Art and Science - Symmetry Poems

The idea behind symmetry is a simple one: what operations can you perform on an object while preserving its appearance or structure? From this simple question, an infinite variety of possible patterns emerge--as the abundant symmetries found in mathematics, art, and nature can attest to. In order to explore the possibilities of symmetry in poetry in a more manageable way, we need to restrict our focus to a certain class of symmetries known as frieze patterns.

Frieze patterns are symmetrical patterns that extend infinitely along one direction, like a number line. Every frieze pattern contains at least one symmetry: translational symmetry, which guarantees that the pattern repeats in space after a finite distance. Shifting the entire pattern by integer multiples of this distance preserves the structure. In addition to translational symmetry, frieze patterns can also contain reflections about horizontal and vertical lines, as well as 180º rotations and glide reflections about horizontal lines (glide reflections are simply reflections combined with translations). An alternative way to visualize frieze patterns is to imagine building them. Start with a simple, asymmetrical shape to use as a seed, and then transform it using rotations, reflections, translations, and glide reflections as needed. Amazingly, with these four types of transformations, it's only possible to build seven distinct symmetric structures out of the same seed.¹ These seven patterns are referred to as symmetry groups, and frieze patterns represent a specific type of structure called a two dimensional line group.²

In an earlier post, I created a graphic representation of the pattern found in a poetic form called the sestina. A few weeks later, I was experimenting with writing a pantoum, a poetic form I had never tried before. I created a similar graphic in order to visualize the repetition pattern and it reminded me of frieze patterns I had seen before. I decided to explore symmetries in poetry by starting with a simple seed that links two lines in different stanzas. Then, I colored each frieze pattern in order to group together lines that were similar.

Two similar lines might:

• Rhyme with each other, as in a sonnet,
• Contain the same number of syllables, as in a limerick or ballad,
• Have the same first word or last word, as in a sestina, or
• Be the exact same line, as in a pantoum or villanelle!

Below are the poetic forms I found for each symmetry group, along with the name of the pattern the poem is based off of and a color and letter coded similarity scheme for each stanza (two blue a's, for instance, represent lines that are similar to each other in some way). Feel free to write your own poems using these styles, or modify the patterns to make up new styles as you see fit. For instance, I like to preserve the rhyme pattern all the way to the end, and then repeat the rhymes I used at the beginning so the poem is cyclical. I haven't tried most of these forms yet, so I don't know which ones work the best! I would love to hear about any discoveries you make.

Hop: A four-line³ sestina.⁴

Step:

Sidle:

Spinning Hop: A pantoum arranged into four line stanzas.

Spinning Sidle:

Jump: A terza rima. 

Spinning Jump:

***

¹ A fun seed to start with is the shape of your footprint. The names of the symmetry groups might give you a hint on how to produce each pattern. 

² For more on the mathematics of symmetry and group theory, I recommend this book.

³ Of course, a sestina doesn't necessarily need to have four lines...

⁴ There are lots of possibilities for a "hop" type poem, because it contains the least symmetry--it simply contains translational symmetry.