# Mathematical forms in poetry 5 – number sequences

Sometime around 1550 BC an Egyptian scribe named Ahmes noted down a method for obtaining the area of a circle, in what is the earliest recorded attempt to evaluate the number we know as 𝜋. The history of 𝜋 (its symbol is the Greek letter pi) is fascinating, as are the many mathematical formulae for determining its value. It features in what is widely regarded as mathematics’ most beautiful expression, a perfect poem in itself, Euler’s identity

e + 1 = 0.

Both 𝜋 and e, the exponential constant (Euler’s number) which also appears in this identity, are transcendental numbers and hence also irrational – they have infinite, non-recurring decimal expansions. To 16 digits, the expansion of 𝜋 is

𝜋 = 3.141592653589793.

Pre-calculator days, mnemonics known as ‘piems’ (pi-poems) were sometimes used to commit to memory the value of 𝜋 to a given number of digits. One I could relate to as an undergraduate is credited to the mathematician and astronomer  James Jeans

`‘How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics!’ `

The Wikipedia article on Piphilology provides many other delightful examples, including several in languages other than English.

Jacques Bens, a founding member of the Oulipo movement, invented the sonnet irrationnel – irrational sonnet –  which has five stanzas of 3, 1, 4, 1 and 5 lines respectively (the first five digits of 𝜋, adding up to a total of 14 lines) and also conforms to specific rhyme and metrical patterns. Rachel Galvin has translated seven of his irrational sonnets into English, the first of which can also be read here in the original French.

A recent variation of the piem is the piku, which has several versions. One is a three-line poem with three syllables in the first line, one in the second and four syllables in the third line. Another version is a haiku with lines of 5, 7 and 5 syllables respectively and the additional constraint that the number of letters in each word corresponds to the equivalent digit in the expansion of 𝜋. Here’s an off-the-cuff piku of my own:

```Sun. I find a beach,
windswept, to wander alone
the foamy wavelets.```

In principle there is no limit to the length of a poem or piece of prose constrained in this way by successive digits of 𝜋. A problem arises, however, with the 33rd digit, which is zero. To overcome this a set of rules have been developed, known as Pilish, which the American mathematician Mike Keith explains here. Keith has produced several remarkable works of constrained writing in Pilish, including a novel, a retelling of Edgar Allan Poe’s ‘The Raven’ and a circle poem using the first 402 digits of 𝜋. He has also created his own elegant version of the piku, written in Pilish with 3, 14 and 15 syllables per line respectively:

```It’s a moon,
A wheel revolving on golden earth, and lotus blossoms.
Mountains embrace windmills, and it all reflects this number, pi.```

A fascinating paper by Dr Tatiana Bonch-Osmolovskaya explores several other applications of 𝜋 in literature, including examples of Russian poetry by herself and others.

Experimental poet and publisher Anthony Etherin frequently makes use of mathematical constraints in his writing. He is the inventor of the aelindrome,  an extension of the palindrome in which words are divided according to a specific numerical sequence into letter units, that are then reversed around a pivot. His aelindrome ‘Asymptote’ is structured around the expansion of the exponential constant e = 2.7182818284590452353…:

```Asymptote

Infinite sets,
to unity,
present critical
enterprises,
over a map.

Ether, I eat ash.

To me,
asymptotic log is key,
logistic to a symptom,
eerie, at a shape ….

The ramp-rise
(so vertical, entire)
sent crypts
to unite
finites in.```

In this poem the first four letter-units, corresponding to the first four digits in the expansion of e, are: [In]2 [finite s]7 [e]1 and [ts to unit-]8. They reflect backward in the final stanza as [-ts to unit]8 [e]1 [finites]7 and [in]2. The poem pivots around the letter-unit [key]3, which corresponds to the 20th digit in the expansion. Etherin has also written aelindromes structured around 𝜋, the golden ratio 𝜑 and √2 .

Like 𝜋 and e,  √2 (the square root of 2) is an irrational number. Legend attributes the discovery of irrational numbers to the Greek mathematician-philosopher Hippasus, who is said to have been drowned as punishment for revealing their existence. Sarah Glaz refers to this legend in her poem ‘√2 = 1.41421…..’, in which the number of lines per stanza correspond to the digits of the decimal expansion:

```√2 = 1.41421.....

We started our voyage on the gulf of Tarentum.

The sea was choppy
and the brothers were restless.
At dawn, we gathered on the deck
intent to solve the conflict like rational men.

Hippasus still refused to keep the secret.

He had discovered that
the diagonal of a square
is incommensurable
with its side.

Alas! Our world had collapsed
and so did our geometric proofs.

Too much to lose, we heaved him overboard.```

As well as irrational numbers there are, of course, many other mathematical sequences that have informed poetic structure. The simple additive sequence 1, 2, 3, … is used to construct ‘snowball poems’, applying an Oulipian constraint in which each line consists of a single word, with each successive word one letter longer. There is a discussion of this form with several interesting examples on JoAnne Growney’s website Intersections – Poetry with Mathematics.

Daniel Tammet has observed that the traditional Japanese poetic forms the haiku and the tanka both use prime number sequences in their structure. The haiku, as we have already seen, has three lines of 5, 7 and 5 syllables to make a total of 17 syllables. The tanka has five lines of 5, 7, 5, 7 and 7 syllables, so 31 syllables in total.  5, 7, 17 and 31 are all prime numbers. In her excellent overview paper ‘Poems structured by integer sequences’, Sarah Glaz discusses this and other applications of prime number sequences in poetry, including Emily Galvin’s poem ‘In the Nick’, noting that Galvin ‘uses the prime number sequence to mark absence rather than presence’.

The Fibonacci sequence has many applications in mathematics and is often found in nature. Starting with 0 and 1, each Fibonacci number is formed by adding the two previous numbers, to give the sequence

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …

It was known to early Indian mathematicians and was used in the study of metrical patterns in Sanskrit poetry. In a previous blog post I have written about Fibs – poems structured according to the Fibonacci sequence.

Another mathematical sequence, formed from the coefficients of binomial expansions, is commonly presented as an array known as Pascal’s Triangle. Here are the first few rows:

Emily Galvin’s poem ‘Pascal’s Triangle’ from ‘Do the Math’ uses this structure in the arrangement of the stanzas. Pascal’s Triangle also provides the architecture for my two poems ‘Midwinter’ and ‘Autumn Sunrise’:

A striking illustration of the creative synergy that can exist between art, poetry and mathematics is artist Carl Andre’s poem ‘On the Sadness’, which is structured according to the Fundamental Theorem of Arithmetic. The theorem states that every integer greater than 1 is either a prime number or can be expressed uniquely as a product of prime numbers. In the poem, Andre assigns a phrase to each prime number, using the conjunctions ‘if’ and ‘then’ to represent the operations of multiplication and exponentiation. The first line of the poem corresponds to the prime integer 47, with the phrase ‘The door is closed’. The second line, corresponding to 2×23, is ‘we are going to die if the moon changes’. The complete poem is included in the anthology Strange Attractors (2008), edited by Sarah Glaz and JoAnne Growney and listed in the References below.

Both Glaz and Growney have also written poems using this structure. Glaz discusses her melodious, elegiac poem ‘January 2009’ in her paper ‘The Poetry of Prime Numbers’.    Here is Growney’s ‘We Are the Final Ones’:

```We Are the Final Ones

we breathe dirty air
oil scums our waterways
we breathe dirty air as we breathe dirty air
bees disappear
we breathe dirty air and oil scums our waterways
climate change affects the poor first
we breathe dirty air as oil scums our waterways
oil scums our waterways as we breathe dirty air
we breathe dirty air and bees disappear
glaciers melt
we breathe dirty air as we breathe dirty air and oil scums our waterways
drought is a serial killer
we breathe dirty air and climate change affects the poor first
oil scums our waterways and bees disappear
we breathe dirty air as we breathe dirty air as we breathe dirty air
What will happen to the polar bears?
```

The poem’s title corresponds to the number 1, the phrase ‘we breathe dirty air’ corresponds to 2 and so on. Multiplication is represented by ‘and’ and exponentiation by ‘as’. In a blog post, Growney provides further explanatory notes of the poem’s structure.

The earliest proof of the Fundamental Theorem of Arithmetic dates back to Euclid, who was born in the 4th Century BC and died in Alexandria.  As in the case of the scribe Ahmes estimating a value for 𝜋 over a millennium earlier, this illustrates how poetry and mathematics can intertwine across centuries, continents and cultures.

Acknowledgements

With thanks to Anthony Etherin, Sarah Glaz, JoAnne Growney and Mike Keith for generously giving permission to quote their work.

References and further reading

Blatner, David (1997) The Joy of π. Penguin Books

Bonch-Olovskaya, Dr Tatiana (2012) ‘Art of π: Mathematical History and Literary Inspiration’, Proceedings of Bridges 2012. Available online at http://archive.bridgesmathart.org/2012/bridges2012-79.pdf

Etherin, Anthony (2017) Aelindromes. Available online at https://anthonyetherin.files.wordpress.com/2017/07/aelindromes.pdf

Etherin, Anthony (2019) Stray Arts (and Other Inventions). Penteract Press

Galvin, Emily (2008) Do the Math. Tupelo Press, Vermont

Glaz, Sarah, and Growney, JoAnne (eds.) (2008) Strange Attractors: Poems of Love and Mathematics. A.K. Peters, Massachusetts

Glaz, Sarah (2011) ‘The Poetry of Prime Numbers’, Proceedings of Bridges 2011. Available online at https://archive.bridgesmathart.org/2011/bridges2011-17.pdf

Glaz, Sarah (2016) ‘Poems structured by integer sequences’, Journal of Mathematics and the Arts, 10:1-4, 44-5.

Glaz, Sarah (2017) Ode to Numbers. Antrim House, Connecticut

Gourévitch, Boris, The world of π. Available online at http://www.pi314.net/eng/index.php

Growney, JoAnne, Intersections – Poetry with Mathematics. Available online at https://poetrywithmathematics.blogspot.com

Keith, Mike, Web Home of Mike Keith. Available online at http://www.cadaeic.net

May D. (2020) ‘Poems Structured by Mathematics’ in: Sriraman B. (eds) Handbook of the Mathematics of the Arts and Sciences. Springer, Cham.

Tammet, Daniel (2012) Thinking in Numbers. Hodder & Stoughton

Posted on 23rd June 2021.

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