## Thursday, September 11, 2014

### Hailstone numbers shape a poem

One of my favorite mathy poets is Halifax mathematician Robert Dawson -- his work is complex and inventive, and fun to puzzle over.  Dawson's webpage at St Mary's University lists his mathematical activity; his poetry and fiction are available in several issues of the Journal of Humanistic Mathematics and in several postings for this blog (15 April 201230 November 2013, 2 March 2014) and in various other locations findable by Google.
Can a poem be written by following a formula?  Despite the tendency of most of us to say NO to this question we also may admit to the fact that a formula applied to words can lead to arrangements and thoughts not possible for us who write from our own learning and experiences.  How else to be REALLY NEW but to try a new method? Set a chimpanzee at a typewriter or apply a mathematical formula.
Below we offer Dawson's "Hailstone" and follow it with his explanation of how mathematics shaped the poem from its origin as a "found passage" from the beginning of Dickens' Great Expectations.

HAILSTONE       by Robert J. MacG. Dawson

My father's family name being Pirrip, and my Christian name Philip, my infant tongue could make of both names nothing longer or more explicit than Pip.

father's name Pirrip, my name my tongue make both nothing  or explicit Pip.

father's family name name being Pirrip, Pirrip, and my my father’s family name being Pirrip, my infant tongue tongue could make make of both both names nothing nothing longer or or more explicit explicit than Pip. Pip. So, I called

family name Pirrip, and my family being my tongue could make both names nothing or more explicit Pip. So called

name and family my could both nothing more Pip.  called

and my both more called

and my Christian my father’s family both names nothing more explicit than called myself Pip, and

my my family names more than myself and

my names than and

names and

and

and my Christian name

my name

name

name being Pirrip and

being and

and…

Hmmm.  Interesting sequences and juxtapositions.  Almost insights.  (And previously published in Rampike 22(1), March 2013.)

And now we have Dawson's explanation of how the poem was made:

The "hailstone sequence" was first defined by Werner Collatz in 1937.  It is defined recursively, by applying the following transformation

if  n is even, replace n by n/2; if n is odd, replace n by 3n + 1

to a given starting number. So, for instance, if we start with 7, we obtain the sequence

7,22,11,34,17,52,26,13,40,20,10,5,16,8,4,2,1,4,2,1,...

which begins to cycle 4,2,1 starting at the 15th term. The sequence resulting from a given "seed" number is known as the "Collatz sequence" or "hailstone sequence," the latter from the chaotic up-and-down movement of hailstones in a thundercloud.

The problem that Collatz posed in 1937 -- which is still unresolved -- is to determine whether every such sequence terminates in the same 4,2,1 cycle. On the one hand, computer searches have checked every number up to nearly six quintillion and found no exceptions. On the other hand, John H. Conway showed in 1972 that a related problem (involving remainders modulo six) is formally undecidable; and it is conceivable that this might be true of the hailstone problem as well.  Mathematician Paul Erdös, who offered a prize for a solution, is said to have said that "Mathematics is not ready for such problems."

I [Dawson speaking here] have created a corresponding recursive algorithm that works on strings of words. The input is a string of words (the "source text") with a specified initial substring that forms the first line of the poem; each subsequent line is found by applying the following rule to its predecessor:
If the number of words in the line is even, take the second, fourth, sixth... words to obtain the next line.
If the number of words in the line is odd, replace each word by the first string of three words in the source text that begins with that word; except for the last word of the line, for which the replacement string should have length four.

The word-counts for the lines of the resulting poem [26,13,40,20,10,5, 16 . . . ] follow the hailstone sequence; additionally, the poem may be considered as a hailstone sequence with extra structure. For most of the possible first lines, the line lengths will eventually cycle 4,2,1,4,... ; and the words will eventually cycle too. (A counterexample to these patterns is the infinite passage beginning "Father, grandfather, great-grandfather, ...")
The texture of the first few lines depends mainly on the length of the first line; two or four times an odd number usually works well. As the poem develops, the influence of repeated words within the source passage increases.

Like Dawson, I too have been moved to poetry by the Collatz conjecture; on 27 March 2011 I posted "A Mathematician's Nightmare," a poem that relies on a slightly different formulation of the conjecture; in my version of the formulas even numbers are divided in half, as above, but for odd numbers two steps are combined -- each from each odd n is calculated (3n+1)/2.