Fibonacci Numbers

Vocabulary flashcards covering key terms from the Fibonacci Numbers and Golden Ratio lecture notes.

Fibonacci Numbers

A sequence defined by F(n) = F(n-1) + F(n-2); commonly starting with F(0) = 0, F(1) = 1. Originates in ancient Indian mathematics (Pingala) and was popularized in Liber Abaci by Leonardo of Pisa (Fibonacci); the sequence appears in nature and relates to the Golden Ratio.

Recurrence Relation

An equation that defines a sequence by relating a term to previous terms; for Fibonacci, F(n) = F(n-1) + F(n-2).

Golden Ratio (phi, ϕ)

An irrational constant approximately 1.618; the limit of F(n)/F(n-1) as n grows and the basis for the proportional rule a:b = (a+b):a.

Golden Angle

Approximately 137.5 degrees; the angle by which successive seeds or leaves are rotated to produce efficient spiral packing in plants.

Fibonacci Numbers in Nature

The sequence appears in natural growing systems such as sunflower seed arrangements, pinecone spirals, and flower petals; related to optimizing growth and resource allocation.

Pingala

Ancient Indian scholar around 200 BCE who used Fibonacci numbers to count syllable patterns in Sanskrit poetry; earliest known appearance of the sequence.

Liber Abaci

Fibonacci's 1202 book that introduced the Fibonacci sequence to Western Europe.

Kepler

German astronomer in the early 1600s who discussed Fibonacci numbers and their connection to the proportions of the pentagon.

Lucas Numbers

A complementary sequence to Fibonacci numbers; part of Lucas sequences; denoted Ln.

Lucas Sequences

A family of sequences that includes the Fibonacci and Lucas numbers, sharing similar recurrence relations.

Indexing Convention for Fibonacci Numbers

Traditional practice starts with F(1) = 1 (often F(2) = 1); modern convention commonly uses F(0) = 0 (with F(1) = 1).

Rabbit Problem

Fibonacci's idealized model of rabbit population growth that yields the Fibonacci sequence; rules include maturation after one month and reproduction after the second month.

Vitruvian Man

Leonardo da Vinci's drawing illustrating human body proportions, associated with golden ratio ideas.

Golden Ratio in Human Anatomy

Applications of the Golden Ratio to human anatomy; examples include overall height to height to the navel, forearm to hand, and facial proportions as discussed in the notes.

Convergence to Golden Ratio

As Fibonacci numbers increase, the ratios of consecutive terms oscillate and converge toward φ, showing a link between a simple arithmetic sequence and a fundamental constant.