I have always been drawn to the visual poetry of mathematics: the crisp clean curves of conic sections; the graceful graphs associated with trigonometric functions; the meditative intricacies of the Mandelbrot Set; even the simple, elongated elegance of the integral sign. I still recall the thrill I experienced as a teenager when I was first introduced to the triangular array of numbers known as Pascal’s Triangle. Mathematically, the array has applications in algebra, combinatorics and probability theory, but it is also intriguing as an object in itself, on account of the many patterns embedded in its structure.

The triangle is named after the 17^{th} century French mathematician and philosopher Blaise Pascal, although it had already been discovered centuries earlier by Persian and Chinese scholars. There are many online resources describing its history, construction, mathematical applications and numerous interesting properties, so I will not go into them here. The Wikipedia article is excellent, albeit quite detailed; simpler descriptions can be found on Britannica and the always-delightful Maths is Fun.

Here are the triangle’s first few rows:

One of the challenges I set myself when I began exploring mathematical constraints in poetry was to write a poem informed by Pascal’s Triangle. The obvious choice, it seemed to me, was in each line to apply a letter count per word corresponding to the numbers along the equivalent row in the triangle. This leads to an immediate problem: what to do with all those ‘ones’ down the sides?

Autumn leaves scattered on the lawn of our garden suggested a solution – I could use a repeated visual motif to represent the number one.

From the sixth line onward I ran into another problem, namely the need for long words. Adjacent ten-letter words are manageable, but the seventh line, containing the sequence 6 15 20 15 6, presents a serious obstacle, while all successive lines would simply be intractable (in English at any rate) given my choice of constraint.

An internet search for 20-letter English words yielded *antiauthoritarianism*, *fundamentalistically*, *superresponsibleness *and other equally ungainly suggestions. ( Words with 35 letters include such monstrosities as *hippopotomonstrosesquipedaliophobia*, meaning *the* *fear of very long words*.)

I decided that hyphens would be my friend, and with this concession I was able to create a seven-line Pascal’s Triangle poem. The result is given below (you can also watch a video of me discussing and reading this poem here).

As I discovered afterwards, there are other approaches to writing a Pascal’s Triangle poem. In 2007 *Mathematics Magazine* published ‘Dearest Blaise’, a seven-line poem by Caleb Emmons that uses a letter count per word without resorting to the use of hyphens, and is bordered by ones. You can read Emmons’ fine tribute to Blaise Pascal, together with an impressive suggestion for a possible eighth line, on JoAnne Growney’s website.

In her poem ‘Pascal’s Triangle’, Emily Galvin takes a very different approach, signifying the triangle’s structure in the wording, line repetitions and arrangement of the stanzas. The number one, for example, is represented by the line *‘One the night that wakes me from my sleep’*; the number two by a two-line stanza beginning with *‘Two the pear tree standing on a hill’*; the number three by three-line stanzas, each commencing with the word *‘Three’*, and so on. It’s one of many brilliant poems in Galvin’s collection *Do the Math: Forms*, published by Tupelo Press.

Following ‘Autumn Sunrise’ I wrote ‘Midwinter’, another Pascal’s Triangle poem – inspired this time by winter in the northeast of Scotland. In this case the constraint uses a letter and character count (i.e. the hyphens contribute to the count so, for example, the ‘count’ for *‘ice-clouds’* is ten).

My intention when I wrote ‘Midwinter’ in 2019 was to complete a seasonal sequence of Pascal’s Triangle poems. However, I was distracted by other projects and set the concept aside until this summer, when David Cohen contacted me to propose a collaboration, as part of a series he was planning for the month of September. David is one of the kindest and most generous people I know, and I’ve written elsewhere about his wonderful Scrabblegrams, which he posts daily on social media. In case you don’t know, a Scrabblegram is a poem or piece of writing that uses only the 100 letter tiles of the board game Scrabble.

We agreed on a collaboration that would feature a seasonal Scrabblegram by David in conversation with complementary Pascal’s Triangle poems by me – which of course meant I had to write poems on the theme of Spring and Summer! A happy coincidence is that, in English, each of the four seasons – Autumn, Winter, Spring and Summer – is a six-letter word, which allowed for a pleasing symmetry in the construction of the poems.

I decided to express the vibrant exuberance of Spring through a range of colours and motifs, including a daffodil, seedlings, new leaves, bird nests, hyacinths, cherry blossom and tulips. (Yes, I know it’s a bit overdone! But so is Spring! I had fun putting it together.)

For summer, I chose the colour green as the theme, but used a sun symbol in the first line to signify light and warmth.

Like ‘Autumn Sunrise’, both these poems employ a letter count (so the hyphens in the last line do not contribute to the constraint).

As with any constrained writing, the joy and the challenge of Pascal’s Triangle poems lie in exploration of the space defined by the chosen bounds. I’m grateful to David Cohen for providing the motivation to complete this seasonal sequence. Here’s his wonderful Scrabblegram take on The Four Seasons; and here’s a link to our season-themed collaboration on Twitter/X, posted on 23^{rd} September 2023 (which was, fittingly, the date of the equinox).

And there’s good news for lovers of constrained writing and wordplay: David has a book of Scrabblegrams coming out soon with Penteract Press. I’ve been lucky enough to have a preview and it’s a delightful read, suitable for any season.