Upcoming ClassesTuesday, Sept. 4th

Fractal Worlds & Chaotic SystemsAssignments 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 PresentationsFirst term paper (if not giving presentation)

Oral PresentationsThe following persons will give oral presentations

on Thursday, September 27th :• Batres, Adan• Boyd, Heidi• Chen, Emily• Kwiatkowski, Dajon• Lebedeff, Christopher• Lipton, ChristopherFor 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 PaintingIn 500 years painting in the Western world evolved

fromthis…

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

Road to Calvary, Martini, 1315 The Annunciation, Botticelli, 1489

The introduction of perspective during the Renaissance made paintings look much more realistic.

16th and 17th CenturyCompositions become more varied; use of

light and shadow is more sophisticated.

Night Watch, Rembrandt, 1642Diana and Callisto, Titian,1559

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.

American Civil War photo, 1864Oldest surviving photograph, 1826

Impressionism in 19th Century

Photographic detail less important than style.

Starry Night, Van Gogh , 1889Rouen Cathedral, Monet, 1894

Cubism & Surrealism in 20th Century

Painters move further away from realism.

Les Demoiselles d'Avignon, Picasso, 1907

The Persistence of Memory, Salvador Dali, 1931

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 DiscoveryOne 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

Composition, W. de Kooning,1955

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)

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 PollocksIn 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.

In 1973, Blue Poles 11 sold for two million dollars* so new Pollocks are worth a fortune.

But is this a genuine Pollock?

* Current value of Blue Poles 11 estimated at 150 million.

Pollock & ChaosIn 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!”

But could Pollock have understand what chaos is? Probably not since scientists only began to understand chaos in the 1970’s.

Nov. 20, 1950

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-SimilarityThese three images appear similar.

Leftmost is photo of an old wall, stripped of wallpaper, in Edgar Allan Poe’s houseOther 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 self-similarity 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 NatureNotice 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.

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

Field of boulders? No, just rocks

Mathematical ConstructionsThese images were designed

using mathematics yet due to their self-similarity they have a natural appearance.

Exactly self-similar coastline Exactly self-similar fern

FractalsThe term fractal was coined in

1975 by Benoît Mandelbrot, an IBM mathematician.

Fractals have (typically):• fine structure at all scales. • self-similar at all scales.• a non-integer dimension.• a natural appearance.

Benoît Mandelbrot

Self-similarity in the Mandelbrot set

Movie: The MandelBrot Set

Fractal-like Objects in Nature

Clouds

Cracks

Vegetables

CoastlinesNorway

How Long is a Coastline?Depends on the size of your ruler; the

shorter the ruler, the longer the coastline.

12x4=48 28x2=56 68x1=68

Measuring the coastline of England

Box CountingInstead 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 ExerciseUse 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

1

2

3 4

5

Fractal ExerciseNow 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.

123 4

5

Fractal ExerciseMost 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:

x (Diameter)

1

2

3 4

5

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 organic-

looking 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).

1

2 3

4 5

Fractal Dimension

To find the dimension of your blob, compute

(Dimension) = ---------------------------------Log( # Small ) – Log( # Big )

Log( 2 )

For example: If the number of big boxes is 30 and the number of small boxes is 80 then

(Dimension) = ----------------------------- = --------------- = Log( 80 ) – Log( 30 )

Log( 2 )

0.9808

0.69311.4150

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.

BigBoxes

SmallBoxes

Pollock’s DimensionThis analysis was performed by

R.P. Taylor and colleagues. They found that Pollock’s paintings are fractals.

Richard P. Taylor

Painting destroyedBy Pollock

Early work

Faking a PollockTaylor tried to create fake

drip paintings in Pollock’s style but couldn’t get the right fractal, self-similar structure.

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.

Newly Discovered Pollocks?

Another Discovered PollockA 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?

ChristinaRodriguez

AlejandroGarcia

JacksonPollock

AlfredoOrtega Hatch

LindseyHuffman

Jennifer Do MichaelSignorelli

AmberHernandez

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.

Hopalong

Lorenz

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