Upcoming Classes
Tuesday, Sept. 4th
Fractal Worlds & Chaotic Systems
Assignments due:
* Topic of first oral presentation or written paper
* Read “Order in Pollock's Chaos”; Scientific American,
December 2002
Thursday, Sept. 6th
Motion, in the real world and in animated worlds
Assignment due:
* Read “It’s All in the Timing and the Spacing”, The
Animator's Survival Kit, R. Williams, Pages 35-51
* Homework #2
Upcoming Deadlines
Thursday, September 13th
First draft of your first term paper or your
oral presentation
Thursday, September 27th
First Set of Oral Presentations
First term paper (if not giving presentation)
Oral Presentations
The following persons will give oral presentations
on Thursday, September 27th :
• Batres, Adan
• Boyd, Heidi
• Chen, Emily
• Kwiatkowski, Dajon
• Lebedeff, Christopher
• Lipton, Christopher
For everyone else, your first term paper is due on
that date.
Quiz
Shortly after the Second World War, a
new style of painting was popularized
by American artists such as Jackson
Pollock and Willem De Kooning.
What is the name of their style?
Extra Credit: SF Museum of Art
Visit San Francisco Museum of Modern Art and
see Abstract Expressionist paintings.
Turn in your ticket receipt ($7 for students). Worth
one homework assignment ; deadline is Oct. 16th
Guardians of the Secret, Jackson Pollock, 1943
Fractal worlds &
Chaotic systems
The Evolution of Painting
In 500 years painting in the Western world evolved
from
this…
Saint Cecilia and Eight
Stories from her Life,
Giotto(?), 1304
Blue Poles 11, Jackson
Pollock, 1952
… to this …
What happened
and what role has
science played?
14th and 15th Century
The introduction of perspective during the
Renaissance made paintings look much
more realistic.
Road to Calvary, Martini, 1315
The Annunciation, Botticelli, 1489
16th and 17th Century
Compositions become more varied; use of
light and shadow is more sophisticated.
Diana and Callisto, Titian,1559
Night Watch, Rembrandt, 1642
18th and 19th Century
Compositions are even more varied.
The Orgy, William Hogarth, 1734
Turkish Bath, Ingres, 1862
Birth of Photography
The first successful permanent photograph
created by Nicephore Niepce in 1826.
Photos become commonplace by 1850.
Oldest surviving photograph, 1826
American Civil War photo, 1864
Impressionism in 19th Century
Photographic detail less
important than style.
Rouen Cathedral, Monet, 1894
Starry Night, Van Gogh , 1889
Cubism & Surrealism in 20th Century
Painters move further away from realism.
The Persistence of Memory,
Salvador Dali, 1931
Les Demoiselles d'Avignon, Picasso, 1907
Wassily Kandinsky
Kandinsky was a pioneer of modern art in the
early 1900’s
Example of Kandinsky’s early work, Old Town II (1902)
An Accidental Discovery
One evening Kandinsky walks
into his studio and stunned
by a beautiful painting that
he doesn’t recognize.
“First I hesitated, then quickly
approached this mysterious
picture, on which I saw
nothing but shapes and
colors…”
He then realizes it’s one of his
own paintings, upside-down.
“I now knew fully well, that the
object harms my paintings.”
"Munich-Schwabing with the
Church of St. Ursula”, 1908
Kandinsky's Composition VII (1913)
Kandinsky and others begin to paint abstract forms
Abstract Expressionism
Abstract Expressionism arises in
America after the Second World
war. It’s roots are in the abstract
paintings of Kandinsky and the
aggressive works of the German
Expressionist movement.
An example of German Expressionism
(Self-Portrait as a Soldier, E.L. Kirchner, 1915)
Composition, W. de Kooning,1955
Abstract Art Humor
Jackson Pollock
In the late 1940’s, Pollock
began to create
paintings not in the
traditional manner, on
an easel with a brush,
but by laying the canvas
on the floor and pouring
(some say dripping) the
paint directly onto it.
Pollock’s One (1950)
Pollock’s Blue Poles 11 (1952)
Pollock?
Newly Discovered Pollocks
In 2003, Alex Matter, whose parents were friends
of Pollock, claimed to have discovered 24
paintings by Pollock among possessions that
Matter’s father had left when he died in 1984.
After the paintings were discovered, Mr. Matter
consulted Ellen G. Landau, one of the world's
most respected authorities on Pollock’s work.
Prof. Landau declared the
paintings to be authentic
but others had doubts.
But is this a
In 1973, Blue Poles 11 sold
for two million dollars* so new
Pollocks are worth a fortune.
* Current value of Blue Poles 11 estimated at 150 million.
genuine Pollock?
Pollock & Chaos
In 1950 Time magazine quotes Italian critic
Bruno Alfieri who describes Pollock's work
as a manifestation of "chaos . . . absolute
lack of harmony . . . complete lack of
structural organization . . . total absence of
technique, however rudimentary . . . once
again, chaos."
In a telegram to the editor Pollock will reply,
"No chaos damn it!”
Nov. 20, 1950
But could Pollock have understand what chaos is?
Probably not since scientists only began to
understand chaos in the 1970’s.
What is a Pollock?
Is it random?
Is it chaotic?
Is it completely
unstructured?
Can we give scientific,
measurable meaning
to these questions?
Yes! Let’s see how.
Detail from Blue Poles 11 (1952)
Self-Similarity
These three images appear similar.
Leftmost is photo of an old wall, stripped of wallpaper, in Edgar Allan Poe’s house
Other two are magnified views of the central section of the photo.
Exact and Statistical Self-Similarity
Mathematical constructs,
such as the ideal “tree”
shown here, can have
exact self-similarity at
every possible scale.
Exact and Statistical Self-Similarity
Mathematical constructs,
such as the ideal “tree”
shown here, can have
exact self-similarity at
every possible scale.
In the natural world selfsimilarity is typically
limited to a few scales
and isn’t an exact
duplicate at each scale.
A real tree and the
wallpaper have
statistical self-similarity
Self-Similarity in Nature
Notice how the
oval shape of the
plant and the
segmentation of
the branches is
duplicated in each
sub-branch, in
each twig, even
down to the
individual leaves
and their veins.
Self-similarity in Geology
Is this cave large
enough for a person to
enter standing up?
Self-similarity in Geology
It’s not a cave, it’s a
small hole.
Field of boulders? No, just rocks
Geologists always place an object, such
as their hammer, is such photos to
establish the scale, which is impossible to
determine otherwise due to self-similarity
Mathematical Constructions
These images were designed
using mathematics yet due to
their self-similarity they have a
natural appearance.
Exactly self-similar coastline
Exactly self-similar fern
Fractals
The term fractal was coined in
1975 by Benoît Mandelbrot,
an IBM mathematician.
Benoît
Mandelbrot
Fractals have (typically):
• fine structure at all scales.
• self-similar at all scales.
• a non-integer dimension.
• a natural appearance.
Self-similarity in the Mandelbrot set
Movie: The MandelBrot Set
Fractal-like Objects in Nature
Clouds
Vegetables
Norway
Coastlines
Cracks
How Long is a Coastline?
Depends on the size of your ruler; the
shorter the ruler, the longer the coastline.
Measuring
the coastline
of England
12x4=48
28x2=56
68x1=68
Box Counting
Instead of using a ruler to find the
length of an island’s coastline (which
is its perimeter), we can lay a grid over
a map of the island and count the
number of boxes on the coast.
Length of the coast is
(Size of box, r) x (Number of boxes, N)
In the three cases we get:
(50)x(35) = 1750 kilometers
(25)x(76) = 1900 kilometers
(12.5)x(168) = 2100 kilometers
Turns out that there is a pattern in
these numbers.
Measuring
coastline of
Iceland
Iceland
Fractal Exercise
Use a CD to draw a circle
on a sheet of graph
paper.
Count the number of large
squares that are on the
perimeter of the circle.
Keep squares connected
(no going diagonal)
Count all the way around
the circle; double check
your count
5
3
2
1
4
Fractal Exercise
Now count the number
of small squares on
the perimeter of the
circle.
Double check your
count (OK to be off
by one square).
Next we’ll collect all the
data in the class.
3
2
1
5
4
Fractal Exercise
Most of you should have
found about 36 or 37
large squares and
around 72-74 small
squares.
Notice that for a circle,
when the size of the
squares is halved, the
number of squares is
doubled.
A circle is not a fractal, the
perimeter has a
circumference given by:
p x (Diameter)
5
3
2
1
4
Dimension of a Circle
For an object of dimension d,
(Number of small boxes)
= 2d x (Number of big boxes)
For a circle, the number of small boxes is twice the
number of big boxes so dimension is d = 1
Every simple curve has dimension d = 1 so they
are not fractals
Fractal Exercise (cont.)
Now draw an organiclooking blob on a fresh
sheet of graph paper.
Count the number of big
and small boxes, as you
did for the circle.
Everyone’s blob is
different so everyone’s
count will be different
(but count carefully and
double check the count).
4
2
3
1
5
Fractal Dimension
To find the dimension of your blob, compute
Log( # Small ) – Log( # Big )
(Dimension) = --------------------------------Log( 2 )
For example: If the number of big boxes is 30 and the number of small
boxes is 80 then
0.9808
Log( 80 ) – Log( 30 )
(Dimension) = ----------------------------- = --------------- = 1.4150
0.6931
Log( 2 )
In this example the blob is a fractal because its
dimension, about 1.4, is between one and two.
Fractal Analysis of Pollock
Start with a known Pollock
Fractal Analysis of Pollock
Take a photo, then scan and digitize pieces.
Fractal Analysis of Pollock
Lay down a grid of big boxes and count the
filled boxes; repeat with small box grid.
Big
Boxes
Small
Boxes
Pollock’s Dimension
This analysis was performed by
R.P. Taylor and colleagues.
They found that Pollock’s
paintings are fractals.
Painting destroyed
By Pollock
Early work
Richard P. Taylor
Faking a Pollock
Taylor tried to create
fake drip paintings in
Pollock’s style but
couldn’t get the right
fractal, self-similar
structure.
Newly Discovered Pollocks?
Newly Discovered Pollocks?
Taylor analyzed the newly discovered Pollock
paintings and this February reported in Nature
that there were "significant differences" between
their patterns and those of known Pollock works.
"Certainly my pattern analysis shouldn't be taken
in isolation but should be integrated with all the
known facts — including provenance, visual
inspection and materials analysis," Taylor said.
Several experts now believe that the paintings are
not by Pollock and could in fact have been
painted by more than one artist, possibly by
Mercedes Matter and her art students, trying to
imitate Pollock's technique.
Another Discovered Pollock
A retired, female truck driver
named Teri Horton, who
purchased a painting from a
San Bernadino thrift store for
$5, thinks that she may have
an original Pollock and is
asking $50 million.
Fractal analysis seems to
uphold her claim but that’s
not been enough to convince
the art world.
Can You Find the Real Pollock?
Amber
Hernandez
Alfredo
Ortega Hatch
Lindsey
Huffman
Christina
Rodriguez
Jennifer Do
Michael
Signorelli
Alejandro
Garcia
Jackson
Pollock
Fractals & Chaos
Fractals are closely
linked to chaos
because chaotic
motion almost always
follows a fractal path.
This makes chaotic
motion irregular but
not entirely random.
Lorenz
Hopalong
Next Lecture
Motion, in the real world and in
animated worlds
Remember:
Read “It’s All in the Timing and the Spacing”, The
Animator's Survival Kit, R. Williams, Pages 35-51
Do Homework Assignment #2