By Christophe Dufour & Ming Au
Finding φ.
The property that defines the golden ratio is:
L
=
L+1
1
L
a. Cross multiplying and rearranging the equation we obtain
the following quadratic equation :
L^2 - L -1 = 0
b. Using the quadratic formula to solve the quadratic
equation in (a). The “positive” root is called the
Golden Ratio:
1 +√[( 1 – 4(1)(-1)] =
2
1 +√(1 + 4) =
2
1 + √5
2
φ2 = 1 + √5 x 1 + √5 ~ 2.618 = φ + 1
2
2
1/φ =
2___ ~ 00.618 = φ - 1
1+ √5
Formation of the Golden Rectangle
A Golden Rectangle is formed by a regular
rectangle that is cut into half's. One of the half is
then cut into another half. One of the quarter equals
1 by ½. The rest of the figure equals √5/2. The
Length of the figure then equals φ.
1 + √5 ~ 1.618 x 1 = 1.618 = φ
2
The Golden Ratio in Art
The Golden Ratio can be seen in the physical structure of the human face
in great paintings such as the Mona Lisa or The sacrament of the Last
Supper; the artist needed a constant which applied to the Golden Ratio
to the drawing.
A Logarithmic Spiral
A special curve that appears in nature in the formation of shells. It is
formed by connecting the corners of the squares of a golden rectangle
with a smooth curve.
The Fibonacci Sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, …
A number in the sequence is obtained by adding the two
previous numbers before it:
For example, 1+1=2, 1+2=3, 2+3=5, 3+5=8, ….
What is the connection between the Fibonacci
F2/F1 1/1 =
Sequence and the Golden Ratio?
F3/F2 2/1 =
The ratios of the numbers in the
F4/F3 3/2 =
Fibonacci sequence tend
F5/F4 5/3 =
toward the Golden Ratio φ.
1
2
1.5
1.667
F6/F5
8/5 =
1.6
F7/F6
13/8 = 1.62