Mathematics of Patterns and Symmetry

Study Notes

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.

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 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

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.