## August 18, 2004

### We Few

Sure, I’m not off drinking *Grand Text Auto*‘s FY 2004 revenues on a cruise ship in the Baltic Sea, but I did get to catch up with Mike Maguire today in Philadelphia and talk about palindromes. Poe, Nabokov, and Cortázar liked ‘em. They play an important role in children’s literature. Mike and I pondered whether nontrivial infinite palindromes can exist; perhaps if the sequence is biinfinite and a first and last element are always defined, as in 123123 … 1230321 … 321321? Mike showed me his very impressive palindrome notebook and told me about his tree-search method of composition. I told him some about how William Gillespie and I composed *2002,* too, although that process seemed far less interesting and principled. (Well, of course it had the *same* principle in one sense.) Collaboratively writing palindromes with someone who can cull your sense from nonsense does seem to have the advantage of preventing linguistic insanity, however, an occupational risk among palindrome composers. Mike hardly seems in need of such measures, however, since — in contrast to *2002* — his poems, such as “Same Nice Cinemas,” are among the most lucid texts in the English language.

August 19th, 2004 at 1:18 am

Hm. The infinite palindrome musing doesn’t wash mathematically. One might create some sort of infinite object that satisfies palindrome properties, but infinite ordinary sequences can’t be reversed and don’t have a last element. So even what one might think of intuitively as a trivial infinite palindrome (an infinite string with every character “a”, for instance) isn’t a palindrome by the usual definitions: it doesn’t have a reverse, its first element is not equal to its last (since its last doesn’t exist).

Infinite palindromes would be too time-consuming to write, anyway.

August 19th, 2004 at 4:56 pm

Re: infinite palindromes, from a resident mathematician (grad student in math): depends what you mean by “infinite”. If you think of a “sequence indexed by the integers”, then yeah, you get palindromes: these are the analogs of “even functions”, like x^2. That is, by a “sequence indexed by the integers” I mean: put some letter or whatever on each of the integer points of a number line (formally, it’s just a function from the integers to whatever characters you’re using). Then a natural definition of a palindrome would be “each number has the same value as its negative” (just like (-x)^2=x^2).

For instance, the sequence: …, -2, -1, 0, 1, 2, … (which sends every integer to itself) is a sequence on the integers, but isn’t a palindrome. A palindromic example would be: …, 2, 1, 0, 1, 2, … (send each integer to its absolute value). More generally, given

anysequence going in one direction, like: a,b,c,d,…, you can make it (more or less uniquely) into a bi-infinite sequence by just flipping it: …,d,c,b,a,b,c,d,… (by more or less, the other option is to double the middle: …,d,c,b,a,a,b,c,d,… — this is also a palindrome in some sense). So in this sense there’s as many infinite palindromes as there are infinite sequences.OTOH, if you want a first and last element, you’re going to have a bit more trouble: one natural way is to index it by the natural numbers (0,1,2,3,…), and then tack on a last element at infinity. This setup isn’t symmetric: there’s a number after 0 (namely 1), but there isn’t a number before infinity (there’s no infinity-1). Now, you could solve this (as you seem to suggest) by starting from both sides, so your sequence would look like: 0,1,2,3,… …,infinity-3,infinity-2,infinity-1,infinity — and then here again any sequence (like a,b,c,d,…) gives a palindrome (via a,b,c,d,… …,d,c,b,a).

I suspect that you’re thinking of something like this last description, as a common way of building long palindromes is to build ‘em from both sides in, as in all the “A man, a plan, …a canal — Panama!” variants (for instance). Mathematically there’s no problem with doing this and stopping at any point (0,1,2,3 — okay, I’m tired, so I get: 01233210), but one could ask if this can be done with English words and whether it could be done grammatically: a silly example would be:

I say “I say “… …” say I” say I.

(Your description actually suggests that you want not only a start and end but also a middle — playing with combinations of the above can give you this, but that’s getting a bit silly…)

August 19th, 2004 at 5:59 pm

The finite sequences we talk about are ordinary (1st 2nd 3rd elements), so I would think the infinite ones should be as well. Sure, infinite objects can be defined that would be palindromic by some definitions, but none of these seem at all interesting, since their quasi-palindromic nature is just a direct result of their definition. (I wasn’t thinking of any “applications” of these musings, by the way — that was a joke.) Palindromes — as usually discussed: sequences that are equal to their own reverse, which is isomorphic to the mathematical definition, natural numbers whose digits read the same forward and backward — are all finite, although I suppose we could take consolation the fact that the total number of them is countably infinite. And, of course, there’s the fact that palindrome verification makes for a great introductory recursive programming project.