Patterns in Nature and Logic

Patterns in Nature

• Patterns are repeated shapes like polygons and circles.
• Tessellations cover a plane by repeating polygons.
  • Tessellations can be regular (one polygon repeated) or semi-regular (two or more polygons repeated).
• Fractals are never-ending patterns created by continuously repeating a process.
  • Examples of fractals include the Sierpinski Triangle, Pascal's Triangle, Fractal Tree, and Koch Snowflake.

### Word Patterns

• Can be found in poetry meters and rhythms.
• Examples include:
  • Plural of nouns
  • Past tense of verbs
  • Analogies
  • Rhyme schemes

### Logical Patterns

• Involve logic reasoning and pattern observing.
• Are identified by:
  • Rotating shapes.
  • Increasing or decreasing numbers of shapes.
  • Alternating patterns, colors, shapes.
  • Mirror images.

### Number Patterns

• Follow a specific sequence or order.
• Examples include:
  • Arithmetic sequences, where the difference between consecutive terms is constant.
  • Geometric sequences, where each subsequent term is found by multiplying the previous term by a constant.
  • Triangular numbers.
  • Cube numbers.
  • Fibonacci numbers.

The Fibonacci Sequence

• Popularized by Leonardo Pisano, also known as Fibonacci.
• Sequence: 1, 1, 2, 3, 5, 8, 13, 21...
• Found in nature:
  • Number of petals in flowers
  • Number of sections in fruits
  • Nautilus shell
  • Hurricanes and tornadoes
  • Arrangement of sunflower seeds
  • Human body

### The Golden Ratio

• Denoted by the Greek letter phi (Ф/φ).
• Named after sculptor Phidias.
• Approximately 1.618033...
• Also known as the Divine Ratio or Divine Proportion.
• Found in:
  • Golden Spiral
  • Golden Rectangle
  • Proportion of the human body

The Language of Mathematics

• Mathematical symbols represent values, operations, groupings, and special relationships.
• Symbols used in mathematics:
  • Operations: +, -, ×, /
  • Stand-in for Values: a, b, c...
  • Special Symbols: =, ..., Σ, ∃, ∀, ∞
  • Grouping: ( ), { }, [ ]
• Letter Conventions:
  • Lowercase letters for variables.
  • Uppercase letters for sets and formulas.

Sets

• An organized collection of objects.
• Ways of describing sets:
  • Set builder notation: {x | x > -2}
  • Roster notation: {1, 2, 3, 4, 5}
• Types of sets:
  • Finite: A set with a specific number of elements.
  • Infinite: A set with an unlimited number of elements.
  • Empty: A set that contains no objects.
  • Unit/Singleton: A set with only one element.
  • Equal: Two sets containing the same elements.
  • Equivalent: Sets with the same number of elements (same cardinality).
  • Disjoint/Non-Intersection: Sets with no common elements.

### Operations on Sets

• Union: The set containing all elements from both sets.
• Intersection: The set containing all common elements between sets.
• Combination: A collection of objects from a set, order doesn't matter.
• Cross Product: The set of all possible ordered pairs from two sets.

### Subsets, Supersets, and Power Sets

• Subset: Every element in set A is also in set B.
• Proper Subset: Every element in A is in B, but there is at least one element in B that is not in A.
• Superset: If A is a subset of B, then B is a superset of A.
• Power Set: The set of all subsets of a set A, denoted as P(A).

### The Set of Real Numbers

• Natural numbers (ℕ): Counting numbers.
• Integers (ℤ): Natural numbers, negative numbers, and zero.
• Rational numbers (ℚ): Numbers that can be represented as a fraction a/b where b is not equal to 0.
• Irrational numbers (ℚ'): Numbers that cannot be represented as fractions.
• Real numbers (ℝ): All rational and irrational numbers.

### The Universal Set

• Contains all possible elements under consideration.
• The union of all sets in a universal set is equal to the universal set.
• The complement of a set is the set of all elements in the universal set that are not in the set.

Explore the fascinating world of patterns in nature, language, and logic with this quiz. From tessellations and fractals to word patterns and number sequences, test your knowledge on how these patterns manifest in various forms. Challenge yourself to identify and understand the underlying principles behind these recurring themes.