Introduction to Patterns, Sequences, Statements, and Fractals (Video Notes)

Vocabulary flashcards covering Patterns, Sequences, Mathematical Statements, and Fractals from the video notes.

Pattern

A regular, repeated or recurring form or design. Examples include floor tiling, textile designs, petals, shells, and natural arrangements.

Sequence

An ordered list of numbers called terms that may repeat, arranged according to a definite rule.

Fibonacci Sequence

A sequence starting with 1 and 1 where each term is the sum of the two preceding ones; 1, 1, 2, 3, 5, 8, 13, …; ratios of consecutive terms approach the Golden Ratio.

Golden Ratio

A number approximately 1.618, the limit of the ratios of successive Fibonacci numbers; associated with aesthetically pleasing shapes.

Recursive Definition

A definition that defines each term of a sequence using some of the preceding terms.

Self-similarity

A property where a shape looks essentially the same from a distance as it does up close.

Fractal

A mathematical set often produced by recursion that exhibits self-similarity and interesting dimensional properties.

Iterated Fractals

Fractals formed by repeating a process over and over.

Initiator

The starting shape used in fractal construction.

Generator

An arranged collection of scaled copies of the initiator used to generate the fractal.

Fractal Generation Rule

At each step, replace every copy of the initiator with a scaled copy of the generator, rotated as necessary.

Sierpinski gasket

A classic fractal formed by repeatedly removing the central triangle; self-similar, containing three scaled copies of itself.

Scale by 1/3

A scaling factor used in fractal construction where each new copy is one third the size of the previous one.

Mathematical Statement

A sentence that is either true or false and may contain words and symbols.

Universal Statement

A statement that is true for all elements in a given set.

Conditional Statement

An if-then statement: if one thing is true, then another thing is true.

Existential Statement

A statement that asserts the existence of at least one object for which the property holds.

Universal Conditional Statement

A statement that is universal and conditional, typically expressed as for all x, if P then Q.

Universal Existential Statement

A statement that combines a universal and existential claim, for all objects there exists something with a given property.

Inclusive Or

The or used in mathematics that allows either or both of the options to be true.

Exclusive Or

The or in everyday language that often means only one of two options; in precise mathematics, or is generally inclusive.

And

A conjunction meaning both A and B must hold.

Venn Diagram

A diagram using circles to illustrate relationships between sets, such as or and and.

Doubling Time

The amount of time required for a quantity to double, used in exponential growth contexts.

Exponential Growth

Growth in which the rate of increase is proportional to the current value, often illustrated by population doubling.