What is a palindrome?

The Oxford English Dictionary describes a palindrome as "a word or a sequence of words that reads, letter for letter, the same backwards as forwards." In extended use it appears in:

• Music. A piece of music in which the second half is a retrograde repetition of the first half; the retrograde itself.
• Numbers. A number, or a date expressed numerically, that is unchanged when the order of its digits is reversed.
• Biology. A nucleic acid sequence that is identical to its complementary sequence when each is read in the same direction (which is usually the direction called 5' - 3').

I will return to these definitions in later blog posts.

To determine whether a sequence of symbols is a palindrome I need to know what the symbols are from which the palindrome is composed, which symbols I can ignore, and when two symbols are considered equal.

The above definitions tell me that the symbols may be single characters, notes, digits, or elements of DNA. But even complete words, or sentences, are used in palindromes. J.A. Lindon, who wrote many palindromes, published the following poem in 1955:

As I was passing near the jail
I met a man, but hurried by.
His face was ghastly, grimly pale.
He had a gun: I wondered why,
His face was ghastly, grimly pale,
I met a man, but hurried by,
As I was passing near the jail.

Reportedly, the first recorded palindromes, written by Sotades in Greece in the third century BC are also on the level of sentences.

A palindrome is a simple kind of mathematical symmetry. Where symmetries are exact and the smallest imperfection disqualifies a symmetry, palindromes in which the symbols are characters are often not exact symmetries. The word "refer" reads "refer" when you read it backwards, but the sequence of words in Adam's introduction "Madam, I'm Adam" reads "madA m'I ,madaM" when reversed, which clearly is not the same. Thus it would not qualify as a palindrome if we are very strict, but when talking about palindromes in languages, capitalization, spaces, and punctuation symbols are almost always ignored, or even changed.

In some situations, even different symbols are considered equal when finding palindromes. In DNA, a double helix is formed by two paired strands of nucleotides that run in opposite directions, and the nucleotides always pair in the same way (Adenine (A) with Thymine (T); Cytosine (C) with Guanine (G)). A (single-stranded) nucleotide sequence is said to be a palindrome if it is equal to its reverse complement. For example, the DNA sequence ACCTAGGT is palindromic because its nucleotide-by-nucleotide complement is TGGATCCA, and reversing the order of the nucleotides in the complement gives the original sequence.