Patterns in Nature - Comprehensive Study Notes

What is a Pattern?

  • A pattern is: anyt h i n g that has repetition (with recurring characteristics)
  • A series of regular or consistent arrangement
  • Follows a specific rule or sequence

Types of Natural Patterns

  1. Symmetry
  • Definition: There is symmetry if an imaginary line is drawn across an object, the resulting parts are mirrors of each other.
  • Examples mentioned: Amazon Lily Pad, Butterfly, Dragonfly
  1. Spiral
  • Definition: A curved pattern that focuses on a center point and a series of circular shapes that revolve around it.
  • Common in nature: plants and some animals
  • Examples mentioned: Aloe polyphylla, Tendrils, Ram’s horns, Chameleon’s tail; also mentions related visuals like navy red flower
  1. Stripes
  • Definition: Stripes are lines or bands that differ in color or tone from an adjacent area.
  • Where seen: various living things, especially animals
  • Examples mentioned: Emperor angelfish, Zebra, Tiger, Heliconius charithonia
  1. Spots and Dots
  • Definition: Distinctive, often circular patterns on surfaces of animals, plants, fungi, and even geological formations; vary in size, color, and distribution
  • Example mentioned: Spotted deer and giraffe
  1. Meander
  • Definition: A series of regular sinuous curves, bends, loops, turns, or windings in a watercourse’s channel; produced by a stream or river swinging from side to side as it flows across its floodplain or shifts its channel within a valley
  1. Cracks
  • Definition: Linear openings that form in materials to relieve stress
  • Significance: The pattern indicates whether the material is elastic or not

Images and quick activities (Pattern identification)

  • Several slides prompt: What kind of natural pattern is suggested in each image?
    • Possible answers include: Spiral, Meander, Symmetry, Spots/Dots, Stripes, Cracks

Patterns in Nature: Why patterns appear (the big idea)

  • Maxim Lavrentovich quote (The Conversation US):
    "The reason patterns often appear in nature is simple: The same basic physical or chemical processes occur in many patterned substances and organisms as they form. Whether in plants and animals or rocks, foams and ice crystals, the intricate patterns that happen in nature come down to what’s happening at the level of atoms and molecules."
  • Key idea: Patterns emerge because the same underlying physical/chemical processes operate across different systems; patterns reflect atomic/molecular interactions.

Patterns in Nature: Two perspectives

  1. Some patterns cannot be seen because they are already parts of human experience
  • Example: the water cycle (evaporation, condensation, precipitation)
  • Understanding this pattern helps forecast weather, manage water resources, and assess ecosystem health
  1. Patterns help us recognize relationships, make predictions, and solve problems
  • In mathematics: patterns form the basis of sequences, functions, and algebraic thinking
  • In nature: patterns explain growth, development, and weather behavior
  • Overall significance: patterns sharpen logical and creative thinking in daily life

Summary (Part 1): Common Natural Patterns

  • 1. Symmetry
  • 2. Spiral
  • 3. Stripes
  • 4. Spots and dots
  • 5. Meander
  • 6. Cracks

Other Types of Patterns

  • Four main types: 1) Logical Patterns, 2) Geometric Patterns, 3) Word Patterns, 4) Number Patterns

Logical Patterns

  • Core idea: logic reasoning and pattern observation are foundational in math education and IQ concepts; categories and classification often precede numeration
  • Who studies them: children (shapes and colors); older students (aptitude tests with sequences)
  • To identify logic patterns, look for four features:
    1) Rotating shapes
    2) Increases or decreases in numbers or sizes of shapes/patterns
    3) Alternating patterns (colors, shapes)
    4) Mirror images or reflections
  • Problem-solving mindset: search for patterns or rules and determine which object follows them
  • Example task types encountered in slides (illustrative):
    • Which frame comes next?
    • Which is the odd one out?
    • Find missing boxes or complete a sequence

Geometric Patterns

  • Definition: geometric patterns are shapes repeated or altered to create a cohesive design; commonly seen in art, tiling, wallpapers, and architecture
  1. Tessellations
  • Repeating patterns of polygons that cover a plane with no gaps or overlaps
  • Types: regular tessellations (regular polygons tiling the plane), semi-regular tessellations (two or more regular polygons), demi-regular (polymorph)
  • Real-world examples shown: honeycombs, scales of fish
  1. Fractals
  • Definition: mathematical constructions with self-similarity; magnified areas resemble the whole but are not identical
  • Key idea: self-similarity across scales
  • Notable examples discussed:
    • Sierpinski Triangle
    • Pascal’s Triangle
    • Fractal Tree
    • Koch Snowflake
  • Real-world examples: broccoli and Romanesco cauliflower; branches of trees; ferns

Geometric Patterns: Specific fractal examples
a. Sierpinski Triangle

  • Start with an equilateral triangle
  • Replace the large triangle with three smaller congruent equilateral triangles sharing a vertex with the large triangle
  • Practical drawing approach: start with an equilateral triangle, mark midpoints of sides, connect them, repeat

b. Pascal’s Triangle

  • Contains the numerical coefficients of binomial expansions
  • Structure: triangular array constructed by summing adjacent elements in preceding rows
  • Connection to fractals: shading odd numbers reveals the Sierpinski pattern

c. Fractal Tree

  • Definition: binary fractal tree created recursively by symmetric branching
  • Process: start at a point, move a certain distance, create two branches, turn at angles, repeat with shorter distances

d. Koch Snowflake

  • Definition: a self-similar figure that looks the same on any scale
  • Construction steps: start with an equilateral triangle; divide each side into three equal parts; construct an equilateral triangle on each middle part; repeat iteratively

Patterns in Geometric Patterns: Fractals and real-world examples

  • Fractal phenomena occur in nature: broccoli, Romanesco cauliflower; branching in trees; fern leaves

Word Patterns

  • Intersection with language arts: patterns exist in poetry, grammar, and morphology; metrical patterns in poems and plural/s verb forms
  • Word patterns connect to mathematical thinking and logical structure
  • Typical Word Pattern examples: 1) Analogy: a relation expressed by parts to reveal a pattern of similarity
    • Example: finger: hand :: toe: foot
      2) Morphology and word pairings (lexical relationships):
    • Example: milk: cow :: wool: sheep
      3) Education and career mappings (illustrative analogies):
    • Basic Education : DepEd :: Higher Education : CHED (in the Philippines)
    • Electrical Engineering : COE :: Legal Management : CCJE
      4) Poetic forms and rhyme patterns:
    • Twinkle, Twinkle Little Star rhyme depiction
    • Rhyme Scheme: pattern of rhymes in lines, denoted by letters (e.g., Roses are red. (A); Violets are blue. (B); Sugar is sweet. (A); And so are you. (B))
    • Haiku: 17 syllables distributed as 5-7-5 across three lines

Number Patterns

  • Core idea: numbers follow sequences or rules that define progression
  • Key categories:
    • Even Numbers: integers divisible by 2
    • Odd Numbers: integers not divisible by 2
    • Prime Numbers: numbers greater than 1 whose only positive divisors are 1 and itself; note: 1 is neither prime nor composite
    • Composite Numbers: integers greater than 1 with more than two positive divisors
    • Arithmetic Sequence: difference between consecutive terms is constant; a<em>n=a</em>1+(n−1)da<em>n = a</em>1 + (n-1)d where d is the common difference
    • Geometric Sequence: each term is obtained by multiplying the previous term by a constant; a<em>n=a</em>1rn−1a<em>n = a</em>1 r^{n-1} where r is the common ratio
    • Triangular Numbers: number of dots to form a triangle; T_n = rac{n(n+1)}{2}
    • Square Numbers: squares of natural numbers; Sn=n2S_n = n^2
    • Cube Numbers: cubes of natural numbers; Cn=n3C_n = n^3

Number Patterns: Examples (as presented in the slides)

  • Example 1: 21, 25, 33, 49, 81, _
    • Differences: +4, +8, +16, +32; pattern of doubling increments
    • Next term: 81 + 64 = 145
  • Example 2: 5, 7, 12, 6, 8, 14, 7, ___, 16, 8, 10, ___
    • Solutions given: 9 and 18
  • Example 3: 09 41 56, 67, 78, 89, 100, → 111, 122
  • Example 4: 1, 10, 100, 1,000, …
    • Next term: 10,000 (multiplying by 10 each step)
  • Example 5: 2, 5, 10, 17, 26, ____
    • Next term: 37
  • Example 6: 2, 3, 5, 9, 17, 33, _
    • Next term: 65
  • Example 7: 1, 2, 6, 24, 120, _
    • Next term: 720
  • Example 8: A1Z, B2Y, C3X, D4W,
    • Next: E5V

Even Numbers, Odd Numbers, Primes, and Composites (quick recap)

  • Even: all integers of the form 2k2k, for k ∈ ℤ
  • Odd: all integers of the form 2k+12k+1, for k ∈ ℤ
  • Prime: defined above; examples include 2, 3, 5, 7, 11, 13, 17, er
    dots
  • Composite: numbers like 4, 6, 8, 9, 10, 12, 14, 15, er
    dots

Summary (Part 2): Four Main Types of Patterns

  • Logic Patterns
  • Geometric Patterns
  • Word Patterns
  • Number Patterns
  • Subtypes and examples:
    • Logic Patterns: Rotating shapes; Tessellations; Analogy (e.g., pattern-based reasoning in IQ/aptitude tests)
    • Geometric Patterns: Fractals (Sierpinski Triangle, Pascal’s Triangle, Fractal Tree, Koch Snowflake); Tessellations; Self-similar growth in nature (broccoli, Romanesco, tree branches, ferns)
    • Word Patterns: Analogy; Morphology; Rhyme schemes; Haiku; Poetic rhythm; Language-based patterns and mathematical thinking
    • Number Patterns: Even/odd/prime/composite numbers; Sequences (arithmetic, geometric, triangular, square, cube); Pattern-based problem solving
  • Additional notes:
    • Rotating shapes exhibit symmetry-related patterns; tessellations tile a plane without gaps or overlaps
    • Fractals emphasize self-similarity at multiple scales
    • Language and poetry provide natural analogies to mathematical pattern thinking (rhyme, meter, syllables)
    • The patterns framework helps in recognizing relationships, making predictions, and solving real-world problems