Nature's Mathematical Patterns

Mathematics

• It involves studying the relationships between numbers, quantities, and shapes
• Includes fields like arithmetic, algebra, trigonometry, geometry, statistics, and calculus
• Mathematics aids in organizing patterns and regularities observed
• Functions as a science of patterns, helping to recognize and generalize patterns

Patterns in Nature

• Patterns are regular, repeated, or recurring forms seen in nature.
• Can be modeled mathematically
• Examples of patterns include symmetries, spirals, waves, foams, and tessellations

Symmetry

• Symmetry is when an object can be divided into mirror-image parts.
• Types of symmetry include line or bilateral symmetry and rotational symmetry

Spiral

• A spiral is a curve that starts from a central point
• Moves progressively farther away as it revolves around the point

Wave

• A wave involves a moving disturbance of one or more quantities, such as sound, water, or light

Foams

• Foams are materials formed by trapping gas within a liquid or solid

Tessellation (Tiling)

• Involves covering a surface with shapes without leaving gaps or overlaps

History of Pattern Study

• Plato, Pythagoras, and Empedocles were early Greek thinkers who studied order in nature
• Joseph Plateau studied soap films and minimal surfaces
• Ernst Haeckel painted symmetrical marine life
• D'Arcy Thompson used math to explain growth patterns
• Alan Turing predicted patterns like stripes and spots

Fractals

• Aristid Lindenmayer & Benoit Mandelbrot demonstrated that plant growth can be modeled by fractals

Fibonacci Sequence

• Numbers where each number is the sum of the two preceding ones
• Example: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

Sequence

• Sequence is an ordered list of numbers
• Numbers are based on a specific rule, with each individual number being a term

Fibonacci

• Leonardo of Pisa introduced the Fibonacci Sequence in the book Liber Abaci

Golden Ratio (Φ ≈ 1.618)

• Ratios of Fibonacci numbers tend to approximate Φ
• Seen in art, nature, and architecture such as the Mona Lisa, Notre Dame, and the Parthenon

Binet’s Formula for Fibonacci

• Fₙ = [(1 + √5)/2]ⁿ – [(1 – √5)/2]ⁿ / √5

Mathematics Purpose

• Provides structure for arranging and understanding regularities
• Allows for prediction of natural behavior and real-life events
• Allows humans to control events and innovating to improve life situations

Arithmetic Sequence

• Found when the same number is added or subtracted consistently
• Example: 3, 6, 9, 12,...

Geometric Sequence

• Found if each term is multiplied or divided by a constant number
• Example: 2, 4, 8, 16,...

Quadratic Sequence

• Occurs when the second differences between terms remain constant
• Example: 1, 4, 9, 16, 25,...

Harmonic Sequence

• Created by taking reciprocals of an arithmetic sequence
• Example: 1, 1/2, 1/3, 1/4,...

Mixed Sequence

• A sequence using both addition and multiplication
• Example: 2, 4, 7, 11, 16,...

Square Number Sequence

• Sequence formed by squaring numbers
• Example: 1² = 1, 2² = 4, 3² = 9, 4² = 16,...

Cube Number Sequence

• Formed by cubing numbers
• Example: 1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64,...

Mathematical Language

• System used to communicate mathematical ideas
• Precise, concise, powerful, clear, and objective

Variable

• A quantity that can change in a mathematical problem or experiment

Mathematical Expression

• Expression constructed with terms that are separated by plus (+) or minus (-) signs
• Literal Coefficient: the variable part of a term
• Numerical Coefficient: a constant number multiplied alongside a variable

Types of Expressions

• Monomial: one term
• Binomial: two terms
• Trinomial: three terms

Mathematical Sentence

• Compares two expressions using a comparison sign (like =, >, <)
• Open Sentence: its truth value is unknown
• Closed Sentence: its truth value (true or false) is known

Universal Statement

• A universal statement asserts that a property holds true for all elements within a set

Conditional Statement

• The conditional statement is stating that if one thing is true, then something else is also true

Existential Statement

• The existential statement is stating that there is at least one element for which the property is true

Definition of a Set

• Organized collection of distinct objects, known as elements
• ∈ indicates "is an element of"
• ∉ indicates "is not an element of"

Ways of Describing a Set

• Roster or Tabular Method displays elements separated by commas within braces {}
• Rule or Descriptive Method details element characteristics using set-builder notation

Kinds of Sets

• Empty / Null / Void Set contains no elements, denoted by ∅ or {}
• Finite Set contains countable elements
• Infinite Set contains uncountable elements
• Universal Set (U) contains all possible elements under consideration

Relationships of Sets

• Equal Sets have exactly the same elements
• Equivalent Sets have the same number of elements
• Joint Sets have at least one common element
• Disjoint Sets have no elements in common