### Later Developments

Paul Cézanne (French, 1839 - 1906)

Henri Matisse (French, 1869 - 1954)

By Source (WP:NFCC#4), Fair use, Link

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 (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

#### Sierpinski Triangle

By Beojan Stanislaus, CC BY-SA 3.0, Link

#### Katsushika Hokusai葛飾 北斎 (Japanese, 1760 - 1849)

##### The Great Wave off Kanagawa

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: $\displaystyle \log_3 4 \approx 1.2619.$

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

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 $S$.

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