From The Archive #3: Fractal Poetry by Michael G. Smith

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 shows that only a few iterations will transform a triangle into a snowflake. I encourage you to watch the PBS Nova special ( 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 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.

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.

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.

Writing Prompt: Poetry Floorplan

I got a request from a reader to re-blog this article. Enjoy.


A Floor Plan for Your Poem: How to Use Stanzas, Titles, Endings

Moving from Room to Room
A first draft of a poem may just be a blurt on the page–but as you revise, you want the poem to make more of a distinct impression. One of the best ways to do this is to use stanzas–and to use the stanzaic arrangement that is tight for your poem.
The word “stanza” in Italian means “room.” This is a fascinating spatial or even architectural way to look at a poem. It means that each stanza is intact, and has its own flavor in terms of both meaning and music. Poems can be written in various traditional stanzaic forms. The choice of stanzaic form is important to each poem–it gives structure, and even mood.
Here are the possible stanzaic arrangements:
0. No stanza. The poem is just arranged as a whole on the page. This is fairly common. It simply uses other techniques to create its flow.
1. One line stanzas. This is difficult, as each line needs to be strong and individual. Chinese-American poet Arthur Sze does this to good effect, perhaps because he is influenced by Chinese poetry which is written in intact vertical lines. Sze writes:
nine purple irises bloom in a triangular glass vase–
a pearl forms an oyster–
she folds a prayer and ties it to a green cryptomeria branch–
(from “Dudyma” in Quipu, Copper Canyon Press)
His use of one line stanzas allows images and thoughts to stand alone, and yet feel connected to a larger whole.
2. 2-line stanzas. Couplets are basic, and solid. They can rhyme or not. Think of them as half of 4–not just obviously, but as doing half the work of a quatrain.
3. 3-line. These are triplets, or tercets. Like tripods, they are stable but also less obviously solid than 2 or 4. Use them for a more tripping or musical effect of flowing from line to line.
4. 4-line stanza. This might be considered the basic unit in English and in other languages as well. Ballads, which are a pan-European form, work as 4-liners. Like a table with four legs, quatrains are solid. They are a good choice for a longer poem or one that tells a story. A classic quatrain opens a traditional ballad. :
Come all you fair and tender ladies
Take warning how you court your men
They’re like the stars on a summer’s morning
First they’ll appear, and then they’re gone.

5. 5-line stanza. You can look at this as a combo of a 2 and a 3. It allows for a lot to happen. Japanese poetry is based on the 5-line stanza–the tanka form. The 3-line haiku is broken out of it. The 5-line stanza feels complete, it can make its own poem. Here is an example of a 5-liner by Elizabeth Searle Lamb:
there is a music
in the fall of white petals
from the peony
onto the camphorwood chest
a bride’s gift sixty years ago

6. 6-line: The sestina is built on 6-line stanzas. You can also consider it as 2 threes or 3 twos. Longer stanzas tend to be built on modular units of shorter ones. For example, 8-line stanzas might best be understood as 2 fours.
An Architectural Plan
To summarize, in English, the most important stanzaic arrangements are 2, 3, and 4. Longer ones tend to be built on shorter ones. To give a poem you are working on an immediate sense of structure, pick one of these and see how you can arrange the poem on the page. Some poems of course are in free verse. Free verse stanzas are just that–stanzas broken for sense or musical quality wherever you like. However, it can be useful to play around with various arrangements–try ending on a couplet, for example, or pairing up quatrains and triplets as if they were geometric shapes or colors of a quilt.
Look at this short poem by Philip Whalen:
Sitting home
Drinking wine
Writing pome
“What do you want
done with that?”

Here a series of funny aphoristic one-line stanzas end on a more solid couplet.
Enjambment is a very useful technique that is often ignored by beginning poets. It simply means that the sense or sentence run over the line and into the next. That is, not every line is a complete thought or grammatical phrase, and you can put a period in the middle of a line. In an example from the Spanish poet Antonio Machado, enjambment in the last two lines lets the poet create a sense of emotional urgency:
The fire coals of a violet twilight
leave smoke behind the black cypresses.
In the shaded summerhouse a fountain
with its stone Eros winged and nude.
He sleeps, silent. In the marble basin
the dead water doesn’t move.

(from Border of a Dream: Selected Poems, translated by Willis Barnstone. Copper Canyon)
Open Field
In arranging your poem physically on the page, there are various options. The old fashioned method is to use left hand margins, and keep the poem flush. Each line can be capitalized, or in a form more favored by contemporary poets, the start of each sentence is capitalized. Some poets like a free form approach, with lines scattered any which way on the page. If you do like this, make sure there is some meaning to the arrangement so that it doesn’t look completely random. For example, indent for emphasis, or to introduce a list or change of mood.
The End and the Beginning
The end of a poem is like the end of a piece of music–after it is silence. The last line is always an important one, but don’t necessarily wrap up the poem’s meaning in a too obvious way. It can be an echo of the meaning of a whole. And happy or sad, the last line is always a little bittersweet because the poem is over.
Titles may come first, but are often written last. Avoid cliched one-word titles that are too abstract, like “Life” or “Soul.” A title can function as a first line to the poem, or can be a contrast to it. Titles can be aphoristic or proverbial, or they can create an impression, like the title of a painting.
A. Write a poem in a particular stanza form. Decide first what you want to do, and then try it to see if it works. You can take an old poem and revise it this way. What pattern might be best?
B. Use enjambment, or run the lines over, to keep the lines an even length. You can even run syntax and sense over from one stanza to another.
C. Settle on what left-hand margins you will use, and standardize where to capitalize.
D. End on a note that works with the mood of the poem.
E. Pick a title that adds to the poem rather than constricting it.
You have now “built” your poem, not just room by room but with a front porch and basement as well! It is ready to hospitably invite in your reader.

This article first appeared in WRITER’S DIGEST. Copyright Miriam Sagan.