Nature of Mathematics: Fibonacci Sequence and Golden Ratio (Vocabulary)

Vocabulary flashcards covering key terms related to the Fibonacci sequence and the golden ratio as discussed in the lecture notes.

Sequence

An ordered list of numbers defined by a fixed rule; terms may repeat.

Fibonacci sequence

An ordered sequence where each term is the sum of the two preceding ones, commonly starting with 0 and 1 (Fn = F{n-1} + F_{n-2}).

Liber Abaci

Fibonacci’s 1202 book that popularized Hindu-Arabic numerals and calculation methods.

Hindu-Arabic numerals

The decimal digits 0–9 used in most of the world today.

Western Arabic-Indic numerals

Digits used in the Western Arabic script, for example: ٠ ١ ٢ ٣ ٤ ٥ ٦ ٧ ٨ ٩.

Eastern Arabic-Indic numerals

Digits used in Eastern Arabic scripts (Persian/Urdu contexts) with different glyphs.

Devanagari numerals

Digits used in the Devanagari script (used in India and Nepal), part of the Hindu-Arabic numeral family.

Rabbit production problem

A classic problem that leads to the Fibonacci sequence, modeling how rabbit pairs reproduce.

Petals on flowers (Fibonacci numbers)

Many flowers have a number of petals that is a Fibonacci number (1, 2, 3, 5, 8, 13, 21, 34, …).

Golden ratio

The irrational number φ ≈ 1.618…, with (a + b)/a = a/b in a golden proportion; relates to successive Fibonacci ratios.

Phi (φ)

The Greek letter used to denote the golden ratio.

Binet’s formula

The closed-form expression for the nth Fibonacci number using φ and its conjugate.

Recursive Fibonacci formula

Fn = F{n-1} + F_{n-2}, used to compute Fibonacci numbers from previous terms.

Fibonacci in nature

Fibonacci numbers appear in natural patterns, such as leaf arrangements and flower structures.

Golden rectangle

A rectangle whose side lengths are in the golden ratio; can be subdivided into a square and a smaller golden rectangle.

Phidias

Ancient Greek sculptor; the golden ratio (phi) is associated with his name.

Golden ratio in art and architecture

Using φ to achieve harmony, balance, and proportion in artworks and buildings.

Great Pyramid ratio

Base-to-height ratio ≈ 1.5717, cited as an approximate golden ratio in architecture.

Notre Dame (Gothic cathedral)

Noted for architectural proportions that reflect golden ratio concepts.

Taj Mahal proportions

Use of the golden ratio in arches and overall architectural proportions.

Eiffel Tower silhouette

Structure whose base widens as it rises, aligning with golden ratio-inspired proportions.

Convergence of Fibonacci ratios

As n increases, the ratio F{n+1}/Fn tends toward the golden ratio φ.