Transcript Part II THE CONCEPT OF AESTHETICS NATURE OF AESTHETICS The Ancient Greeks The ancient Greeks are known for their pioneering work in Geometry. Constructing rectangles based.

Part II
THE CONCEPT
OF AESTHETICS
NATURE OF
AESTHETICS
The Ancient Greeks
The ancient Greeks are known for their pioneering work in Geometry.
Constructing rectangles based on geometric projections outside of a
square, the Greeks defined what today we call the Golden Rectangle
or the Golden Section .
NATURE OF
AESTHETICS
The Golden Rectangle
The ancient Greeks considered the golden rectangle to be “beautiful”
and applied it to their art and architecture. The Parthenon, illustrated
here, represents the ultimate achievement of the classic Greek – and
subsequently Western – idea of beauty.
Its shape is a golden rectangle.
NATURE OF
AESTHETICS
The Golden Rectangle
A law established by the ancient architect, Vitruvius, states: "For a space
divided into two parts to be agreeable and aesthetic, between the smallest and
largest parts there must exist the same relationship as between the larger part
and the whole space.“ Vitruvius was describing the golden rectangle.
Construction of a Golden Rectangle:
a: Draw a square and find the midpoint of the base.
b. Draw an arc with a radius that extends from the midpoint to the corner. Extend the
base of the square until it meets the arc.
c: The aspect ratio of the resulting rectangle is 0.6180339 to 1, the golden ratio. A golden
section can be found where the side of the original square crosses a diagonal.
a
b
c
arc
Golden Section
midpoint
NATURE OF
AESTHETICS
NAUR
E
Phi, The Golden Ratio, Φ
Phi (pronounced “fee,”
symbol: Φ)
is calculated as:
√5 – 1
Φ=
2
= 0.61803398874989484820458683
436563811772030917980576 …
Phi has natural symmetry and the aesthetic beauty of Phi
can be seen in these curious mathematical relationships:
Φ =1-Φ
2
1
=1+Φ
Φ
1
2 = 2 + Φ
Φ
NATURE OF
AESTHETICS
The Italians
In a book completed in 1202, an Italian
mathematician by the name of Leonardo
Fibonacci stumbled upon the golden ratio
when he explored a sequence of numbers first
mentioned in 1150 by Indian mathematicians.
Thereafter called Fibonacci Numbers, each
value in the sequence is derived by adding
together the two previous values. As the
numbers progress, Fibonacci found that the
ratio of any number to its larger neighbor
closely approaches the golden ratio:
0.6180339.
10946
Leonardo Fibonacci
= 0.6180339
17711
Fibonacci Numbers:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711 …
NATURE OF
AESTHETICS
Fibonacci Numbers
Much later, botanists were
astonished to discover
Fibonacci numbers appearing
throughout nature. It seemed
as if Fibonacci Numbers were
Nature’s Formula.
Spirals seen in the arrangement of
seeds in the head of this sunflower
number 34 in a counterclockwise
direction and 55 in a clockwise
direction. 34 and 55 are the ninth and
tenth Fibonacci numbers respectively.
The flower itself has 34 petals.
Fibonacci Numbers:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711 …
NATURE OF
AESTHETICS
Fibonacci Numbers
The seed pods form 3 spirals
clockwise and 5 counterclockwise in
the pine cone of the Giant Sequoia.
As the Nautilus grows, its shell
structure describes a mathematical
curve called a logarithmic spiral,
whose radius at each complete turn
approximates the "Fibonacci Number"
series, or the so-called "golden ratio."
This Black-eyed Susan has 21 petals.
Fibonacci Numbers:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711 …
NATURE OF
AESTHETICS
Fibonacci Numbers
Number of Petals:
Flower
3 petals (or 2 sets of 3)
Lily, Iris
5 petals
buttercup, wild rose, larkspur, columbine
(aquilegia), vinca
8 petals
delphinium, coreopsis
13 petals
ragwort, marigold, cineraria
21 petals
aster, black-eyed susan, chicory
34 petals
plantain, daisy, pyrethrum, sunflower
55 petals
daisy, the asteraceae family
Fibonacci Numbers:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711 …
NATURE OF
AESTHETICS
The Golden Ratio In Art
Leonardo da Vinci used the Golden
Ratio in much of his artwork.
In what is perhaps his most famous
work, The Mona Lisa (seen here
with golden section lines drawn in
red), da Vinci drew diagonal lines
from the paintings corners to its
golden section points along the
edge.
Notice how the subject’s limbs tend
to follow these lines and how key
stress points (anatomical joints and
features) appear at intersections
(highlighted by circles).
The Mona Lisa
1503-5, Leonardo da Vinci, Oil on Panel
NATURE OF
AESTHETICS
The Golden Ratio In Art
The “Rule of Thirds” is a
simplification of the Golden
Section.
In Madonna And Child With St.
John The Baptist, Jacopo
Bassano placed key focal points
(the eyes) along diagonals and
their intersections.
Renaissance artists and those to
follow employed the golden
section to stress what they felt
were important elements of their
work.
Some even used it to relay
hidden meaning.
Madonna And Child With St. John The Baptist
1570, Jacopo Bassano, Oil on Canvas
NATURE OF
AESTHETICS
Summary:
THE CONCEPT OF AESTHETICS
The ancient Greeks thought the golden rectangle and its
corresponding golden section to be the most aesthetically
pleasing shape.
In a natural series of numbers described by Leonardo
Fibonacci in 1202, each value is a golden ratio of its next
larger neighbor.
Fibonacci numbers and hence, the golden ratio, are found
everywhere in the structure of living things.
Artists have used the golden ratio in their work to align key
elements of their design.
In Part I we saw how the
brain has evolved neural
processes which enhance
the perception of objects in
the visual field. Honed over
hundreds of thousands of
years, these processes have
helped to ensure our very
survival.
In Part II we visited the
concept of aesthetics and its
relationship to the golden
section and Phi, a number
intimately entwined with
nature.
Please view Part III