I used to loathe coordinate geometry at school, mainly because we had to calculate, plot and draw the graphs by hand. My geometry notebooks were full of wobbly parabolas and ellipses that staggered uncertainly from point to point rather than flowing in one smooth, continuous curve.
It took me a while to appreciate the beauty of these shapes and their geometric companions. The parabola and ellipse form part of a class of curves known as conics, which were studied extensively by the Greek mathematician Apollonius of Perga (c. 240 BC – 190 BC). Apollonius – described by his contemporaries as ‘the Great Geometer’ – wrote a series of books on conics, seven of which survive either in the original Greek or in 9th century Arabic translations. Facts about the life of Apollonius are sparse, but his books give the impression of a man who valued friendship and was generous in sharing his insights. Here’s the preface to the revised edition of his first book, addressed to his friend Eudemus (tr. Boris Rosenfeld):
If you are restored in body, and other things go with you to your mind, well; and we too fare pretty well. At the time I was with you in Pergamum, I observed you were quite eager to be kept informed of the work I was doing in conics. And so I am sending you this first book revised.
Conics, as Apollonius goes on to elucidate in detail, are formed by the intersection of a plane and a double cone. They include the parabola, circle, ellipse and hyperbola as well as ‘degenerate’ cases – the point, line and pair of intersecting lines. The construction method is neatly visualised and explained here, and is reflected in the title of my poem ‘Five ways to slice a double cone’.
Daniel W. Galef wrote his witty, elegant sonnet ‘One Straight Line Addresses Another Traveling in the Same Direction on an Infinite Plane’ while reading Apollonius on conics:
We’ll never meet, my love! We’re parallel!— and cursed to range across our fated file parted by an inch or by an ell as totally as by a statute mile. If this flat plane were spheric, we might bend and cross our paths. (You’d make a prime meridian.) But space cannot protract, my breadthless friend, the laws that bind us—moral or Euclidian. I’d wish that we ourselves could bend, could curve, but fear: What if we curved, but not by much, and then—So close! With one Lucretian swerve we’d near, and near, and near—and never touch. A hopeless quest, eternal and quixotic— your form just out of reach, love asymptotic.
Poets past and present have been inspired by conics, whether as symbol, metaphor or in terms of their appearance on the page. John Donne’s ‘A Valediction: Forbidding Mourning’ draws on the image of a geometry compass set to represent the relationship between two lovers, one of whom is about to leave on a journey. Here are the last four stanzas:
Our two souls therefore, which are one, Though I must go, endure not yet A breach, but an expansion, Like gold to airy thinness beat. If they be two, they are two so As stiff twin compasses are two; Thy soul, the fixed foot, makes no show To move, but doth, if the other do. And though it in the center sit, Yet when the other far doth roam, It leans and hearkens after it, And grows erect, as that comes home. Such wilt thou be to me, who must, Like th' other foot, obliquely run; Thy firmness makes my circle just, And makes me end where I begun.
Terrance Hayes uses circle imagery in a very different way in ‘How to Draw a Perfect Circle’. With its layers of meaning, finely judged juxtapositions of sensuality and violence and Hayes’ exceptional command of language, the poem is complex and powerful, compassionate and disturbing.
Another member of the conics family, the parabola, features in poems by A. D. Hope, Andrei Vosnesensky and in this graceful concrete poem by Sarah Glaz, ‘Do You Believe in Fairy Tales?’
Here the parabolic shape suggests the outline of a moon or planet, enhancing the magical description of a land ‘of new beginnings’, with its ‘red sky,/ pale purple sun,/ magenta clouds’.
In addition to his work on conics, Apollonius wrote a number of other treatises which sadly have not survived. According to later writers, in one of these, ‘Tangencies’, Apollonius stated and solved the following problem (his solution, like the book itself, has been lost):
construct a circle that is tangent to (i.e. touches) three given circles in a plane.
This became known as the Problem of Apollonius. The problem attracted the interest of Renaissance mathematicians including René Descartes, who considered the special case of mutually touching circles. In correspondence with Princess Elisabeth of Bohemia in 1643, Descartes discussed a theorem connecting the radii of four circles that are all tangent to each other.
Descartes’ theorem can be neatly expressed in terms of the ‘bends’ of the circles, where bend is defined as the reciprocal of the radius: bend = 1/radius. The value of the bend can be positive or negative depending on whether a circle bends away from, or towards, the others. If we label the bends of the four tangential circles a, b, c, and d respectively, then the theorem states:
Subsequently several mathematicians independently formulated and solved this theorem, which for some reason never became widely known. In 1936 it was rediscovered by Frederick Soddy. Soddy – awarded the Nobel Prize in Chemistry in 1921 for his investigations into radioactivity and isotopes – was a man of wide-ranging interests who also worked on statistical mechanics, finance and economics. He even turned his hand to poetry: delightfully, he expressed his solution to the theorem in verse form. Soddy’s poem, entitled ‘The Kiss Precise’, was published in the journal Nature in June 1936. It opens with the lines:
For pairs of lips to kiss maybe
Involves no trigonometry.
'Tis not so when four circles kiss
Each one the other three.
The poem can be read in full here. The final stanza extends the theorem to spheres.
We have seen that, given three mutually tangential circles, it is possible to construct a fourth circle touching each of the other three. The process can be repeated to create further circles that are tangent to any three of these four circles. Successive applications will generate an aesthetically pleasing fractal, as Paul Bourke neatly explains in his 2003 paper ‘Apollony Fractal’.
This fractal provides the framework for my concrete poem ‘Circles within Circles’.
Would Apollonius have approved of the many poetic interpretations of his work? I like to think the answer is yes.
With thanks to Daniel Galef and Sarah Glaz for generously giving me permission to quote their poems.
Further Reading
Marian Christie (2021), Fractal Poems. Penteract Press.
Daniel W. Galef (2019) ‘Two Sonnets’. Arion: A Journal of Humanities and the Classics
Vol. 27, No. 2 (Fall 2019), pp. 103-104. Available at: https://www.jstor.org/stable/arion.27.issue-2
Sarah Glaz (2012) ‘Mathematical Pattern Poetry’. Bridges 2012 Conference Proceedings. Available at: http://m.archive.bridgesmathart.org/2012/bridges2012-65.pdf
Sarah Glaz (2017) Ode to Numbers. Antrim House, Connecticut.
Dana Mackenzie (2010) ‘A Tisket, a Tasket, an Apollonian Gasket’ in American Scientist, Volume 98. Available at: https://www.tcd.ie/Physics/research/groups/foams/media/gasket.pdf
Mark Pollicott (2014)’Apollonian Circle Packings’. Available at: https://homepages.warwick.ac.uk/~masdbl/apollo-29Dec2014.pdf