The Golden Ratio

The Golden SectionThe Divine Proportion

The Golden Mean

Golden ratio φ = 1 + √ 5 = 1.61803399..

2

Golden Rectangle

1

φ

1

φ -1

Golden Triangle

72 72

36

Ratio of A to B is the golden ratio

A = 1.618 B

A B

The Great Pyramid of Giza2560 BC

-Side lengths approximately 230m-Base covers 53 000 m^2 -Sides angled at 51.5 degrees.

2

1

φ

√φ

1^2 + (√φ)^2 = φ^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

I AF I = φI FD I

A

B

C

DE

F

Pentagon, each side of length 1.

Saint JeromeLeonardo 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?