Mathematics Chapter 1: Patterns

• The nature of mathematics is covered in Chapter 1.
• Francis R. Quarre, LPT, prepared the content.

Patterns

• Patterns are regular, repeated, or recurring designs.
• Patterns help identify relationships and form generalizations.
• Patterns allow finding logical connections and making predictions.

Logic Patterns

• Logic patterns deal with the characteristics of various objects.
• Logic patterns deal with order.

Number Patterns

• Number patterns are a sequence of numbers following a specific order or rule.
• Number patterns establish a relationship with numbers.
• Examples of number patterns include repeating, growing/increasing, and shrinking/decreasing sequences.
• Repeating patterns example: 10, 20, 30, 40, 50, 60,...
• Growing/increasing patterns example: 14, 15, 17, 20, 24, 29,...
• Shrinking/decreasing patterns example: 118, 98, 73, 43, 8, -32,...

Word Patterns

• Word patterns involve the arrangement of letters within words.
• Word patterns involve the structure of words in a sentence.

Types of Patterns

• Spiral patterns occur naturally in plants and natural systems.
• Spiral patterns reflect order and growth, studied by mathematicians, and used as symbols and architectural forms.
• Fractal patterns are non-linear patterns that infinitely repeat in different sizes.
• Fractal patterns are uniform in shape but may appear in varied sizes.
• Tessellation patterns are repeating patterns of the same shape without any overlaps or gaps.
• Tessellation patterns form or arrange small squares in a checkered or mosaic pattern, example, Drexel University
• Foam is formed from repeating spheres.
• Foam is approximately self-similar on smaller scales, making it fractal.

Symmetry

• Bilateral symmetry has similar anatomical parts arranged on opposite sides of a median axis.
• Only one plane can divide an individual with bilateral symmetry into essentially identical halves.
• Rotational symmetry is when a shape looks the same after some rotation by a partial turn.
• The smallest angle a figure can be rotated while preserving its original formation is called the angle of rotation.

Number Pattern in Humans

• Population growth is a number pattern in humans.
• Mathematics models population growth using the formula:
• A = Pert where:
  • A is the size of the population after it grows
  • P is the initial number of people
  • r is the rate of growth
  • t is time
  • e is Euler's constant with an approximate value of 2.718

Sequence

• A Fibonacci Sequence is a sequence of numbers.
• The man behind the Fibonacci sequence is Italian mathematician Leonardo Pisano Bigollo, nicknamed "Fibonacci."
• The Fibonacci sequence can be defined mathematically:
  • Fn = Fn-1 + Fn-2 where:
    • Fn is the Fibonacci number
    • Fn-1 is the previous term
    • Fn-2 is the term before Fn-1
• Binet’s Formula gives the nth term of the Fibonacci Sequence.