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

Golden ratio φ = 1 + √ 5 = 1.61803399.. 2 Golden Rectangle Golden Triangle φ -1 36 φ 1 72 72 1

Ratio of A to B is the golden ratio A = 1.618 B

The Great Pyramid of Giza 2560 BC -Side lengths approximately 230m -Base covers 53 000 m^2 -Sides angled at 51.5 degrees. 1^2 + (√φ)^2 = φ^2 1+ 1.618 = 2.618 √φ φ 1 2

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 = φ I FD I

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 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?