Giza (Egypt): The perimeter of the pyramid divided by twice its vertical height evaluates close to Phi (ϕ).

6. The Link Between Fibonacci & The Golden Ratio

The Fibonacci sequence and the Golden Ratio are deeply intertwined:


  • Dividing any term in the Fibonacci sequence by its preceding term yields a ratio that approaches the Golden Ratio (= 1.618) as the numbers get larger:


  • 5 div 3 = 1.666...

  • 8 div 5 = 1.600...

  • 13 div 8 = 1.625..

  • 55 div 34 = 1.6176...

  • 144 div 89 = 1.6179...

7. Real-World Applications

  • The Golden Spiral: Built using squares with widths equal to consecutive Fibonacci numbers (1x1, 1x1, 2x2, 3x3, 5x5,....). A curve drawn through the corners creates a Golden Spiral, seen in seashells, hurricanes, and spiral galaxies.

  • Plant Growth (Phyllotaxis): Pinecones, sunflowers, and pineapples grow seeds/scales in opposing spirals. The counts of clockwise vs. counter-clockwise spirals are almost always consecutive Fibonacci numbers (e.g., 21 and 34) to maximize packing efficiency.