A term used in the terminology of fractals is self-similarity: a self-similar object has exactly (or approximately) the same shape as a part of itself. A variety of objects in the real world, such as ferns and coastlines, are approximately self-similar: parts of them show the same statistical properties at many scales. At the end of this post are a couple of diagrams that illustrate how a fractal may be developed. But first, experience the generative beauty of self-similarity via a poem by Maryland poet Greg McBride. Mathematician Benoit Mandelbrot (1924-2010), quoted in McBride's epigraph, often is nicknamed "the father of fractals."
Fractals by Greg McBride
Clouds are not spheres,
mountains are not cones . . . .
Sotted quirks of English shores resemble
mountain edges; so, pondered Mandelbrot,
this self similarity, continent
and sea, needs naming, and offered "fractal":
as branches branch, so too leaves vein, so too
veins branch; just so, some random element
prompts recursion, give or take, of grander
shapes in ever-smaller scale, as wayward
thought or mood—dark dog of late afternoon—
drives one's lines beyond the classically smooth
to unconstructed, un-Euclidean
lines, the resistant, frictional. The line:
self-resemblant unit, of some metered
lilt and length, shaped in similarity,
yet divergent worlds each of them creates,
and then another follows on: fractal
within fractal cluttering the edges,
curbing procession of word upon word
upon other word, bits of hot tinder
that crackle into ember, that smolder,
consonantal, vocalic, emotive—
fire cascading in a gouache of color.
Greg McBride is founder and ongoing editor of the online poetry journal Innisfree.
Here, as promised above, are diagrams describing the development of a particular fractal -- this one called the Koch Snowflake.
|Starting with a triangle, taking steps toward the fractal Koch Snowflake . . .|
|The Koch Snowflake happens this way: at each step, replace each line segment by the shape shown to the right.|