# The Golden Ratio The Golden Section The Divine

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The Golden Ratio The Golden Section The Divine Proportion The Golden Mean

Golden ratio φ = 1 + √ 5 2 Golden Rectangle = 1. 61803399. . Golden Triangle 36 φ -1 φ 1 72

A B Ratio of A to B is the golden ratio A = 1. 618 B

The Great Pyramid of Giza 2560 BC -Side lengths approximately 230 m -Base covers 53 000 m^2 -Sides angled at 51. 5 degrees. 1^2 + (√φ)^2 = φ^2 √φ φ 1 2 1+ 1. 618 = 2. 618

The Parthenon

Notre Dame

CN Tower Base to observation deck 342 m Base to spire 553. 33 m 553. 33 = 1. 618 = φ 342

B Pentagon, each side of length 1. A C F E D I AF I I FD I = φ

Saint Jerome Leonardo Da Vinci (1451 -1519 )

Crucifixion (1502) Raphael

Ognissanti Madonna (Madonna of Glory) Giotto di Bondone (1267 -1337)

Bathers Seurat (1859 -1891)

Vetruvian Man Leonardo Da Vinci

Mona Lisa Leonardo Da Vinci

Golden Ratio on the Body Height / Head to fingertips / Head to elbow Elbow to fingertips / Elbow to wrist Head to fingertips / Head to naval Elbow to wrist / Length of hand

Fibonacci Sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …. . 1/1=1 2/1=2 3 / 2 = 1. 5 5 / 3 = 1. 6666. . . 8 / 5 = 1. 6 13 / 8 = 1. 625 21 / 13 = 1. 615385. . . 34 / 21 = 1. 619048. . . 55 / 34 = 1. 617647… 144 / 89 = 1. 617978… Φ = 1. 61803399

Ratio of Clockwise spiral to counterclockwise spirals 34 : 55, 89 : 55, 144 : 89

Nautilus Shell

Golden ratio in art, architecture & nature…. believable or not?