The day of mathematical poetry

posted by @ulaulaman about #mathematics #poetry #BridgesConference

Bridges: Mathematical Connections in Art, Music, and Science (discovered via Mr Honner) is an annual conference where mathematicians, scientists and artists explore the mathematical connections between these subjects. In particular today is the poetry day, and I decide to publish an interesting scientific poetry, that was submitted and published on Nature!
Frederick Soddy, who winned the Nobel Prize in Chemistry in 1921, composed The Kiss Precise, in which he rediscovered the Descartes' Circle Theorem

originally proved by Rene Descartes, which involves the radii of four mutually tangent circles

He wrote his result in verses:

For pairs of lips to kiss maybe
Involves no trigonometry.
'Tis not so when four circles kiss
Each one the other three.
To bring this off the four must be
As three in one or one in three.
If one in three, beyond a doubt
Each gets three kisses from without.
If three in one, then is that one
Thrice kissed internally.
Four circles to the kissing come.
The smaller are the benter.
The bend is just the inverse of
The distance from the center.
Though their intrigue left Euclid dumb
There's now no need for rule of thumb.
Since zero bend's a dead straight line
And concave bends have minus sign,
The sum of the squares of all four bends
Is half the square of their sum.^{(1, 2)}

The most interesting thing is the generalization of the result, that was submitted also⁽⁷⁾ in poem by Thorold Gosset:

And let us not confine our cares
To simple circles, planes and spheres,
But rise to hyper flats and bends
Where kissing multiple appears,
In n-ic space the kissing pairs
Are hyperspheres, and Truth declares,
As n + 2 such osculate
Each with an n + 1 fold mate
The square of the sum of all the bends
Is n times the sum of their squares.^{(3, 4)}

It's interesting to observe that Soddy's poetry is the oldest poem submitted to a scientific journal, being older than The Detection of Shocked Co/ Emission from G333.6-0.2 by J. W. V. Storey, published in 1984 on the Proceedings of the Astronomical Society of Australia and discovered by Maria Popova.

A generalization of Descrates' Circle Theorem⁽⁹⁾

But, following Sarah Glaz⁽²⁾, the oldest use of mathematics in poetry was made by Archimedes with The Cattle Problem that you can read in english translation in Solving the Pell Equation by H. W. Lenstra Jr. (pdf). About this specific problem, you can find a simple statement, and the solution (part 1 and part 2) by Chris Rorres.

A manuscript containing this problem was discovered by Lessing in the Wolffenbüttel library, and published by him in 1773. It is now generally credited to Archimedes. In twenty-two Greek elegiac distichs, the problem asks for the number of white, black, dappled, and brown bulls and cows belonging to the Sun god, subject to several arithmetical restrictions.⁽⁵⁾

We can reduce the problem like a system of 7 equations with 8 unknowns, so the system is indeterminare and we can find an infinity of solutions. The best we can do is find a parametric solution, in other words we can write the number of the oxen depending by an integer or real parameter.
An aid in determining solutions is surely came from the computation. In particular I report to you the paper by Hugh C. Williams, R. A. German and Charles Robert Zarnke⁽¹⁰⁾: they used a couple of IBM, the 7040 and the 1620 to be precise!
Another interesting and particularly curios example of mathematics in poetry is A square in verse of a hundred monasillbles only: Describing the sense of England's happiness by Henry Lok. In this case we have a series of words written in a square pattern like sudoku. Various interpretations of the text are possible, for example the following by Roche:

God makes kings rule for heaue[n]s; your state hold blest
And still stand will their shields; fear yields best rest.⁽⁶⁾

It seems also that some others poems are embedded in Lok composition⁽⁶⁾.
I would conclude the post with this poetry, [883], by Emily Dickinson (via JoAnne Growney)

The Poets light but Lamps—
Themselves—go out—
The Wicks they stimulate—
If vital Light

Inhere as do the Suns—
Each Age a Lens
Disseminating their
Circumference—

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Links and papers:
Mathematical poetry (blog)
Mathematical poetry: A small Anthology
Mathematical poetry (a collection of poems with math)
Mathematics and poetry: The right connection by David Within and Michelle Piwko (pdf)
The math poem: Incorporating mathematical therms in poetry by Rod Keller and Doris Davidson (pdf)
Cai Tianxin (translated by Robert Berold and Gu Ye). Mathematicians and Poets, Notices of the AMS vol.58, n.4 (2011)
Ernest Hilbert. Without a Net: Ernest Hilbert on Optic, Graphic, Acoustic, and Other Formations in Free Verse

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(1) Soddy, F. (1936). The Kiss Precise, Nature, 137 (3477) 1021. DOI: 10.1038/1371021a0
(2) Glaz, S. (2011). Poetry inspired by mathematics: a brief journey through history, Journal of Mathematics and the Arts, 5 (4) 183. DOI: 10.1080/17513472.2011.599019 (pdf)
(3) Thorold Gosset (1937). The Kiss Precise: The Hexlet Nature, 139 (3506), 62-62 DOI: 10.1038/139062a0⁽⁸⁾
(4) The text of the original poem with the generalization are published on ac-noumea.nc
(5) H. W. Lenstra Jr.. Solving the Pell Equation. Notices of the AMS, vol.49, n.2 (2002)
(6) JoAnne Growney. Mathematics in Poetry, Volume 6. October 2006
(7) I find also The hexlet by Gosset, that seems a usual mathematical paper.
(8) Following Lagarias, Mallows, Wilks⁽⁹⁾, I accredited the paper to Gosset
(9) Lagarias, J.C., Mallows, C.L. & Wilks, A.R. (2002). Beyond the Descartes Circle Theorem, The American Mathematical Monthly, 109 (4) DOI: 10.2307/2695498 (arXiv)
(10) H.C. Williams, R.A. German, C.R. Zarnke (1965). Solution of the Cattle Problem of Archimedes Mathematics of Computation, 19 (92), 671-674 DOI: 10.1090/S0025-5718-65-99945-X