Common Mistakes in Finding Fractals:

Things that look like fractals but aren't

Here we illustrate three common errors made when seeking fractals in art and nature.

First, recalling that no physical fractal can exhibit scaling over infinitely many levels, nevertheless to make a plausible claim of fractality, a pattern must be repeated on at least a few levels. The "skulls within skulls" of Dali's Visage of War are repeated three (maybe four) times. By contrast, two fists do not make a covincing Cantor set. The decomposition of this picture into two pieces, the two fists, does not continue to even one more level. The fists are not split into smaller pieces. The more levels of the pattern, the more convincing the fractality of the picture. Here is an analogous example based on the Sierpinski tetrahedron.

Second, a repeating pattern alone is not sufficient to guarantee fractality. A checkerboard or a brick wall has a repeating pattern, but the repetition is with respect to translation, whereas for fractals the appropriate transformation is magnification.

Third, repeating a pattern under magnification is not sufficient to guarantee fractality. For example, a spiral (A below) is symmetric under magnification about its center point, but about no other point. Iterating this process, the limiting shape is just a single point. In order to produce a fractal, at each level the decomposition must involve at least two scaled copies. Nested dolls (often Russian, but not in this instance, B) are another example of a non-fractal involving a single scaling transformation, as is the cat bottle (C). On the other hand, the cow picture (D) is fractal, as would be more obvious if the cow's left earring were turned toward us. Thanks to Nial Neger for finding the cat and cow pictures. Click on all but the spiral to magnfy in a new window.

A B C D