Opening Lines to an Infinite Poem

Let's talk about how we build our world.

I wrote these 108 lines using a simple constraint. The second line must appear exactly once in every second line, the third in every third line, and so on. This means the phrase “we build,” the second line, must appear in every even-numbered line. Lines containing other lines this way can’t introduce any new words—all you can choose is the order in which their parts appear and their punctuation.

This is simple enough that probably someone’s done it already. I’ve found poems that do this in reverse, so that the poem steadily falls apart, with the earliest being Carl Andre’s 1963 On the Sadness, which ends with the line “we are going to die” and includes it every even line. Later poets seem to have stuck to the bleak tone when copying or tweaking the format.

But I’m not here for bleak. If you do it with the numbers ascending, you get a progression. The poem can start out minimalist and get gradually more and more maximalist. And it doesn’t really ever have to end. The 109th line can be chosen freely, as can the 113th, so you can take the poem in any direction you want. So I’ll declare this to be an infinite poem, and I also hereby release these opening lines into the public domain.

You don’t need math to read or write these poems, but to write finite commentary on an infinite poem, I’ll need some number theory. As a consequence of the rules, the lines introducing new words are the prime-numbered lines. As Euclid first proved, this means the poem never devolves into pure repetition. In fact, somewhere around line 15,485,863 (the millionth prime) it must start coining new words, since there are only about a million in the history of the English language.

However, as proven in 1896, the poem does get more repetitive as it goes along. Prime numbers get steadily rarer, so the poet needs to work harder to create new concepts from the existing building blocks. We’re at our freest when two primes are close together, allowing us to use multiple new lines to reframe our old ones. We know, as of 2014, that 247-line blocks framed by novel lines occur infinitely often. We’re still trying to prove that there are infinitely many 3-line blocks where only the middle one is repetitious.

Compatibility With Other Forms

Say we call any poem that follows this constraint a “recurrent poem.” What other forms can a recurrent poem fit?

Because longer and longer lines need to appear, any form with a fixed meter is impossible for the full run of the poem. But can we embed other forms into an infinite recurrent poem? Assuming, to make it interesting but tractable, that each prime-numbered line has the same meter and length. This question sent me to the internet, and eventually to Eggleton, Kimberly, and MacDougall (2012), which tells me that there’s at least one run of six consecutive numbers with exactly 10 prime factors, meaning an infinite poem could contain a fixed-meter sestet. Sonnets remain an open question.

The question of rhyme schemes is currently breaking my brain and I might need to come back to it. I conjecture that you can embed any nontrivial rhyme scheme.

Why write a recurrent poem?

There’s two ways to write a poem a machine could never have written. One is to use your uniquely human soul and free will. The other, much easier way is to embed simple math problems into the poem format. Large Language Models are terrible at math. So there. For now.

But I actually just wrote it because it was fun to write.