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Week 12
Fractals

Later Developments



Paul Cézanne 047 Paul Cézanne (French, 1839 - 1906)

La danse (I) by Matisse.jpg
PD-US, Link
Henri Matisse (French, 1869 - 1954)

Pablo Picasso, 1910-11, Guitariste, La mandoliniste, Woman playing guitar, oil on canvas.jpg
By Pablo Picasso - http://www.pablo-ruiz-picasso.net/work-3193.php, PD-US, Link

Woman in Hat and Fur Collar.jpg
By Source (WP:NFCC#4), Fair use, Link

PicassoGuernica.jpg
By PICASSO, la exposición del Reina-Prado. Guernica is in the collection of Museo Reina Sofia, Madrid. Source page: http://www.picassotradicionyvanguardia.com/08R.php (archive.org), Fair use, Link
Pablo Picasso (Spanish, 1881 - 1973)


Abstract Expressionism



Robert Motherwell's 'Elegy to the Spanish Republic No. 110'.jpg
Fair use, Link
Robert Motherwell (American 1915 - 1991)


Mark Rothko (Russian-America, 1903 - 1970)



Willem de Kooning (Dutch-American, 1904 - 1997)



Jackson Pollock (America, 1912 - 1956)








The Connoiseur, Norman Rockwell (American, 1894 - 1978)






Fractals



Koch Snowflake


Koch Snowflake 7th iteration

Von Koch curve

KochSnowGif16 800x500 2

Fractal Fern


Fractal fern explained

Sierpinski Triangle


Sierpinski triangle.svg

By Beojan Stanislaus, CC BY-SA 3.0, Link

Triangle sierpinski animat

Sierpinski-zoom4-ani

Sierpinski Carpet


Sierpinski carpet 6

Katsushika Hokusai
葛飾 北斎 (Japanese, 1760 - 1849)


The Great Wave off Kanagawa
Great Wave off Kanagawa2

As we see in the Koch snowflake example, curves which exhibits "self-similarity" seem to have different lengths at different scales of magnification.

In mathematical terminology, these curves have non-integer "dimension". Roughly speaking, their geometry lies somewhere between between a line and a plane.

The Kock snowflake has dimension: \[ \displaystyle \log_3 4 \approx 1.2619. \]

Koch Snowflake 7th iteration

The Sierpinski triangle, which may be constructed by "hollowing out" a filled-in triangle (which has dimension 2), has dimension: \[ \displaystyle \log_2 3 \approx 1.5850. \]
Sierpinski triangle.svg

By Beojan Stanislaus, CC BY-SA 3.0, Link


Box-Counting Dimension

One way to compute the dimension of a set $S$ is the so-called "Box Counting" method:
  1. Position the set $S$ in question in a rectangular box.
  2. Subdivide the rectangular box into squares of the same width $s$.
  3. Count the number $N$ of boxes which has non-empty intersection with $S$ under examination.
  4. Compute $\displaystyle \log_{\frac{1}{s}} N$.
  5. Repeat the procedure with smaller and smaller boxes. In many cases, as the size of the boxes tend to zero, the quantity computed in Step 4 approaches a fixed number. This number is the (fractal) dimension of the set $S$.

Great Britain Box

Computing the dimension of the coastline of Great Britain via the box-counting method.