Imagine a Fractal

California poet Carol Dorf is also a math teacher and is poetry editor of the online journal TalkingWriting.  In the most recent issue of Talking-Writing is this fascinating poem by Brooklyn poet, Nicole Callihan, "How to Imagine a Fractal."  Enjoy Callihan's poetic play with recursion and infinite nesting -- be lulled by the back and forth of forever.

Carol Dorf's work has appeared in this blog:

  Her fan-letter to the author of a math book is here 

and a poem about fear of math is posted here.

How to Imagine a Fractal     by Nicole Callihan 

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."

A Fractal Poem

    A fractal is an object that displays self-similarity -- roughly, this means that the parts have the same shape as the whole -- as in the following diagram which shows successive stages in the development of the "box fractal" (from Wolfram MathWorld). 

   
Michigan poet Jack Ridl and I share an alma mater (Pennsylvania's Westminster College) and we recently connected when I found mathematical ideas in the poems in his collection Broken Symmetry  (Wayne State University Press, 2006); from that collection, here is "Fractals" -- offering us a poetic version of self-similar structure:

       Fractals    by Jack Ridl

       On this autumn afternoon, the light  
       falls across the last sentence in a letter,
       just before the last movement of Brahms’ 
       Fourth Symphony, a recording made more 
       than 20 years ago, the time when we were  
       looking for a house to rehabilitate, maybe  

Fractals -- poems and photos

     Marc Frantz and Annalisa Crannell have written about mathematics and art (Viewpoints:  Mathematical Perspectives and Fractal Geometry in Art: Princeton University Press, 2011) and now Frantz (who is both a mathematician and an artist, a painter) has collaborated with a poet -- Robin Walthery Allen --  to develop a collection entitled Dance of Eye and Mind (not yet published).  I am honored to present a poem-photo pair from this exquisite collection.

     What is in us that must reach the top,
     that longs to look down upon the world as if a god?
     Don’t we know that in this infinite space
     the same rocks at the seashore know the secret of each peak?

In Praise of Fractals

     Philosopher Emily Grosholz is also a poet -- a poet who often writes of mathematics. Tessellations Publishing has recently (2014) published her collection Proportions of the Heart:  Poems that Play with Mathematics (with illustrations by Robert Fathauer) and she has given me permission to present one of the fine poems from that collection.

In Praise of Fractals     by Emily Grosholz

               Variations on the Introduction to
               The Fractal Geometry of Nature by Benoit Mandelbrot
               (New York: W. H. Freeman and Company, 1983)

Euclid’s geometry cannot describe,
nor Apollonius’, the shape of mountains,
puddles, clouds, peninsulas or trees.
Clouds are never spheres, 

Fractal Geometry

Lee Felice Pinkas is one of the founding editors of cellpoems -- a poetry journal distributed via text message.  I found her poem,"The Fractal Geometry of Nature" in the Winter/Spring 2009 Issue (vol.14, no 1) of Crab Orchard Review.

The Fractal Geometry of Nature       by Lee Felice Pinkas

               Most emphatically, I do not consider
               the fractal point of view as a panacea. . .
                                             --Benoit Mandelbrot (1924-2010)

Father of fractals, we were foolish
to expect a light-show from you,

hoping your speech would fold upon itself
and mimic patterns too complex for Euclid. 

From order to chaos -- a sonnet

Fractals    by Diana Der-Hovanessian

                             Euclid alone has looked on beauty bare
                                                    --Edna St. Vincent Millay

Euclid alone began to formulate
the relation of circle, plane and sphere
in equations making it quite clear
that symmetry is what we celebrate. 

Butterfly Effects

An equation or system of equations is said to be "ill-conditioned" if a small change in input data can produce a very large change in the output.  This inverse relationship between input and output has become popularly known by the phrase "butterfly effect."  Two poets from Eastern Pennsylvania, Gary Fincke and Harry Humes, have written poems about this phenomenon. 

The Length of a Coastline

In the nineties, fifteen or so years ago, when I began posting mathematical poems on the Internet, two of my earliest connections were Ken Stange, a poet and polymath and professor of psychology at Ontario's Nipissing University, and his daughter Kate, then a teen.  Kate publicized her love of mathematics and poetry by creating an online collection,"Mathematical Poetry:  A Small Anthology" which she has continued to maintain for many years--during which she has completed undergraduate and graduate studies in mathematics. 

Poems starring mathematicians - 6 (Mandelbrot)

More familiar than the name Benoit Mandelbrot are images, like the one to the left, of the fractal that bears his name.  Born in Poland (1924) and educated in France, Mandelbrot moved to the US in 1958 to join the research staff at IBM. A fractal is a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole, a property called self-similarity.