The Fibonacci Sequence and The Golden Ratio

Vocabulary-style flashcards covering the life of Fibonacci, the specific properties of the Fibonacci Sequence, its relation to the Golden Ratio, and real-world examples in nature, art, and music.

Leonardo Pisano Bogollo

Better known as Fibonacci ($1170$-$1250$), an Italian mathematician who revived ancient mathematics and introduced the Hindu-Arabic decimal system and Arabic numerals to Europe.

Liber abaci

The book written by Fibonacci used to introduce Hindu-Arabic numerals and the decimal system to Europe.

The Rabbit Problem

A mathematical problem asking how many pairs of rabbits can be produced from one pair in a year if every month each pair begets a new pair that becomes productive from the second month on.

Fibonacci Sequence

The series of numbers $0, 1, 1, 2, 3, 5, 8, 13, 21, 34, \text{...etc.}$ where each term is the sum of the two preceding numbers.

$$a_N = a_{N-2} + a_{N-1}$$

The mathematical formula for the Fibonacci sequence where each term ($a_N$) is the sum of the two previous terms, starting with $a_0 = 0$ and $a_1 = 1$.

Golden Ratio ($\phi$)

The ratio of any two successive Fibonacci numbers, approximately equal to $1.618034\text{...}$, which is approached as the numbers in the sequence get larger.

Sunflowers

In nature, these exhibit seed head spirals in two interlocking sets, where the number of spirals in each direction is consistently a pair of consecutive Fibonacci numbers, commonly $34$ and $55$.

Pinecones

A nature example showing an interlocking-spiral structure on a woody scale-covered surface where the scales running in opposite directions are Fibonacci numbers.

Pineapples

Fruits with hexagonal scales forming spirals typically in sets of $8$, $13$, and $21$ running in three different directions.

Piano Octave

A musical structure containing $13$ keys ($8$ white and $5$ black), with black keys grouped in sets of $2$ and $3$, all of which are Fibonacci numbers.

Parthenon and Great Pyramid of Giza

Examples of ancient architecture that have long been associated with Golden Ratio proportions.

Leonardo da Vinci

An artist frequently cited for using Golden Ratio proportions in works such as the Vitruvian Man and the Mona Lisa.

Vitruvian Man

An illustration by Leonardo da Vinci used to depict ideal human proportions which often reflect the Golden Ratio.

Mathematics (Generalization)

A tool to quantify, organize and control the world, predict phenomena, and figure out "WHY" rather than just "solving for X."