Palindromes, Squares, and Other Repetitions in Natural Languages

Jeffrey Shallit, School of Computer Science, University of Waterloo

Mathematicians and computer scientists are interested in various kinds of repetitions in strings of symbols, or words. In this page I give some examples in natural languages. I don't list all possibilities, but rather the best ones according to a vague personal metric that depends on length, commonality of use, and surprise. I don't allow any sort of punctuation, such as apostrophes or hyphens.

Palindromes

A palindrome is a string that reads the same forwards and backwards.

Here are some palindromes in English:

radar
redivider (one who divides more than once)
repaper
rotator
deified

Finnish holds the world record for the longest palindrome:

saippuakivikauppias (literally, "soap-stone-seller", or in other words, a lye-dealer)

Spanish has

reconocer (the verb, "to recognize")

Danish has

regninger
sneppens
snerrens

Swedish has

rattar ("steering wheels"; also slang for "breasts")

Italian has

onorarono ("to give honor")
avallava ("agree")

French has

serres
essayasse
ressasser ("to repeat the same thing over and over")

I was recently informed about what is supposedly the longest palindrome in German, namely, reliefpfeiler ("pilaster" in English).

Two Palindromes Concatenated

Examples in English include emesis, gigolo, papaya, usurer, and pipette.

Words whose reversal is another word

Examples in English include drawer, which becomes reward.

Squares

A square is a word of the form xx, where x is a nonempty string of symbols.

The best examples in English are

hotshots
beriberi
testes
murmur
tartar
atlatl

Example in French include chercher
froufrou
rentrent
.

An example in Finnish is valtavalta.

An example in German is

nennen (the verb "to call")

An example in Italian is

restereste ("you would remain").

Some examples in Spanish include

enarenar ("to run aground")
adorador ("worshipper")
arar ("to plow")

Examples in Danish include

farfar
mormor
purpur

Some examples in Dutch include

kerker
tenten
enen

The yellow-headed caracara, a type of bird, has both a common name that is a square, and a Latin name (Milvago chimachima) that is a square.

Higher Powers

A cube is a word of the form xxx, where x is a nonempty word. Similarly, one can define fourth powers, fifth powers, etc. Examples of repetitions like this are quite rare in natural languages.

English has the sort-of-word

shshsh (an admonition to be quiet)

The uncommon word

tratratratra (an extinct lemur from Madagascar)
is a fourth power. Sometimes it is spelled tretretretre.

Overlaps

An overlap is a word of the form axaxa, where a is a single letter, and x is a (possibly empty) word.

English has the overlaps

kinnikinnik (a native American tobacco substitute)
alfalfa
entente (taken from French)

In addition to entente, French has

tentent (as in ils tentent, "they are trying")

Dutch has

koekoek

Spanish has

adoradora ("one who adores")

Hungarian has

szeszes

as in the phrase szeszes italok, alcoholic spirits.

Fractional Powers

A fractional power is a repetition y of the form (xx...x)x', where (xx...x) means 1 or more repetitions of x, and x' is a prefix of x. The exponent of the repetition is the length of y divided by the length of x.

For example, the words

tormentor
underfund
ingesting
are all 3/2-powers. The German word schematische is also a 3/2-power.

The words

entrenchment
entanglement
physiography
educated
keepsake
meantime
reassure
template
usurious
are 4/3-powers.

The words onion
magma
salsa
verve

are 5/3-powers.

The unfamiliar word uricosuric (increased excretion of uric acid in the urine) is also a 5/3-power.

The words

alchemical
antinomian
hydropathy
meddlesome
phonograph
photograph
repertoire
strategist

are 5/4-powers.

The word brekekekex, devised by Aristophanes to represent frogs croaking, contains a 7/2-power.

The words alfalfa and entente, mentioned previously, are 7/3-powers. The words lophophore and Mississippi contain 7/3-powers.

The word ingoing is a 7/4-power. A 7/4-power in French is cherche.

The word outshout is an 8/5-power.

The word kinnikinnik, mentioned above, is a 9/4-power.

The unusual word tarantara is a 9/5-power. The word fractionation contains a 9/5-power.

The unusual taratantara (a word representing the sound of a trumpet) is an 11/7 power. A more familiar 11/7 power is abracadabra.

Abelian Squares

An Abelian square is a word of the form x x', where x and x' have the same length and x' is a permutation of x. Of course, an ordinary square is an Abelian square, as is a palindrome of even length, so in this section we restrict our attention to Abelian squares that are not squares or palindromes.

English has the Abelian squares
mesosome
reappear
intestines

as well as the less familiar words indenied, notition, vetitive, horseshoer, and meteorometer.

French, however, is positively overflowing with abelian squares. To list just a few:
resseres (as in "tu resserres", "you make tighter")
entaillaient (as in "ils entaillaient", "they make a deep cut")
entassassent (as in "qu'ils entassassent", subjunctive of "entasser", "to put in the same place")
entrainerait (conditional of verb "entrainer", "to train" (as for an athletic event))
ressassera (from "ressasser", to repeat the same thing over and over)
retaillerait (conditional of verb "retailler", "to retailor")
taillait (imperfect of "tailler", "to tailor")
tarirait (conditional of "tarir", "to cease flowing")
tenaillaient (imperfect of "tenailler", "to nail down")
trairait (from "traire", "to milk a cow")

German has the very nice 14-letter Abelian square interessierten.

Dutch has entten (grafted); gedoogde (tolerated); gieriger (more miserly); intentie (intention); meenemen (to take along); and overvoer (overfeed).

Abelian cubes

An abelian cube is like an abelian square, except that there is yet another block that is a permutation of the first two. The only example I know in English is deeded.

The inside-out transformation

This reversible transformation alters a word by taking the first half, reversing it, and concatenating it with the second half (also reversed). For example, stifle becomes itself; damned becomes madden; and raptor becomes parrot.