Golden Mean Presentation: BCC - 2D Design

1. The
Golden
Mean
The
Mathema)cal
Formula…
of
Life?

2. The
Golden
Mean
Also known as:
• The Golden Ratio
• The Golden Section
• The Golden Rectangle
• The Golden Number
• The Golden Spiral
• Or the Divine Proportion

3. The
Golden
Mean
The Golden Mean is a ratio which
has fascinated generation after
generation, and culture after culture.

4. The
Golden
Mean
It can be expressed succinctly in the
ratio of the number “1” to the
irrational 1.618034.
1.618034
1

5. The
Golden
Mean
• The golden ratio is 1·618034. It is often
represented by a Greek letter Phi Φ.
Phi
pronuncia)on:
Linguis)c
purist
might
opt
for
the
original
Greek
fee,
most
mathema)cians
know
phi
as
fi.
• The Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8,
13, ... (add the last two to get the next and so on)
• The golden ratio and Fibonacci numbers also
exist in nature… more on that in a bit.

6.
The Golden Ratio

7. One Way to Understand It Is
A
M
B
A B
A M
The line AB is divided at point M so that
the ratio of the two parts,
the smaller MB to the larger AM is the
same as the ratio of the larger part AM to
the whole AB. Does that make sense?

8. What?

9. OK try this

10. Fibonacci Sequence Expressed as a
Golden Rectangle
1+ 1 =2
1
2 2
1

11. Fibonacci Sequence Expressed as a
Golden Rectangle
1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 37
1+1= 2 2+3=5 3+5=8 5+8=13

12. So the Golden Ratio is…
Given a rectangle having sides in
the ratio 1:phi, phi is defined such
that partitioning the original
rectangle into a square and new
rectangle results in a new
rectangle having sides with a ratio
1: phi. Such a rectangle is called
a golden rectangle, and
successive points dividing a
golden rectangle into squares lie
on a logarithmic spiral. This figure
is known as a whirling square.

13. Whirling Square

14. The
Golden
Mean
&
Aesthe1cs
• Throughout history, the ratio for length to
width of rectangles of 1: 1.61803 39887
49894 84820 has been considered the
most pleasing to the eye

15. The
Parthenon
The exterior
dimensions of
the Parthenon in
Athens

16. The
Parthenon
The exterior
dimensions of
the Parthenon in
Athens, built in
about 440BC …

17. The
Parthenon
The exterior
dimensions of
the Parthenon in
Athens, built in
about 440BC …
form a perfect
golden rectangle.

26. Golden Spiral
Logarithmic Spiral: The Whirling Square.
Note that each new square has a side which is as long as the sum of
the latest two square's sides.

27. Leonardo
Da
Vinci
• Leonardo Da Vinci called it
the "divine proportion" and
featured it in many of his
paintings.

28. Leonardo
Da
Vinci
• Leonardo Da Vinci called it
the "divine proportion" and
featured it in many of his
paintings.
• In the famous "Mona Lisa".
Try drawing a rectangle
around her face. You can
further explore this by
subdividing the rectangle
formed by using her eyes as
a horizontal divider, etc.

29. Vitruvian
Man
• Leonardo did an entire
exploration of the
human body and the
ratios of the lengths of
various body parts.
Vitruvian Man
illustrates that the
human body is
proportioned
according to the
Golden Ratio.

30. Look at your own hand:
You have ...
• 2 hands each of which has ...
• 5 fingers, each of which has ...
• 3 parts separated by ...
• 2 knuckles
And…

33. The Golden Ratio
A
M
B
A B
A M

38. The
Golden
Mean
is
also
Found
in
Nature

40. On many plants, the number of
petals is a Fibonacci number:
buttercups have 5 petals;
lilies and iris have 3 petals;
some delphiniums have 8;
corn marigolds have 13 petals;
some asters have 21 whereas
daisies can be found with 34,
55 or even 89 petals.

41. The Golden Spiral can be seen in the
arrangement of seeds on flower heads.

46. Pine cones show
the Fibonacci
Spirals clearly.
Here is a picture
of an ordinary
pinecone seen
from its base
where the stalk
connects it to
the tree.

52. And In Shells

56. And In Insects

59. And In Space

66. So?
Where and how can I use it?

67. In Compositions

69. A Pretty Picture

70. A Pretty Balanced Picture

75. In Design

83. Project
8
• Objectives
We will explore geometric relationships within
the Picture plane derived from the Golden
Section, or the Golden Rectangle. This system
provides a way to place and organize shapes
based on using mathematical ratios and rational
proportions.
We will also explore how to direct the viewer s
eye by contrasts of value, shape and size.

84. But
first
an
in-­‐class
project
to
warm
up