Mathematics in Poetry

Once a professor of mathematics at Bloomsburg University in Pennsylvania, JoAnne Growney now lives in Maryland and writes poetry. This is a draft of a now-revised article, MATHEMATICS IN POETRY, that will soon appear in JOMA, The Journal of Online Mathematics and Its Applications . This space will continue to collect math-poetry ideas and links that mathematical poets and poetic mathematicians will want to follow.

As mathematicians smile with delight at an elegant proof, others may be enchanted by the grace of a poem. An idea or an image expressed in just the right language--so that it could not be said better--is a treasure to which readers return. Particularly thrilling for me is to read a work from a poet who is fluent in the language of mathematics and uses mathematical images to make a poem vivid. I begin with a few of my favorites.

Rita Dove served as poet-laureate of the United States for the term 1993-95; here is the first stanza of her poem, "Geometry," a poem that captures the ecstasy that accompanies discovery.

(Copyright considerations have kept me from quoting Dove's poem and certain other poems in their entirety; Section 3.1, below, supplies print-sources for this and other poems. Additionally, extensive information about all poets mentioned herein can be obtained online using Mozilla Firefox or Google or another search engine.)

Poet Laureate of the United States during 1988-1990, Howard Nemerov (1920-91) served as a combat pilot during World War II. Here are lines of a poem that gives the essence of a mathematical model.

Born in Reading, Pennsylvania, Wallace Stevens (1879-1955) was, as the fragment below illustrates, one of the great imagistic voices of the twentieth century--his inventive poetry in stark contrast to the paperwork of his employment as an executive in a Hartford Insurance Company.

Winner of the Nobel Prize in Literature in 1996, Polish poet Wislawa Szymborska (1923- ) is skilled at using specific details with wit and irony and offering new insights, often moral in nature. Extremely popular in her native Poland, she is persistent in her defense of individual rights. Here are the opening lines of one of her poems.

A challenge that any word-lover may enjoy is to create a 26-word poem whose words begin with succeeding letters of the alphabet. Here is an "analytic geometry" poem I wrote in response to that challenge. (Reader, you might try it too. Beyond mathematical topics, there are many themes with varied vocabularies that yield interesting results: for example, food or gardening--either flower or vegetable--or travel or bird-watching or a visit to the zoo.)

One evident connection between mathematics and poetry is that of counting--poems have a chosen number of stanzas, stanzas have a number of lines, lines have a number of syllables. Beyond that, there is the shape of a poem. Often it is more-or-less a rectangle; sometimes the shape is not at all rectangular. For example, In Alice's Adventures in Wonderland, Lewis Carroll tells a tale of a mouse’s tail using a long-tail-shaped arrangement of words. Such a poem--which is offered in the shape of its subject--is called a concrete poem. A more general category is visual poetry--in which the shape of the poem in some general way relates to the meaning of the poem.

The Mouse's Tale by Lewis Carroll

"It is a long tail, certainly," said Alice, looking down with wonder at the Mouse's tail; "but why do you call it sad?" And she kept on puzzling about it while the Mouse was speaking, so that her idea of the tale was something like this :

My stanza, "More than Counting," offered below, is a type of visual poem; its layout and content are related. This stanza also shows the sound-effect of different line lengths. If you read it aloud, you may notice how your pace changes as the length of line changes and you "feel" the shape of the poem.

For poets as with musicians and mathematicians there is a sub- or semi-conscious inner ear that hears and keeps track of properties such as the number, the quality, and the duration of the sounds in a poem. In the four-line stanza just above, the bold-uppercase syllables designate accented or stressed syllables and there are four of them in each line. Poetry in which accents are counted is called accentual verse; if syllables are counted in each line we have syllabic verse. Poems written in English are more likely to be accentual that syllabic, but the traditional Japanese forms such as Haiku count syllables and much of the poetry written in Romance languages (French, Italian, Spanish) is syllabic rather than accentual.

Although there need not be an underlying pattern for a poem, in those works that please the ear it is likely that some pattern exists. Still, just as we do not need to know the key of a musical composition to enjoy it, similar ignorance is acceptable with poetry. We will go on to explore a few square poems (in which the number of lines and the number of syllables per line are the same) and briefly consider other syllabic patterns.

The Bear Cave (a square poem of Romania)

Twenty-five years ago at Chiscau,

marble quarry workers discovered—

trapped by an earthquake in a wondrous,

enormous cave—bones of one hundred

ninety bears, Ursulus spelaeus

(now extinct). Cold rooms of cathedral

splendor now render tourists breathless

while the insistent drip of water

counts the minutes. There is no safe place.

Invite someone to someone read "The Bear Cave" aloud and, as you listen, consider the question, Can my ear hear that a nine-syllable line consists of nine syllables? I doubt that many can answer, Yes. Instead, the effect may be that the poem seems attentively organized rather than unplanned. I find a guarded, measured quality in the uniform structure of the square poem: these are not simply words but carefully arranged thoughts--very different from the fragile imagistic Haiku--and here are two from Basho (Japan, 1644-1694) :

The hollyhocks They don’t live long

lean toward the sun but you’d never know it--

in the May rain the cicada's cry.

Discussion of syllabic verse could not be complete without mention of the fine American poet Marianne Moore (1887-1972). Born in Missouri, she majored in biology at Bryn Mawr and spent much of her life in New York. She loved baseball. Counted syllables were a consistent ingredient of Mooore’s work. Her poem, "The Icosasphere" has three six-line stanzas in which the lines vary in syllable-count and yet the stanzas have nearly identical pattern. This sort of counting was the underpinning of Moore’s legacy which combined imagination and irony with the careful observation and articulation of the scientist.

The Icosasphere by Marianne Moore (Syllable counts are given at the end of each line.)

                "In Buckinghamshire hedgerows                                                                                      ( 7)
                   the birds nesting in the merged green density,                                                            (11)
                      weave little bits of string and moths and feathers and thistledown,                    (15)
                          in parabolic concentric curves" and,                                                                      (10)
                   working for concavity, leave spherical feats of rare efficiency;                                (18)
                       whereas through lack of integration,                                                                         ( 9)

                avid for someone's fortune,                                                                                                ( 7)
                   three were slain and ten committed perjury,                                                                 (11)
                      six died, two killed themselves, and two paid fines for risks they'd run.               (14)
                            But then there is the icosasphere                                                                          ( 9)
                   in which at last we have steel-cutting at its summit of economy,                             (18)
                       since twenty triangles conjoined, can wrap one                                                      (11)

                ball or double-rounded shell                                                                                              ( 7)
                   with almost no waste, so geometrically                                                                         (12)
                       neat, it's an icosahedron. Would the engineers making one,                               (16)
                            or Mr. J. O. Jackson tell us                                                                                     ( 9)
                   how the Egyptians could have set up seventy-eight-foot solid granite vertically? (21)
                      We should like to know how that was done.                                                            ( 9)

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3. Notes and Tidbits

3.1.1 The complete text of the poems sampled above in Section 1 may be found in these collections.

"Geometry" by Rita Dove, from Selected Poems, Vintage Books, New York, 1993, page 17.

"Figures of Thought" by Howard Nemerov, from The Western Approaches, University of Chicago Press, 1975.

"Six Significant Landscapes" by Wallace Stevens, from The Collected Poems of Wallace Stevens, Vintage Books, New York, 1990, pages 73-5.

"Pi" by Wislawa Szymborska, translated by Stanislaw Baränszak and Clare Cavanagh, from View With a Grain of Sand: Selected Poems, New York: Harcourt Brace, 1995, 129-30.

3.1.2 The four poems listed in 3.1.1 also are included in NUMBERS AND FACES: A Collection of Poems with Mathematical Imagery, published June 2001 by the Humanistic Mathematics Network.

These poems are included in the collection: How I Won the Raffle by Dannie Abse, My Number by Sandra Alcosser, Reservation Mathematics by Sherman Alexie, Numbers and Faces by W. H. Auden, Thirty-six Poets and Fibonacci by Judith Baumel, The Inclined Plane by Nina Cassian, Numbers by Mary Cornish, Geometry by Rita Dove, The Parallel Syndrome, The Fraction Line, and Brief Reflections on Logic by Miroslav Holub, To Myself by Abba Kovner, Suicide by Federico Garcia Lorca, Figures of Thought by Howard Nemerov, Ode to the Numbers by Pablo Neruda, The One Girl at the Boys’ Party by Sharon Olds, Algebra by Linda Pastan, Arithmetic by Carl Sandburg, Six Significant Landscapes (III, VI) by Wallace Stevens, A Large Number, Pi, and A Word on Statistics by Wislawa Szymborska, The Calculation by David Wagoner.

An electronic copy of this anthology of twenty-four poems may be obtained from its editor, JoAnne Growney, by an e-mail request.

3.1.3 Additional collections of poems with mathematical imagery:

Against Infinity: An Anthology of Contemporary Mathematical Poetry, edited by Ernest Robson and Jet Wimp, Primary Press, Parker Ford, PA 1979.

Imagination’s Other Place: Poems of Science and Mathematics, compiled by Helen Plotz, Thomas Y. Crowell, NY, 1955.

My Dance is Mathematics, poems by JoAnne Growney, available Spring 2006 from Paper Kite Press, Wilkes-Barre, PA

A small online anthology is maintained by Katherine Stange, a mathematics doctoral student at Brown University.

A project now in the editing stage is the publication of a collection of mathematical love poems gathered by Sarah Glaz (Department of Mathematics, University of Connecticut, and JoAnne Growney .

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3.2.1 Lewis Carroll's mouse tale is from Chapter 3 of Alice’s Adventures in Wonderland, available online through the Classical Library website. A Google-search using either “concrete poetry” or “visual poetry” leads to a number of interesting examples. One engaging site is maintained by Michael P. Garofalo. This example from Garofalo’s site illustrates the use of idea and image instead of meter and rhyme in concrete poetry.

3.2.2 One of the best known of the early "visual" poets was the French experimental poet, Guillaume Apollonaire, who wrote a number of poems in diagrammatic form that he called "Calligrammes." University of California Press (1980 published Calligrammes: Poems of Peace and War (1913-1916 , a bilingual volume. One Apollonaire's best known poems is "Heart, Crown, and Mirror."

3.2.3 For a given line of verse, readers do not necessarily all agree which syllables will be stressed, For example, for the line

"BUT, even WITH a much LONGer LINE,"

a reader might choose not to stress the initial word--instead stressing the first syllable of "E-ven."

3.2.4 UbuWeb, a website used above for Apollonaire's "Heart, Crown, and Mirror" also contains Henry Lok’s square poem. A text interpretation of the manuscript photo appears on page 166 of Thomas P. Roche, Jr.'s Petrarch and the English Sonnet Sequences (New York: AMS Press, 1989). Roche's Appendix G goes on to point out the complexity of the structure within the square. For example, the first and final "pillars" give a two-line verse:

God makes kings rule for heaue[n]s; your state hold blest

And still stand will their shields; fear yields best rest. [Roche, 550]

Embedded in the poem also are other poems, found by tracing the patterns of sub-squares or crosses.

3.2.5 The square poems and "ABC" appear in My Dance is Mathematics by JoAnne Growney, available Spring 2006 from Paper Kite Press, Wilkes-Barre, PA.

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3.2.6 Translation of poetry is a topic far from the focus of this article, but in passing we mention that syllable count is one of the most difficult things to preserve when a poem is translated from one language to another. Some readers will be aware, for example, of an ongoing debate about whether the 5-7-5 syllable requirement for the Japanese Haiku is a suitable constraint to bring into the English translation. Gabriel Prajitura , a mathematician at SUNY-Brockport, is a poet and a lover of poetry. He has translated "The Bear Cave" into Romanian.

3.2.7 My source for the Haiku of Basho (1644-1694, Japan) is The Essential Haiku (Ecco Press, 1994), Edited by Robert Haas, pp 44-5. Haas gives us the delicate images in English without insistence on the 5-7-5 syllable count.

3.2.8 My source for “The Icosasphere” by Marianne Moore (1887-1972) poem is an old collection, The Complete Poems of Marianne Moore (New York, Penguin Books, 1981). More recent publications also are available.

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3.3. A few more things

3.3.1 Counting rhyme schemes

There are a myriad of combinatorial questions that maybe posed about poetry--question such as, For a Shakespearean sonnet, how many ways are there of arranging the sonnet's lines so that the rhyme scheme is preserved? Although the problem might be fun to solve, the answer is of little interest--and more interesting result, how many of these make sense? cannot be computed.

Of interest, however, is that the Catalan and Bell numbers are involved when we count rhyme schemes. For a stanza of n lines, the number of possible rhyme schemes is given by the nth Bell number and the number of planar rhyme schemes is given by the nth Catalan number. (A rhyme scheme is planar if when pairs of end-line rhyming words in a stanza are joined by arcs, no arcs intersect.) For a stanza of three lines, there are five rhyme schemes, all of which are planar. For a stanza of four lines, fourteen of the fifteen possible schemes are planar and the only non-planar rhyme scheme occurs when lines 1 and 3 rhyme and lines 2 and 4 rhyme--as in these opening lines of a love poem "Geometry" by X. J Kennedy (POETRY, October-November 2002, p 38).

They say who play at blindman's bluff

and strive to fathom space

That a straight line drawn long enough

Regains its stating place.

Martin Gardner has written extensive and readable introductions to the Catalan and Bell numbers. We learn of the Bells in Chapter 2 of Gardner's collection Fractal Music, Hypercards and More (W. H. Freeman, 1992); rhyme schemes are mentioned on page 31. An introduction to the Catalan numbers is given in Chapter 20 of Time Travel and Other Mathematical Bewilderments (Freeman, 1988); see page 63 for mention of rhyme schemes and a diagram that compares planar and non-planar varieties.

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3.3.2 A collection of mathematical love poems and book of poetry organized by the Fibonacci numbers

There are uncounted love poems that employ the imagery of mathematics to express their passion; consider, for example, these final lines from a poem entitled "Gravity & Levity" by Bin Ramke (Conjunctions 35; American Poetry: States of the Art (Bard College, 2000, ed. Bradford Morrow)).

This is a bigger world than it was once

it expands an explosion it can't help it it has

nothing to do with us with whether we know or

not whether our theories can be proved

whether or not a mathematician

knew a better class of circles

(he has a name, Taniyama, a Conjecture)

than was ever known before before—

not circles, elliptic curves. Not doughnuts.

Not anything that is nearly, only is, such

a world is hard to imagine, harder to live in,

harder still to leave. A little like love, Dear.

I have been working with Sarah Glaz, a mathematician at the University of Connecticut, to gather and edit a collection of mathematical love poems and this effort has led, for example, to Ramke’s poem above and to "Geometry" by X. J. Kennedy--and also to a fine book, alphabet, by Inger Christensen, and translated from Danish by Susanna Nied. alphabet is structurally based on the Fibonacci sequence--after a one-line title and a one-line stanza are stanzas of two lines, then three, then five and eight. Eventual stanzas explore the sequence in different ways and the impact of the structure is strong and surprising.

There is no way to end a consideration of the links between mathematics and poetry. They go on and on. But I stop here and invite you to contact me with your ideas and comments.

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