📐 What Is the Golden Rectangle?
The two figures above show different rectangles. In the first figure, the third rectangle from the left is a golden rectangle. Its length-to-width ratio equals the mythical golden ratio, often considered the most aesthetically pleasing. This was suggested by an experiment conducted in 1876 by Gustav Fechner on 374 subjects. However, these results have since been questioned.
🏛 The Golden Ratio in Architecture
The golden ratio appears in many constructions, such as the Parthenon, shown in the second figure above.
🧮 Mathematical Definition
The golden ratio is the solution to the equation: x2 = x + 1. This equation has two solutions:
- The golden ratio: (1 + √5)/2 ≈ 1.6180
- Its inverse (up to sign): (1 – √5)/2
🔍 Why This Equation?
Dividing both sides of the equation by x gives: x/1 = (x + 1)/ x.
If x represents a segment length, this expresses that the ratio of the larger part (x) to the smaller part (1) is the same as the ratio of the whole (x + 1) to the larger part (x). This symmetry is what makes the golden ratio so harmonious.
🌿 The Golden Ratio in Nature
The golden ratio is also the limit of the ratio between two consecutive terms in the Fibonacci sequence, where each term is the sum of the two previous ones (un-1 + un-2)
1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 …
As the sequence progresses, the ratio between consecutive terms approaches the golden ratio:
1 (1/1) 2 (2/1) 1.5 (3/2) 1.66 (5/3) 1.6 (8/5) 1.625 (13/8) 1.615 (13/8) 1.615 (21/13) 1.619 (34/21) 1.617 (55/34) 1.618 (89/55) 1.618 (144/89) 1.618 (233/144)
These numbers appear in nature:
- Daisies often have 34, 55, or 89 petals
- Pinecones show 5 spirals in one direction, 8 in the other
- Sunflowers display 21 spirals clockwise and 34 counterclockwise
🎨 A Universal Fascination
The golden ratio—sometimes called the divine proportion—has inspired countless works in:
- Architecture and painting
- Music, where time ratios reflect it
- Design, from credit cards to furniture
- And of course, nature
Though its reputation may be exaggerated, it remains a historical curiosity. Recent studies show that Fechner’s findings reflect a collective average, but individual preferences vary widely. In fact, people often prefer more square-like shapes (McManus, Cook, Hunt).