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. 

Art and Science - Sestina Numbers

Earlier this year, I wrote a sestina which was then published in Totem, Caltech's literary magazine. Sestinas are one of my favorite types of poems, because they use a complicated repetition scheme that gives structure to the poem without using rhyme. Knowing this, one of my friends pointed out to me a generalization of the structure of sestinas that produces a sequence of numbers with interesting and surprising properties.

Sestinas have 39 lines each, with six stanzas and a three line ending. Ignoring the last three lines, each of the six stanzas use the same six words at the end of each line. This is what makes writing a sestina difficult--you have to pick six versatile words and avoid repeating the same message in each stanza! However, in each stanza, the six words are in different orders. The pattern by which the words get shuffled to produce the next stanza is the same between each stanza: the last word of the previous stanza is always the end word for the first line of the next. By the end of the poem, five shuffles later, repeating the shuffling procedure will produce the original order of the six words. Here's a picture of how it works:

Mathematicians call this reshuffling a permutation. If you've been looking closely, you might have noticed how it works for this particular permutation. To to figure out where the n^th word goes, first figure out if n is in the first half or second half of the stanza. For n in the first half, the n^th word will be the 2n^th word in the next stanza. For n in the second half, the n^th word will be in the 2*(6-n)+1^th position. The paper linked to below suggests a good way to think of the permutation: as a shuffle (alternating between the first three numbers and the second three numbers) with the second group of  "cards' turned upside down (so that the last word becomes the first word). This type of permutation can be generalized for any number m stanzas, but the number will only be a sestina number if after m permutations, the original order is obtained. 

The picture below shows a braid pattern that represents each permutation. The black lines designate the final word ordering for each stanza. Notice how, if you wrapped the picture around on itself, the colors would connect to each other in the same order as they started (representing the order of the first stanza). For knot theorists, this means sestinas form links with six loops. 

As it turns out, sestina numbers have a lot of interesting mathematical properties related to prime numbers. For instance, if s is a sestina number, then 2s+1 is a prime number! Many sestina numbers are also prime numbers, too. One method for proving an infinite number of sestina numbers exist depends on the truth of the Reimann Hypothesis, an unsolved problem in mathematics that is deeply connected to the distribution of prime numbers. Unexpectedly, sestina numbers are related to both beautiful poetry and beautiful mathematics. 

Read more about sestina numbers (also called Queneau Numbers) here. 

Read more about sestinas here.