# Mathematical research published in the form of poems

The article

Friedrich Wille: Galerkins Lösungsnäherungen bei monotonen Abbildungen, Math. Z. 127 (1972), no. 1, 10-16

is written in the form of a lengthy poem, in a style similar to that of the works of Wilhelm Busch.

Are there any other examples of original mathematical research published in a similar form?

• Free version: gdz.sub.uni-goettingen.de/dms/load/img/… Dec 29 '13 at 16:34
• The same author (F. Wille) has a survey paper on the subject: MR0690246 Dec 29 '13 at 16:51
• I don't think this counts, but for your amusement: lel.ed.ac.uk/~gpullum/loopsnoop.html Dec 29 '13 at 17:17
• NOTE to people answering this question: To format poetry without the ugly hack of code formatting or the other ugly hack of leaving a double spacing the lines, simply add two spaces to the end of each line. This will preserve linebreaks in the rendered output.
– TRiG
Dec 30 '13 at 12:07
• Two comments: (1) A wonderful/relevant book is Poetry and Mathematics by Scott Buchanan. Of course, if all that is wished is connecting the two topics, then there are many other articles. The most recent that comes to (my) mind is: Alice Major's Word Shapes and Rhymescapes: Translating Translation Symmetry into Music and Poetry. (2) I encourage others not to up-vote answers they cannot understand. In particular, the answer consisting of Chinese "poetry" does not come with any justification; so, unless you can read the classical Chinese there, a vote is not much more than a guess... Jan 1 '14 at 6:00

A famous example is Tartaglia's solution of the equation of degree 3, which he gave to Cardano (after much discussion) in the following form :

Quando chel cubo con le cose appresso

se agguaglia  qualche numero discreto

trovan dui altri differenti in esso.

Dapoi terrai questo per consueto

Che'l lor produtto sempre sia eguale

Al terzo cubo delle cose neto,

El residuo poi suo generale

Delli lor lati cubi ben sottratti

Varra la tua cosa principale...

Frederick Soddy, "The Kiss Precise." Nature 137, 1021, 1936. (See, e.g., this Wikipedia article.)

Celebrating $$b_1^2 +b_2^2 + b_3^2 + b_4^2 = \frac{1}{2}(b_1+b_2+b_3+b_4)^2$$ where $b_i$ is the i-th "bend":

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.

To spy out spherical affairs
An oscular surveyor
The sphere is much the gayer,
And now besides the pair of pairs
A fifth sphere in the kissing shares.
Yet, signs and zero as before,
For each to kiss the other four
The square of the sum of all five bends
Is thrice the sum of their squares.

The Kiss Precise (Generalized) 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.

• Wasn't there also an MO answer in such form about balls? Dec 29 '13 at 17:24
• Thank you Joseph. I was actually thinking of mathoverflow.net/a/50163/35626 , but I'll settle for your response. Dec 30 '13 at 1:21
• @TheMaskedAvenger: Ha! You are two steps ahead of me. But: That was posted two years before you joined MO. Gotcha!? :-) Dec 30 '13 at 2:05
• Not according to my stated record (see the candidate statement for the moderator elections). Using a common conflation, I started reading MO years before making this user profile right after the migration. In any case, Google has stored your wonderful contributions for future entities to enjoy. I am looking forward to a mathematically prosperous 2014 here on MO with you and others. Dec 30 '13 at 2:13

Some of ancient Chinese mathematics literatures are written in the forms of poems.(Source from WIkipedia.) Here just outline a few examples:

(1) The Mathematical Classic of Sunzi

e.g. 1:

e.g. 2:

(2) The Nine Chapters on the Mathematical Art 九章算術

composed by several generations of scholars from the 10th–2nd century BCE

How to evaluate AREA?

(3) Book on Numbers and Computation 算數書

witten around 202 BC and 186 BC

(4) Zhou Bi Suan Jing - The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven. 周髀算經

witten and organized in the Zhou Dynasty (1046 BCE—256 BCE), further compilation and addition in the Han Dynasty (202 BCE – 220 CE)

• Perhaps(?) one of the debated reasons that why it is hard to further develop the old Chinese mathematics into a more scientific/algebraic form in the past ancient eras, is due to the writings/paragraphs are too poetic. Dec 30 '13 at 0:34
• I think this answer is just false. Neither the Nine Chapters nor the Suan Shu Shu ("Book on Numbers and Computation") are written as poems. If you want to look towards India, however, you'll find much better examples. Dec 30 '13 at 18:54
• "Happy New Year! Well the post is fascinating, but completely wrong. The first two examples (the first two pictures) are actually the same page of the same book (the Nine Chapters)--it's just that one is a more modern facsimile of the manuscript--I am not sure why the person couldn't supply more examples if he wanted to claim that "almost all of Chinese math is poetry"? But the examples they gave (including the "additional" ones in the responses) are not poetry... Jan 7 '14 at 4:05
• "...I'd say that the additional example from Sun Zi is "regulated prose"; that is to say, it is "high" literary Chinese written with attention to parallelism, tonality, parts of speech, and other aesthetic conventions that one also finds in poetry. Prose written this way is considered elegant and sophisticated. The fact that Sun Zi's prose is regulated (using a fixed number of characters for each line, making the lines rhyme where possible but following no particular pattern of rhymes) makes the text easier to read (and remember) for the ancient reader, and the author seem more learned... Jan 7 '14 at 4:05
• @Idear: only your example 1 has anything close to foot prosody. All the other examples are in standard classical literary prose; they are not even regulated in the sense of the quote by Marty. The demonstration for "how to calculate area", for example, is not "colloquial". But it is so in the same way that modern mathematical proofs are not colloquial: it has omitted steps, uses jargon and symbols (areas are named and "colored" instead of the modern labeling by roman letters). Heck, it even ends with "the result obviously follows". Jan 8 '14 at 9:15

I am not sure whether this qualifies, but anyway there is the following coin problem

Twelve coins are identical in appearance, but one coin is either heavier or lighter than the others, which all weigh the same. Identify in at most three weightings the bad coin and determine whether it is heavier or lighter than the others using only a pan balance.

There was a solution given by the Blanche Descartes collective in the form of a poem published in the magazine of the CU mathematical society (the reference is Nr.13, 1950)

F set the coins out in a row
And chalked on each a letter, so,
To form the words "F AM NOT LICKED"
(An idea in his brain had clicked).
And now his mother he'll enjoin:
MA DO LIKE
ME TO FIND
FAKE COIN

Ian Stewart has written on this in an article called To Find Fake Coin in the SIAM.

 Taking Three as the subject to reason about—
A convenient number to state—
We add Seven, and Ten, and then multiply out
By One Thousand diminished by Eight.

"The result we proceed to divide, as you see,
By Nine Hundred and Ninety and Two:
Then subtract Seventeen, and the answer must be
Exactly and perfectly true.


Lewis Carroll, putting $$[(3+7+10)\times(1000-8)]/992-17=3$$ into verse, in Fit the Fifth of The Hunting of the Snark.

• This was original research? Dec 29 '13 at 19:08
• Todd, if you can find a previous statement of that result in the literature, poetic or prose, I'll take it down. Dec 29 '13 at 19:10
• There's also a theorem about $342-173$, set to music, with the lyrics at dmdb.org/lyrics/lehrer.tw3.html#10 Dec 29 '13 at 19:18
• Very droll, Gerry. :-) Dec 29 '13 at 19:53
• @Mark, so, change it to $\int t^2\,dt$. Jun 20 '17 at 23:17

A very important networking algorithm (the spanning tree protocol) can very well be considered graph theory.

In the original work (Radia Perlman, An algorithm for distributed computation of a spanningtree in an extended LAN, SIGCOMM '85 Proceedings of the ninth symposium on Data communications) the algorithm is summarized as follows:

**Algorhyme**

I think that I shall never see
a graph more lovely than a tree.
A tree whose crucial property
is loop-free connectivity.
A tree that must be sure to span
so packets can reach every LAN.
First, the root must be selected.
By ID, it is elected.
Least-cost paths from root are traced.
In the tree, these paths are placed.
A mesh is made by folks like me,
then bridges find a spanning tree.

• You can listen to author of STP playing on piano and her daughter Dawn Perlner (voice) performing at MIT's Lincoln Laboratory, a musical version of the poem, set by the author's son Ray Perlner youtube.com/watch?v=iE_AbM8ZykI Dec 30 '13 at 14:45

Most of ancient Indian mathematics are in the form of poems.

Couple of ones that immediately comes to mind are Knight's tour on a chess board, Madhava series, Baudhayana theorem

• Can you talk a bit about why you think "most of Ancient [Chinese] mathematics" is "in the form of poems"? Jan 4 '14 at 7:03
• @BenjaminDickman I am sorry, I do not have the evidence. Guess I meant only Indian mathematics. Jan 4 '14 at 15:26
• Thanks for editing. I cannot speak to Indian mathematics, for my background in Asian languages is restricted to Chinese. For what it's worth, I think an earlier answer here makes the same questionable/unjustified claim about the poetry of Ancient Chinese mathematics: mathoverflow.net/a/153109/22971 Jan 4 '14 at 20:06
• My understanding is that there is no prose in Sanskrit. When I studied the language in college, we read encyclopedias and histories and they, at least, were written in verse.
– ncr
Jan 5 '14 at 20:07

Maybe it is not really a research in the form of a poem. But it is a poem in order to help to memorize a mathematical result: The Persian (Iranian) mathematician Jamshid Kashani (1380–1429) of 15th century computed $\pi$ up to $16$ decimal places and he held the world record for about $180$ years (the best approximations before him were up to $7$ decimals by Chinese mathematicians and $11$ decimals by Indian Madhava). He computed : $2\pi=6.2831853071795865$. Then he wrote a poem to memorize this in his "Treatise On Circumference". The poem originally in Persian reads:

شش و دو هشت وسه یک هشت و پنج سه صفری

به هفت و یک زا و نه پنج و هشت و شش پنج است

The translation is roughly just the name of the above $17$ digits (including $6$) that are put together in such a way that the rhythm of the poem in Persian makes it smooth and easy to memorize. (There is only one non-trivial point which is in the second line: the word "Za"(زا) is supposed to mean $7$ (Haft in Persian) and represents the second $7$ in the decimal representation. The reason is that in Abjad arithmetic, one associates $6$ to $Z$ (ز) and $1$ to $a$ (ا) so $Za=Z+a=6+1= 7$ ).

This method of memorizing the decimal places of $\pi$ later was used also in Europe for example in a an English poem with beginning "How I like" or another one in French with beginning "Que j'aime" in which the number of letters of words are in one to one correspondence with decimals of $\pi$.

A rhymed mnemonic that "the homomorphic image of a group is isomorphic to the factor group over the homomorphism kernel", well-known to Russian math students:

Гомоморфный образ группы
Со времён капитализма
Изоморфен фактор группе
По ядру гомоморфизма.

• Does the second line indicate that the poem stems from the communist era? Jun 12 '15 at 16:47
• @StefanKohl: yes, I've heard of it in late 1980s. Jun 12 '15 at 17:56
• I've heard 'до победы коммунизма' for the second line. Apr 1 '16 at 19:32

The Archimedes Cattle problem should be mentioned here, see, for example http://web.archive.org/web/20070124203443/http://www.mcs.drexel.edu/~crorres/Archimedes/Cattle/Statement.html

Of course, the question can be asked whether this qualifies as a new result. (There is no way Archimedes or anyone else could write the answer).

In a modern interpretation, the mathematical result announced in this work is that every Pell's equation $x^2-ny^2=1$ where $n$ is a positive nonsquare integer has integral solutions with $y \not= 0$. The first published proof is due to Lagrange.

There is also a beautiful poem by J. C. Maxwell:

http://www.poemhunter.com/poem/a-problem-in-dynamics/

It is not a real research paper, but I would qualify it as a graduate-level problem:-) Thanks to Lasse Rempe who brought this ti my attention.

I just found this excellent blog-post:

http://geomblog.blogspot.co.il/2013/10/focs-reception-sing-along.html

It contains a poem on complexity classes, based on Dylan's song "Man gave names to all the animals", with the same tune but different lyrics:

Chorus:
Theorists gave names to all the classes
in the beginning, in the beginning
Theorists gave names to all the classes
in the beginning long time ago

Verse 1:
They saw a woman slim and tall
she went by fast but that's no crime
uh I think I'll call her PTIME

Verse 2:
They saw a kid who soldiered on
he got no rest from dusk to dawn
he kept sweating in the same place
uh I think I'll call him PSPACE

Verse 3:
They saw a blind man comb his shack
to find his walkstick in the black
but once he found it he could see
uh I think I'll call him NP

Verse 4:
They saw a boy who walked and walked all night
never veered to the left or right
was kind of slow but not that bad
uh I think I'll call him PPAD

Verse 5:
There was a beast and on its back
it carried Heisenberg and Dirac
it was as weird as a beast can be
uh...


Here is Samuel Taylor Coleridge on Euclid's first proposition (not original research, but I think the other answers have set ample precedent). Rather than type it all in, I grabbed this from www.poemhunter.com:

This is now--this was erst,
Proposition the first--and Problem the first.

I.
On a given finite Line
Which must no way incline;
To describe an equi--
--lateral Tri--
--A, N, G, L, E.
Now let A. B.
Be the given line
Which must no way incline;
The great Mathematician
Makes this Requisition,
That we describe an Equi--
--lateral Tri--
--angle on it:
Aid us, Reason--aid us, Wit!

II.
From the centre A. at the distance A. B.
Describe the circle B. C. D.
At the distance B. A. from B. the centre
The round A. C. E. to describe boldly venture.
(Third Postulate see.)
And from the point C.
In which the circles make a pother
Cutting and slashing one another,
Bid the straight lines a journeying go,
C. A., C. B. those lines will show.
To the points, which by A. B. are reckon'd,
And postulate the second
For Authority ye know.
A. B. C.
Triumphant shall be
An Equilateral Triangle,
Not Peter Pindar carp, not Zoilus can wrangle.

III.
Because the point A. is the centre
Of the circular B. C. D.
And because the point B. is the centre
Of the circular A. C. E.
A. C. to A. B. and B. C. to B. A.
Harmoniously equal for ever must stay;
Then C. A. and B. C.
Both extend the kind hand
To the basis, A. B.
Unambitiously join'd in Equality's Band.
But to the same powers, when two powers are equal,
My mind forbodes the sequel;
My mind does some celestial impulse teach,
And equalises each to each.
Thus C. A. with B. C. strikes the same sure alliance,
That C. A. and B. C. had with A. B. before;
And in mutual affiance,
None attempting to soar
Above another,
The unanimous three
C. A. and B. C. and A. B.
All are equal, each to his brother,
Preserving the balance of power so true:
Ah! the like would the proud Autocratorix do!
At taxes impending not Britain would tremble,
Nor Prussia struggle her fear to dissemble;
Nor the Mah'met-sprung Wight,
The great Mussulman
Would stain his Divan
With Urine the soft-flowing daughter of Fright.

IV.
But rein your stallion in, too daring Nine!
Should Empires bloat the scientific line?
Or with dishevell'd hair all madly do ye run
For done it is--the cause is tried!
And Proposition, gentle Maid,
Who soothly ask'd stern Demonstration's aid,
Has prov'd her right, and A. B. C.
Of Angles three
Is shown to be of equal side;
And now our weary steed to rest in fine,
'Tis rais'd upon A. B. the straight, the given line.

Prof. Werner Gueth, an economist known for inventing of the Ultimatum game, recently wrote a poem summarizing his research work. It is called "Poem with Endnotes and References", published in Homo Oeconomicus 30(1): 113-118:

I think many of the answers have devolved from the original question to just quoting math-related poetry (as opposed to research). I shall join the devolution only to comment that I'm surprised nobody has yet listed Jon Saxton's somewhat famous limerick:

A dozen, a gross, and a score
Plus three times the square root of four
Divided by seven
Plus five times eleven
Equals nine squared and not a bit more.

$(12+144+20+3√4)/7 +5 \cdot 11 = 9^2 + 0$

• By the way -- in a similar spirit, one can also say: "What is the order of ${\rm A}_8$? -- Seven gross scores." ... May 9 '14 at 22:41
• This limerick made an appearance on a recent episode of the TV program, QI. May 31 '16 at 23:51

Certainly not a piece of research but lovely; if pressed, I would pretend it is a survey article :D

P.S. I consider myself lucky: first time I heard it was sung by Saunders Mac Lane. Those who have heard him sing will agree one hardly could wish any better performance.

https://www.math.upenn.edu/~pemantle/songs/zeta.new

"Where are the zeros of zeta of s?", to the tune of "Sweet Betsy from Pike";
words by Tom Apostol

Where are the zeros of zeta of s?
G.F.B. Riemann has made a good guess,
They're all on the critical line, said he,
And their density's one over 2pi log t.

This statement of Riemann's has been like trigger
And many good men, with vim and with vigor,
Have attempted to find, with mathematical rigor,
What happens to zeta as mod t gets bigger.

The efforts of Landau and Bohr and Cramer,
And Littlewood, Hardy and Titchmarsh are there,
In spite of their efforts and skill and finesse,
(In) locating the zeros there's been no success.

In 1914 G.H. Hardy did find,
An infinite number that lay on the line,
His theorem however won't rule out the case,
There might be a zero at some other place.

Let P be the function pi minus li,
The order of P is not known for x high,
If square root of x times log x we could show,
Then Riemann's conjecture would surely be so.

Related to this is another enigma,
Concerning the Lindelöf function mu(sigma)
Which measures the growth in the critical strip,
On the number of zeros it gives us a grip.

But nobody knows how this function behaves,
Convexity tells us it can have no waves,
Lindelöf said that the shape of its graph,
Is constant when sigma is more than one-half.

Oh, where are the zeros of zeta of s?
We must know exactly, we cannot just guess,
In order to strengthen the prime number theorem,
The integral's contour must not get too near 'em.


New verses:

Now Andy has bettered old Riemann's fine guess
by using a fancier zeta (s).
He proves that the zeros are where they should be,
provided the characteristic is p.

There's a moral to draw from this sad tale of woe
which every young genius among you should know:
if you tackle a problem and seem to get stuck,
just take it mod p and you'll have better luck.


Errett Bishop is known for his principled opposition to the law of excluded middle. His opposition was so principled in fact that he viewed it (or more precisely the classical mathematician relying upon it) as The Devil, and composed the following verse to bring the point home:

The devil is very neat. It is his pride \ To keep his house in order. Every bit \ Of trivia has its place. He takes great pains \ To see that nothing ever does not fit. \ And yet his guests are queasy. All their food, \ Served with a flair and pleasant to the eye, \ Goes through like sawdust. Pity the perfect host! \ The devil thinks and thinks and he cannot cry.

This is on page 14 in Bishop, E. Schizophrenia in contemporary mathematics. Errett Bishop: reflections on him and his research (San Diego, Calif., 1983), 1--32, Contemp. Math., 39, American Mathematical Society, Providence, RI, 1985.