My late, beloved brother Martin had a globetrotting habit and a penchant for quirky and impractical gifts. On one occasion he arrived to stay with me bearing an inordinately heavy stone square, which he had seen on a trip to Italy and decided would make the perfect present for his Classics-loving, mathematically minded sister.
It was a replica of the famous SATOR square, a five word palindrome that can be read from top to bottom, bottom to top, left to right and right to left:
There has been much scholarly speculation about the origins, meaning and significance of the SATOR square (which, incidentally, is integral to the recent Christopher Nolan film, Tenet), but viewed as a form of mathematical poetry its elegant ingenuity cannot be disputed. One possible translation of the inscription is: The farmer Arepo holds wheels as his work, presumably for the purpose of ploughing the land.
The SATOR square is a special case of an acrostic, a poetic form where the first letter or word of each line spells out a message. Another remarkable acrostic is the nineteenth-century poem ‘I Often Wondered’:
I often wondered when I cursed,
Often feared where I would be –
Wondered where she’d yield her love
When I yield, so will she.
I would her will be pitied!
Cursed be love! She pitied me…
This poem is generally attributed to the mathematician Charles Lutwidge Dodgson (who, under the pseudonym Lewis Carroll, wrote Alice’s Adventures in Wonderland and Through The Looking Glass), although we do not know this for certain. Like the SATOR square, it can be read both across and down. Read vertically, the first words of each line give the opening line, the second words give the second line, and so on.
‘I Often Wondered’ is a 6 x 6 array: it has six lines, with six words in each line. The square form can, however, also be defined by syllable count, as in a remarkable poem written in 1597 by Henry Lok, in honour of Queen Elizabeth 1. The poem has 10 lines, each containing 10 single-syllable words. It can be read in the standard way, line by line. However, by choosing different starting points and routes through the word square, it can also be read in a number of other ways.
A search for ‘square poems’ on JoAnne Growney’s brilliant blog Intersections – Poetry with Mathematics will provide links to a number of examples of the form, including several delightful poems by Growney herself. Her 5 x 5 syllable square poem ‘Counting the women’ (reproduced here with her permission) is a wry comment on the paucity of women in mathematics and other STEM subjects:
When I look around the room – if I don't know in one glance how many women are there with me, I smile.
In ‘Orders of Magnitude’, H. L. Hix extends the concept of the square poem to 100 ‘decimals’, where each decimal is a 10-line poem with 10 syllables per line. You can read extracts from the sequence, which features in his collection Rational Numbers, here.
Recent years have seen the emergence of innovative forms of square poetry, including the grid poem, the Latin square puzzle poem and its more challenging variation, the Graeco-Latin square puzzle poem.
The grid poem was conceived by Brian Isett and consists of a 3 x 3 array of cells, each of which contains a word or phrase. The poem can be read in two ways, either horizontally (by row) or vertically (by column), generating shifts in perspective that give rise to different interpretations.
A Latin square is an n x n array of cells with n distinct characters, each of which occurs exactly once in each row and each column. Two examples of order 3 are shown below.
Mathematician and poet Lisa Lajeunesse has devised a form of puzzle poetry based on the Latin square that has characteristics in common with Sudoku. Instructions on how to design and solve a Latin square puzzle poem, as well as some sample puzzles, are given here. Below is a simple one of mine for you to try (solution at the end of this post):
When two Latin squares are combined in such a way that each cell entry is distinct, they form a Graeco-Latin square:
Inspired by Isett’s grid poem approach, Lajeunesse has extended the concept of the puzzle poem to Graeco-Latin squares. In this case, the words within the square can be rearranged according to two schemes: one, which reads horizontally from left to right using the number system, gives the first stanza of the poem while the second rearrangement uses the lettering system and reads vertically, from top to bottom. In a paper presented at the Bridges Math Art Conference 2019, Lajeunesse explains the method in detail, together with examples and practical suggestions for writing one’s own Graeco-Latin square poem. The constraints of the form make the construction of a coherent poem fiendishly difficult, but at the same time enormously fun both to create and to solve. (Incidentally, it’s also a useful method for alleviating writer’s block!)
Combinatorics also play a role in Bob Cobbing’s witty concrete take on the square poem. His poem consists of 12 lines, each of which is a permutation of the five words ‘This is a square poem’. Through Cobbing’s artful choice of font, spacing and punctuation, the poem forms a geometric square.
By its nature, visual poetry lends itself to geometric associations. Artist and visual poet Laura Kerr drew inspiration from Pythagoras – ‘There is geometry in the humming of the strings, there is music in the spacing of the spheres’ – to create the stunning square poem below (reproduced here with her permission):
The square and its three-dimensional companion the cube have a particular significance in ancient theories of the fundamental elements (Earth, Water, Fire, Air and Space/Ether) that were thought to constitute all matter. Plato assigned the cube to the element Earth, on the grounds that the cube was inherently the most stable of the five Platonic solids. In the tantric theory of subtle energy, Earth is represented by the yellow square contained in the symbol for the Muladhara chakra. My poem ‘Earth Geometry’ invokes these geometric objects in its structure: the poem has 22 = 4 stanzas, 42 = 16 lines and 43 = 64 syllables.
Let us assign the cube to earth. (Plato)
In this Muladhara Lotus is the square region of Prthivi. (Purnananda)
Rather delightfully, recent research supports the Platonic belief that the basic structure of earth is the cube. The farmer Arepo, working the soil with his wheeled plough, may well have had his own thoughts on the matter.
Solution to Latin square puzzle:
atoms, indivisible, move in the void, so Epicurus believed
References and further reading:
Birken, Marcia and Coon, Anne C. (2008) Discovering Patterns in Mathematics and Poetry, Amsterdam, Rodopi.
Fishwick, Duncan (1964) ‘On the Origin of the Rotas-Sator Square.’ The Harvard Theological Review, vol. 57, no. 1, 1964, pp. 39–53. http://docshare02.docshare.tips/files/16340/163404657.pdf
Glaz, Sarah (2012) ‘Mathematical Pattern Poetry.’ Bridges 2012 Conference Proceedings http://m.archive.bridgesmathart.org/2012/bridges2012-65.pdf
Coxson, G. E. ‘JoAnne Growney’s Poetry-With-Mathematics Blog — An Appreciation,’ Journal of Humanistic Mathematics, Volume 2 Issue 2 (July 2012), pages 140-150. DOI: 10.5642/ jhummath.201202.12. Available at: https://scholarship.claremont.edu/jhm/vol2/iss2/12
Hix, H. L. (2000) Rational Numbers, Truman State University Press
Lajeunesse, Lisa (2019) ‘Graeco-Latin Square Poems.’ Bridges 2019 Conference Proceedingshttp://archive.bridgesmathart.org/2019/bridges2019-35.pdf
Lajeunesse, Lisa (2018) ‘Poetry Puzzles’. Bridges 2018 Conference Proceedings http://archive.bridgesmathart.org/2018/bridges2018-645.pdf
May D. (2020) ‘Poems Structured by Mathematics’. In: Sriraman B. (eds) Handbook of the Mathematics of the Arts and Sciences. Springer, Cham.
Motoyama, Hiroshi (2001) Theories of the Chakras. New Age Books, Delhi
Plato, Timaeus and Critias, tr. Desmond Lee (1965). Penguin Classics, Middlesex