Fractals in Nature and Mathematics

One way we see, model and construct/deconstruct the world is by the use of traditional (Euclidean) shapes of geometry, i.e. circles, squares, pyramids, spheres. For example, we can design and build a house (or deconstruct one) using such regular shapes. However, how would an irregular- or fragmented- shaped object, such as a mountain chain, coastline, tree, a snowflake, the Grand Canyon, a sprig of broccoli be modeled or constructed? If one looks closely at such objects, they might see that they can be broken apart into a shape that is a “reduced copy of the whole”. For example, the two-dimensional surface of mountain range can be modeled as a collection of triangles. Mathematically this is accomplished by taking a “seed” reflective of the whole, plugging it into an algorithm and continually iterating (repeating) the algorithm again and again. In this process, called recursion, the seed feedbacks on itself, further adding to and refining the resulting object.

The mathematical objects created by such iterative processes are called fractals,

a term coined by Benoît Mandelbrot in 1975. Fractals have their root in fractus, meaning broken or fractured. The dimensionality of a fractal is fractional, i.e. 1.39 or 1.67, and often between 1 and 2. Since the number of fractions is much greater than the number of whole numbers (0, 1, 2…) the number of fractals, and hence fractal dimensions and objects, dwarfs our habitual 2-D and 3-D views of the universe.

A well-known example of fractal mathematics used to model a a snowflake begins with an equilateral triangle (a triangle whose sides have the same length). The middle third of every side of the triangle is replaced with a pair of lines that form an equilateral “bump”. The result is called a Koch snowflake when iterated. A video of the process http://en.wikipedia.org/wiki/Fractal shows that only a few iterations will transform a triangle into a snowflake. I encourage you to watch the PBS Nova special (http://www.pbs.org/wgbh/nova/physics/hunting-hidden-dimension.html) about the history of fractal mathematics and to see how other objects are made.

To summarize, a fractal is a mathematical object that begins with a seed and grows, or as a friend of mine said, “as above, so below”. And, much like poetry and art, fractal mathematics is a geometry that finds order out of chaos.

Fractals in Art and Poetry

The recursive nature of fractal mathematics is often employed in media, animation, and digital art. Computer screensavers are one common example. Fractals are also found in “traditional” art. Computer analysis has shown fractal patterns appear in the paintings of Jackson Pollock. Why might Pollack’s paintings have fractal regions embedded in them?

Intentionally and unintentionally poets write fractal poems. A fractal poem is one that employs a seed phrase, uses symmetry and repetition of word-types and sounds, and considers dimensionality. Diana Der Hovanessian, with a nod to Edna St. Vincent Millay, beautifully describes the transition from Euclidean geometry to fractal geometry in her surprising poem Fractals (see http://aurorastyle.blogspot.com/2010/09/fractals-by-diana-der-hovanessian.html). Fractals is iteration on the traditions of mathematics and poetry!

Below are four other fractal poems that exhibit different aspects of fractals. Consider the following questions while you read the poems or say them aloud:

What is the seed, or seeds, of each poem?

What words or word-types add texture and dynamics to the poem and help move it? Consider the multiplicity of meanings that some of the words have;

How does the scale of the poem progress? Does it move outward, inward, or in another direction? What does this reveal to the reader about the author’s life or scene?

How does the author “escape” the fractal iteration and end the poem? What might this hint about the things that lie beyond the poem?

Escaping the Mandelbrot Set

Robin Chapman

She says

The coffee is fine

Though it could have been stronger

And cream would be nice.

She says

The weather today

Is, yes, fine, though cold

For summer and more rain likely tonight.

She says

The summer’s going well,

Of course awfully fast and won’t last

Long enough to get done what she’d planned.

She says

The marriage was ten good years

And then ten bad, and she’s learned

A lot since, though of course it’s lonely.

She says

Buying a new cappuccino maker,

Espresso roast, and best jam for her bread

Is frivolous, but we only have one life.

Published in Images of a Complex World: The Art and Poetry of Chaos,

Robin Chapman and Julien Clinton Sprott, World Scientific Publishing Co. Pte., Ltd., 2005, p. 116.

Migratory

Daniel Stewart

If no one; if nothing, if not me; if burning; if wing; if organ

(failing); eyes: mouths. Vise.

If trap knows teeth. If nerve knows cruel. If heart knows blood. If

water tongues fire.

Hiss of water. If hand kneads fire. If instinct (:desire). If seasonal,

flower. If I the liar.

I the liar. If God’s lips glazed with pollen’s praise.

If flight is ache is palm frond’s wince in salt and wind.

If whale, fin and foam, wave and spray.

If Man-O-War, promise sting.

If Monarch, splinter chrysalis into rein.

Published in Prairie Schooner, Spring 2011.

Fractal

Arthur Sze

Stopped at an intersection,

ruminating on how, in

a game of go, to consider all

the possible moves until

the end would take a computer

longer than the expected

lifetime of the universe,

you flit from piccolo

to stovepipe in a letter,

to scrutinizing faces

while standing in line

at the post office, to weather

forecast – a snowflake

has an infinite number of possible shapes –

consider, only last weekend,

a wasp threaded along a

screen door in south light,

mark the impulse to – not

see this, do that – water

leafing pear trees along

a curved driveway, relax

the intricate openwork mesh

of spring, recall lifting

a packet of flax seed

off the counter, and, checking

for an expiration date,

note – red light, green light –

sow when danger of

frost is past, then go, go.

Published in The Ginkgo Light, Copper Canyon Press, 2009, p. 55.

Purification Fractal

Michael G. Smith

Grand Canyon National Park

Booted, at the rim

step down forward

start reverse

step a minute, a day

sixty-thousand years

step, stepped, steeped

dry tawny Toroweap

my eyes weep

weeping, last night’s rain

pooled in sepia Esplanade

here is camp

tinaja, sandstone,

limestone, sandstone

salty sea bones

here is rattlesnake

here memory falls

form forming warping

worked Amerindian stone

working tool

clatter clattered

conglomerate rubble

sandstone, limestone

ocean silted

Red Wall limestone

rubbled, rubble

blistered, blistering

feet step, stepped,

layered, layers

faulted, avalanched

bloody pricks

of desert bricklebrush

proffered prickly pear

thunder thundered river

from sandstone wall

here is two-night camp

scorpion, scorpion-eating

pallid bat, hear pack rat

scurry, cottonwood rustle

silent great-horned owl,

hear morning camp-robber

raven flap flap

we hit the trail ahead

slick rain slickened

tumbled down

Cardenas ball-bearing

lava, Shinumo shale

narrow ledge

exposed exposure

above flowing

Tapeats ribbon

flowing into early,

early energy, pressured

heated folded

folded heated

folded river-severed

black, blackened, buried

two-billion-year old schist

gracing our handholds

released again

to movement

While it may seem these poems are no different than other “stream-of-consciousness” or abstract poems, there can be no doubt they affect the reader’s senses differently, such as haiku does. I, for one, get chills and tingles when I read and recite such poems. Or write one. Such results do not surprise me – the human circulatory system with its beating heart at the center, beginning and end is the quintessential fractal system found in Nature.

Michael, this was just fabulous. A nice addition to last night’s Poetry Safari, that got me off on a kind of fractal of dimensions. An article I’ll treasure (and is already packed to go to FL with me). Gassho.