Similar, self-similar -- fractals, a poem

      In geometry two objects are said to be similar if they have the same shape --- which happens if their angles are the same size and occur in the same sequence. For example, any pair of triangles with angles 30, 60, and 90 degrees are similar; also, the lengths of pairs of corresponding sides of these triangles have the same ratio.
      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 . . . .

                                  —Benoit Mandelbrot

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.

To learn more about fractals see, for example,  Wikipedia.  More poetry postings in this blog for fractals (and other topics of interest) may be found if you click here and use the search box.