Tuesday, November 18, 2014

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, 
nor mountains cones, nor Ponderosa pines;
bark is not smooth; and where the land and sea
so variously lie about each other
and lightly kiss, is no hyperbola.

Compared with Euclid’s elementary forms,
Nature, loosening her hair, exhibits patterns
(sweetly disarrayed, afloat, uncombed)
not simply of a higher degree n
but rather of an altogether different
level of complexity:
the number of the scales of distances
describing her is almost infinite.

How shall we study the morphology
of the amorphous? Mandelbrot
solved the conundrum by inventing fractals,
a lineage of shapes
fretted by chance, whose regularities
are all statistical, like Brownian motion,
whose fine configurations
turn out to be the same at every scale.

Some fractal sets are curves
(space-filling curves!) or complex surfaces;
others are wholly disconnected ‘dusts’;
others are just too odd to have a name.
Poincaré once observed,
there may be questions that we choose to ask,
but others ask themselves,
sometimes for centuries, while no one listens.

Questions that ask themselves without repose
may come to rest at last in someone’s mind.
So Mandelbrot in time
designed his fractal brood to be admired
not merely for its formal elegance
as mathematical structure,
but power to interpret, curl by curl,
nature’s coiffure of molecules and mountains.

What gentle revolution of ideas
disjoins the nineteenth century from ours!
Cantor’s set of nested missing thirds,
Peano’s curve of fractional dimension,
Mandelbrot’s fractals, counter the old rule
of simple continuity,
domesticating what short-sightedly
was once considered monstrous.

Nature embraces monsters as her own,
encouraging the pensive mathematician
to find anomaly
inherent in the creatues all around us.
The masters of infinity,
Cantor, Peano, Hausdorff, and Lebesgue,
discovered sets not in the end transcendent
but immanent, Spinoza’s darling Cause.

Imagination shoots the breeze with Nature,
and what they speak (mathematics) as they flirt
reveals itself surprisingly effective
in science, a wrought gift
we don’t deserve or seek or understand.
So let us just be grateful,
and hope that it goes on, although our joy
is always balanced by our bafflement.
This poem was first published in The Hudson Review.  Here is a link to a video presentation by Grosholz that includes the poem, read at a Banff workshop.  Additional poetry video-casts from Banff are available at this link.

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