Mathematics of Patterns and Symmetry

Introduction

• Mathematics is the study of numbers, forms, and relationships.
• Patterns are repeated arrangements of numbers, shapes, colors, objects, and events.
• Symmetry is the balance of an object where the resulting parts are mirror images of each other.
• Leonardo Davinci's Virtuvian Man shows the proportion or symmetry of the human body.
• Rotational symmetry occurs when an object is rotated on its own axis, and the shape remains the same.
• Bilateral symmetry occurs when the right and left sides of an item are are mirror images of each other
• Radial symmetry is a type of balance where parts of an object are arranged around a central point.
• Angle of rotation is a geometric transformation that rotates every point of a figure in a plane by a given angle and direction around a fixed point.
• Order of Rotation is the number of times it can be rotated around a full circle and still look the same.
• Packing problems are a class of optimization methods involving attempting to pack objects together into containers.
• Exponential growth is a pattern of data that shows greater increases with passing time, creating the curve of an exponential function.
• A sequence is defined as an ordered list of terms that may have repeated values.
• A series is defined as the sum of the elements of a sequence.
• The Fibonacci Sequence follows the rule that each number is equal to the sum of the preceding two numbers.
• The Golden Ratio is the sum of two numbers next to each other in the Fibonacci sequence.

Statements

• An existential statement says that there is at least one thing for which the property is true, even though a given property may or may not be true.
• A universal conditional statement (for all; if) is a statement that is both universal and conditional.
• A universal existential statement (for all; there is) is a statement that is universal because its first part says that a certain property is true for all objects of a given type, and it is existential because the second part asserts the existence of something.
• An existential universal statement (there is; for all) is a statement that is existential because its first part asserts that a certain object exists and is universal because its second part says that the object satisfies a certain property for all things of a certain kind.

Notation

• Notation is using symbols for representing operations, unspecified numbers, relations, and any other mathematical objects and assembling them into expressions and formulas.
• Roster notation is where every element that is more than one is separated by a comma symbol.
• Set-builder notation is used to “build” sets by classifying numbers according to their properties.
• A subset of a set is a part of the set or the whole set itself.
• The Cartesian product is the collection of all ordered pairs obtained by the product of two non-empty sets.
• Ordered pairs are pairs of elements taken from each set; it is represented by an INPUT or OUTPUT
• Symbols:
  • Z = “Zahlen”; Integers
  • Z¯= Negative Integers
  • Z+ = Positive Integers
  • N = Natural Numbers
  • R = Real Numbers
  • Q = “Quotient”; Rational Numbers
  • ∈ = Elements
  • ⊂ = Proper Subset
  • ⊆ = Subset

Relations and Functions

• A variable is a symbol for a number where the value is not known.
• A universal statement says that a certain property is true for all elements in a set.
• A conditional statement says that if one thing is true then some other thing also has to be true.
• An arrow diagram of a relation tells us how the relation assigns elements of each set. If two elements are connected by an arrow, then they are related.
• For a relation to represent a function, every input must be assigned to exactly one output.
• A function is a relation which derives one OUTPUT for each given INPUT.
• Properties of Functions:
  • For every element X in A, there is an element Y in B.
  • A graph displays data in a visual format.