Patterns, Fibonacci Sequence, and Golden Ratio: Lecture Notes

A set of vocabulary flashcards derived from lecture notes on patterns in nature, logic, geometry, and the mathematical properties of the Fibonacci Sequence and Golden Ratio.

Symmetry

The balanced arrangement of parts of an object or organism such that they correspond or match across a line, around a point, or after a rotation (Mirror Image).

Stripes

A pattern formed by a curve that winds around a central point while gradually moving farther away or closer to it.

Spiral

A pattern consisting of alternating parallel bands or lines that differ in color, texture, or composition.

Cracks

A pattern of fractures or splits that develop when a material experiences too much stress, shrinking, drying, or pressure.

Meanders

Winding, curving, or looping patterns formed by the continuous bending of rivers, streams, or other flowing systems as they erode and deposit material over time.

Logical Patterns

Sequences, arrangements, or relationships that follow a consistent rule, allowing one to predict, analyze, or determine what comes next.

Geometric Patterns

Arrangements of shapes, lines, angles, or figures that repeat according to a specific mathematical rule or design, often exhibiting regularity, symmetry, and repetition.

Tessellations

Patterns formed by repeating one or more geometric shapes that fit together perfectly without any gaps or overlaps, covering a flat surface completely.

Regular Tessellation

A type of tessellation that repeats exactly one geometric shape.

Semiregular Tessellation

A type of tessellation that repeats two or more geometric shapes.

Fractals

Geometric patterns that exhibit self-similarity, repeating the same pattern at different scales; examples include the Sierpinski Triangle, Fractal Tree, and Koch’s Snowflake.

Pascal’s Triangle

A triangular arrangement of numbers where each number is the sum of the two numbers directly above it, used to represent binomial coefficients, probability, and combinatorics.

Arithmetic Sequence

A sequence of numbers in which the difference between consecutive terms is constant.

Common Difference

The constant difference between consecutive terms in an arithmetic sequence.

Geometric Sequence

A sequence of numbers in which each term is obtained by multiplying the previous term by the same nonzero constant.

Common Ratio

The nonzero constant used in a geometric sequence to obtain the next term from the previous one.

Triangular Numbers

Numbers that can be arranged in the shape of an equilateral triangle, where each number is the sum of the first $n$ natural numbers.

Square Numbers

Numbers obtained by multiplying a whole number by itself.

Cube Numbers

Numbers obtained by multiplying a whole number by itself twice.

Leonardo Pisano / Leonardo of Pisa

The mathematician who introduced the Fibonacci Sequence in $1202$ through his book Liber Abaci.

Liber Abaci

The book in which the Fibonacci Sequence was introduced using a rabbit population problem.

Fibonacci Sequence

A sequence where each term is the sum of the two preceding terms (e.g., $1, 1, 2, 3, 5, 8, ...$).

Golden Ratio ($\phi$)

An irrational mathematical ratio of approximately $1.61803398875$ where the ratio of the whole to the larger part equals the ratio of the larger part to the smaller part.

Golden Rectangle

A rectangle whose length-to-width ratio is the Golden Ratio (approximately $1.618:1$).

Golden Spiral

A spiral formed by drawing quarter-circle arcs inside consecutive squares of a golden rectangle.

Human Body Proportions (Golden Ratio)

Approximate ratios close to $1.618$ found in the human body, such as the total height relative to the height from the feet to the navel.