the golden ratio the golden section the divine proportion the golden mean
TRANSCRIPT
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?