Palindrome

"Palindromes" redirects here. For the film, seePalindromes (film).

The 4th-century Greek Byzantine palindrome:ΝΙΨΟΝ ΑΝΟΜΗΜΑΤΑ ΜΗ ΜΟΝΑΝ ΟΨΙΝ(Wash Your Sins, Not Only Your Face) on a mosaic in theMonastery of Maleviin Greece.

Apalindromeis a word,number,phrase, or other sequence of symbols that reads the same backwards as forwards, such asmadamorracecar,the date "22/02/2022"and the sentence:" A man, a plan, a canal –Panama".The 19-letterFinnishwordsaippuakivikauppias(asoapstonevendor), is the longest single-word palindrome in everyday use, while the 12-letter termtattarrattat(fromJames JoyceinUlysses) is the longest in English.

The wordpalindromewas introduced by English poet and writerHenry Peachamin 1638.The concept of a palindrome can be dated to the 3rd-century BCE, although no examples survive. The earliest known examples are the 1st-century CE Latinacrosticword square,theSator Square(which contains both word and sentence palindromes), and the 4th-century Greek Byzantine sentence palindromenipson anomemata me monan opsin.

Palindromes are also found in music (thetable canonandcrab canon) and biological structures (mostgenomesincludepalindromic gene sequences). Inautomata theory,the set of all palindromes over anAlpha betis acontext-freelanguage, but it is notregular.

Etymology

The wordpalindromewas introduced by English poet and writerHenry Peachamin 1638.It is derived from the Greek rootsπάλιν'again' andδρóμος'way, direction'; a different word is used in Greek, καρκινικός 'carcinic' (lit.crab-like) to refer to letter-by-letter reversible writing.

Historical development

ASator square(in SATOR-form), on a wall in the medieval fortress town ofOppède-le-Vieux,France

The ancient Greek poetSotades(3rd-century BC) invented a form ofIonic metercalled Sotadic orSotadeanverse, which is sometimes said to have been palindromic,since it is sometimes possible to make a sotadean line by reversing a dactylic hexameter.

A 1st-century Latin palindrome was found as a graffito atPompeii.This palindrome, known as theSator Square,consists of a sentence written in Latin:sator arepo tenet opera rotas'The sower Arepo holds with effort the wheels'. It is also anacrosticwhere the first letters of each word form the first word, the second letters form the second word, and so forth. Hence, it can be arranged into aword squarethat reads in four different ways: horizontally or vertically from either top left to bottom right or bottom right to top left. Other palindromes found at Pompeii include "Roma-Olina-Milo-Amor", which is also written as an acrostic square.Indeed, composing palindromes was "a pastime of Roman landed gentry".

Nipson anomēmata mē monan opsinpalindrome, on a font atSt Martin, Ludgate

Byzantinebaptismal fontswere often inscribed with the 4th-century Greek palindrome,ΝΙΨΟΝ ΑΝΟΜΗΜΑΤΑ(orΑΝΟΜΗΜΑ)ΜΗ ΜΟΝΑΝ ΟΨΙΝ( "Nipson anomēmata mē monan opsin") 'Wash [your] sin(s), not only [your] face', attributed toGregory of Nazianzus;most notably in the basilica ofHagia SophiainConstantinople.The inscription is found on fonts in many churches in Western Europe:Orléans(St. Menin's Abbey);Dulwich College;Nottingham(St. Mary's);Worlingworth;Harlow;Knapton;London(St Martin, Ludgate); andHadleigh (Suffolk).

A 12th-century palindrome with the same square property is theHebrewpalindrome,פרשנו רעבתן שבדבש נתבער ונשרףperashnu: ra`avtan shebad'vash nitba`er venisraf'We explained the glutton who is in the honey was burned and incinerated', credited toAbraham ibn Ezrain 1924,and referring to thehalachicquestion as to whether a fly landing in honey makes the honeytreif(non-kosher).

The palindromic Latin riddle "In girum imus nocte et consumimur igni"'we go in a circle at night and are consumed by fire' describes the behavior of moths. It is likely that this palindrome is from medieval rather than ancient times. The second word, borrowed from Greek, should properly be spelledgyrum.

In English, there are many palindromewordssuch aseye,madam,anddeified,but English writers generally cited Latin and Greek palindromic sentences in the early 19th century;thoughJohn Taylorhad coined one in 1614: "Lewd did I live, & evil I did dwel" (with theampersandbeing something of a "fudge"). This is generally considered the first English-language palindrome sentence and was long reputed, notably by the grammarianJames "Hermes" Harris,to be theonlyone, despite many efforts to find others.(Taylor had also composed two other, "rather indifferent", palindromic lines of poetry: "Deer Madam, Reed", "Deem if I meed".) Then in 1848, a certain "J.T.R." coined "Able was I ere I saw Elba", which became famous after it was (implausibly) attributed toNapoleon(alluding to his exile on Elba).Otherwell-known English palindromesare: "A man, a plan, a canal – Panama" (1948),"Madam, I'm Adam" (1861),and "Never odd or even" (1930).

cartoon using the palindrome "Madam, I'm Adam"

Types

Characters, words, or lines

See also:Reversible poem

The most familiar palindromes in English are character-unit palindromes, where the characters read the same backward as forward. Examples arecivic,radar,level,rotor,kayak,madam,andrefer.The longest common ones arerotator, deified, racecar,andreviver;longer examples such asredivider,kinnikinnik,andtattarrattatare orders of magnitude rarer.

There are also word-unit palindromes in which the unit of reversal is the word ( "Is it crazy how saying sentences backwards creates backwards sentences saying how crazy it is?" ). Word-unit palindromes were made popular in therecreational linguisticscommunity byJ. A. Lindonin the 1960s. Occasional examples in English were created in the 19th century. Several in French and Latin date to theMiddle Ages.

There are also line-unit palindromes, most oftenpoems.These possess an initial set of lines which, precisely halfway through, is repeated in reverse order, without alteration to word order within each line, and in a way that the second half continues the "story" related in the first half in a way that makes sense, this last being key.

Sentences and phrases

Ambigramof the palindrome "Dogma I am God"

Palindromes often consist of a sentence or phrase, e.g., "A man, a plan, a canal, Panama", "Mr. Owl ate my metal worm", "Do geese see God?", or "Was it a car or a cat I saw?". Punctuation, capitalization, and spaces are usually ignored. Some, such as "Rats live on no evil star", "Live on time, emit no evil", and "Step on no pets", include the spaces. For more examples, seeList of English palindromic phrases.

Names

Some names are palindromes, such as thegiven namesHannah, Ava, Aviva, Anna, Eve, Bob, and Otto, or thesurnamesHarrah, Renner, Salas, and Nenonen. Lon Nol(1913–1985) was Prime Minister of Cambodia.Nisio Isinis a Japanese novelist andmangawriter, whose pseudonym ( tây đuôi duy tân,Nishio Ishin) is a palindrome when romanized using theKunrei-shikior theNihon-shikisystems, and is often written as NisiOisiN to emphasize this. Some people have changed their name in order to make it palindromic (including as the actorRobert Treborand rock-vocalistOla Salo), while others were given a palindromic name at birth (such as the philologistRevilo P. Oliver,the flamenco dancerSara Baras,the runnerAnuța Cătună,the creator of theEden ProjectTim Smit,and the Mexican racing driverNoel León).

There are also palindromic names in fictional media. "Stanley Yelnats" is the name of the main character inHoles,a 1998 novel and2003 film.Five of the fictionalPokémonspecieshave palindromic names in English (Eevee,Girafarig, Farigiraf, Ho-Oh, and Alomomola), as does the region Alola.

The 1970s pop bandABBAis a palindrome using the starting letter of the first name of each of the four band members.

Numbers

Main article:Palindromic number

Main article:Periodic continued fraction

Palindromic license plate number

The digits of a palindromic number are the same read backwards as forwards, for example, 91019;decimalrepresentation is usually assumed. Inrecreational mathematics,palindromic numbers with special properties are sought. For example, 191 and 313 arepalindromic primes.

WhetherLychrel numbersexist is an unsolved problem in mathematics about whether all numbers become palindromes when they are continuously reversed and added. For example, 56 is not a Lychrel number as 56 + 65 = 121, and 121 is a palindrome. The number 59 becomes a palindrome after three iterations: 59 + 95 = 154; 154 + 451 = 605; 605 + 506 = 1111, so 59 is not a Lychrel number either. Numbers such as 196 are thought to never become palindromes when this reversal process is carried out and are therefore suspected of being Lychrel numbers. If a number is not a Lychrel number, it is called a "delayed palindrome" (56 has a delay of 1 and 59 has a delay of 3). In January 2017 the number 1,999,291,987,030,606,810 was published in OEIS asA281509,and described as "The Largest Known Most Delayed Palindrome", with a delay of 261. Several smaller 261-delay palindromes were published separately asA281508.

Every positive integer can be written as the sum of three palindromic numbers in every number system with base 5 or greater.

Dates

A day or timestamp is a palindrome when its digits are the same when reversed. Only the digits are considered in this determination and the component separators (hyphens, slashes, and dots) are ignored. Short digits may be used as in11/11/1111:11or long digits as in2 February 2020.

A notable palindrome day is this century's 2 February 2020 because this date is a palindrome regardless of thedate format by country(yyyy-mm-dd, dd-mm-yyyy, or mm-dd-yyyy) used in various countries. For this reason, this date has also been termed as a "Universal Palindrome Day".Other universal palindrome days include, almost a millennium previously,11/11/1111,the future12/12/2121,and in a millennium03/03/3030.

In speech

A phonetic palindrome is a portion ofspeechthat is identical or roughly identical when reversed. It can arise in context where language is played with, for example in slang dialects likeverlan.In theFrench language,there is the phraseune Slave valse nue( "a Slavic woman waltzes naked" ), phonemically/ynslavvalsny/.John Oswalddiscussed his experience of phonetic palindromes while working on audio tape versions of thecut-up techniqueusing recorded readings byWilliam S. Burroughs.A list of phonetic palindromes discussed byword puzzlecolumnist O.V. Michaelsen (Ove Ofteness) include "crew work" / "work crew", "dry yard", "easy", "Funny enough", "Let Bob tell", "new moon", "selfless", "Sorry, Ross", "Talk, Scott", "to boot", "top spot" (also an orthographic palindrome), "Y'all lie", "You're caught. Talk, Roy", and "You're damn mad, Roy".

Longest palindromes

The longest single-word palindrome in theOxford English Dictionaryis the 12-letteronomatopoeicwordtattarrattat,coined byJames JoyceinUlysses(1922) for a knock on the door.TheGuinness Book of Recordsgives the title to the 11-letterdetartrated,thepreteriteand past participle ofdetartrate,a chemical term meaning to removetartrates.The 9-letter wordRotavator,a trademarked name for an agricultural machine, is listed in dictionaries as being the longest single-word palindrome. The 9-letter termredivideris used by some writers, but appears to be an invented or derived term; onlyredivideandredivisionappear in the Shorter Oxford English Dictionary; the 9-letter wordMalayalam,a language of southern India, is also of equal length.

According toGuinness World Records,theFinnish19-letter wordsaippuakivikauppias(asoapstonevendor), is the world's longest palindromic word in everyday use.

English palindrome sentences of notable length include mathematicianPeter Hilton's "Doc, note: I dissent. A fast never prevents a fatness. I diet on cod",and Scottish poetAlastair Reid's "T. Eliot, top bard, notes putrid tang emanating, is sad; I'd assign it a name: gnat dirt upset on drab pot toilet."

In English, two palindromic novels have been published:Satire: Veritasby David Stephens (1980, 58,795 letters), andDr Awkward & Olson in Osloby Lawrence Levine (1986, 31,954 words).Another palindromic English work is a 224-word long poem, "Dammit I'm Mad", written byDemetri Martin."Weird Al" Yankovic's song "Bob"is composed entirely of palindromes.

Other occurrences

Classical music

Centre part of palindrome in Alban Berg's operaLulu

Joseph Haydn'sSymphony No. 47in G is nicknamed "the Palindrome". In the third movement, aminuetandtrio,the second half of the minuet is the same as the first but backwards, the second half of the ensuing trio similarly reflects the first half, and then the minuet is repeated.

The interlude fromAlban Berg's operaLuluis a palindrome,as are sections and pieces, inarch form,by many other composers, includingJames Tenney,and most famouslyBéla Bartók.George Crumbalso used musical palindrome to text paint theFederico García Lorcapoem "¿Por qué nací?", the first movement of three in his fourth book ofMadrigals.Igor Stravinsky's final composition,The Owl and the Pussy Cat,is a palindrome.

The first movement fromConstant Lambert'sballetHoroscope(1938) is entitled "Palindromic Prelude". Lambert claimed that the theme was dictated to him by the ghost ofBernard van Dieren,who had died in 1936.

British composerRobert Simpsonalso composed music in the palindrome or based on palindromic themes; the slow movement of hisSymphony No. 2is a palindrome, as is the slow movement of hisString Quartet No. 1.His hour-longString Quartet No. 9consists of thirty-two variations and a fugue on a palindromic theme of Haydn (from the minuet of his Symphony No. 47). All of Simpson's thirty-two variations are themselves palindromic.

Hin und Zurück( "There and Back": 1927) is an operatic 'sketch' (Op. 45a) in one scene by Paul Hindemith, with a German libretto by Marcellus Schiffer. It is essentially a dramatic palindrome. Through the first half, a tragedy unfolds between two lovers, involving jealousy, murder and suicide. Then, in the reversing second half, this is replayed with the lines sung in reverse order to produce a happy ending.

The music ofAnton Webernis often palindromic. Webern, who had studied the music of the Renaissance composerHeinrich Isaac,was extremely interested in symmetries in music, be they horizontal or vertical. An example of horizontal or linear symmetry in Webern's music is the first phrase in the second movement of thesymphony,Op. 21. A striking example of vertical symmetry is the second movement of thePiano Variations,Op. 27, in which Webern arranges every pitch of thisdodecaphonicwork around the central pitch axis of A4. From this, each downward reaching interval is replicated exactly in the opposite direction. For example, a G♯3—13 half-steps down from A4 is replicated as a B♭5—13 half-steps above.

Just as the letters of a verbal palindrome are not reversed, so are the elements of a musical palindrome usually presented in the same form in both halves. Although these elements are usually single notes, palindromes may be made using more complex elements. For example,Karlheinz Stockhausen's compositionMixtur,originally written in 1964, consists of twenty sections, called "moments", which may bepermutedin several different ways, including retrograde presentation, and two versions may be made in a single program. When the composer revised the work in 2003, he prescribed such a palindromic performance, with the twenty moments first played in a "forwards" version, and then "backwards". Each moment, however, is a complex musical unit, and is played in the same direction in each half of the program.By contrast,Karel Goeyvaerts's 1953 electronic composition,Nummer 5(met zuivere tonen)is anexactpalindrome: not only does each event in the second half of the piece occur according to an axis of symmetry at the centre of the work, but each event itself is reversed, so that the note attacks in the first half become note decays in the second, and vice versa. It is a perfect example of Goeyvaerts's aesthetics, the perfect example of the imperfection of perfection.

Inclassical music,acrab canonis acanonin which one line of the melody is reversed in time and pitch from the other. A large-scale musical palindrome covering more than one movement is called "chiastic", referring to the cross-shaped Greek letter "χ"(pronounced /ˈkaɪ/.) This is usually a form of reference to the crucifixion; for example, theCrucifixusmovement of Bach'sMass in B minor.The purpose of such palindromic balancing is to focus the listener on the central movement, much as one would focus on the centre of the cross in the crucifixion. Other examples are found in Bach's cantata BWV 4,Christ lag in Todes Banden,Handel'sMessiahand Fauré'sRequiem.

Atable canonis a rectangular piece of sheet music intended to be played by two musicians facing each other across a table with the music between them, with one musician viewing the music upside down compared to the other. The result is somewhat like two speakers simultaneously reading theSator Squarefrom opposite sides, except that it is typically in two-part polyphony rather than in unison.

Biological structures

Main article:Palindromic sequence

Palindrome ofDNA structure
A: Palindrome, B: Loop, C: Stem

Palindromic motifs are found in mostgenomesor sets ofgeneticinstructions. The meaning of palindrome in the context of genetics is slightly different, from the definition used for words and sentences. Since theDNAis formed by two paired strands ofnucleotides,and the nucleotides always pair in the same way (Adenine(A) withThymine(T),Cytosine(C) withGuanine(G)), a (single-stranded) sequence of DNA is said to be a palindrome if it is equal to its complementary sequence read backward. For example, the sequenceACCTAGGTis palindromic because its complement isTGGATCCA,which is equal to the original sequence in reverse complement.

A palindromicDNAsequence may form ahairpin.Palindromic motifs are made by the order of thenucleotidesthat specify the complex chemicals (proteins) that, as a result of thosegeneticinstructions, thecellis to produce. They have been specially researched inbacterialchromosomes and in the so-called Bacterial Interspersed Mosaic Elements (BIMEs) scattered over them. In 2003, a research genome sequencing project discovered that many of the bases on theY-chromosomeare arranged as palindromes.A palindrome structure allows the Y-chromosome to repair itself by bending over at the middle if one side is damaged.

It is believed that palindromes are also found in proteins,but their role in the protein function is not clearly known. It has been suggested in 2008that the prevalent existence of palindromes in peptides might be related to the prevalence of low-complexity regions in proteins, as palindromes frequently are associated with low-complexity sequences. Their prevalence might also be related to anAlpha helicalformation propensity of these sequences,or in formation of protein/protein complexes.

Computation theory

Inautomata theory,asetof all palindromes in a givenAlpha betis a typical example of alanguagethat iscontext-free,but notregular.This means that it is impossible for afinite automatonto reliably test for palindromes.

In addition, the set of palindromes may not be reliably tested by adeterministic pushdown automatonwhich also means that they are notLR(k)-parsableorLL(k)-parsable.When reading a palindrome from left to right, it is, in essence, impossible to locate the "middle" until the entire word has been read completely.

It is possible to find thelongest palindromic substringof a given input string inlinear time.

Thepalindromic densityof an infinite wordwover an Alpha betAis defined to be zero if only finitely many prefixes are palindromes; otherwise, letting the palindromic prefixes be of lengthsn_kfork=1,2,... we define the density to be

$d_{P}(w)=\left({\limsup _{k\rightarrow \infty }{\frac {n_{k+1}}{n_{k}}}}\right)^{-1}\.$

Among aperiodic words, the largest possible palindromic density is achieved by theFibonacci word,which has density 1/φ, where φ is theGolden ratio.

Apalstaris aconcatenationof palindromic strings, excluding the trivial one-letter palindromes – otherwise all strings would be palstars.

Notable palindromists

• Dmitry Avaliani(1938–2003)
• Howard W. Bergerson(1922–2011)
• Hugo Brandt Corstius(1935–2014)
• Noam Dovev(b. 1974)
• Anthony Etherin(b. 1981)
• Simo Frangén(b. 1963)
• Baby Gramps(active 1964–present)
• Pasi Heikura(b. 1963)
• Peter Hilton(1923–2010)
• Su Hui (poet)(fourth century CE)
• Velimir Khlebnikov(1885–1922)
• J. A. Lindon(c. 1914–1979)
• Demetri Martin(b. 1973)
• Leigh Mercer(1893–1977) best known for devising the palindrome "A man, a plan, a canal: Panama!"
• Georges Perec(1936–1982)
• Mark Saltveit(b. 1961)

See also

• Ambigram
• Anagram
• Ananym
• Anastrophe,different word order
• Antimetabole
• Backmasking
• "Bob" ( "Weird Al" Yankovic song)
• Chiasmus
• Constrained writing
• Eodermdrome
• "I Palindrome I"byThey Might Be Giants
• List of English palindromic phrases
• List of palindromic places
• Mirror writing
• Palindroma,a genus of spiders with palindromic species names
• Palindromic number
• Palindromic polynomial
• Pangram
• Tarzan yell
• Yreka, California,which has the palindromicYreka BakeryandYrella Gallery

Further reading

• Word Ways: The Journal of Recreational Linguistics.Greenwood Periodicals et al., 1968–.ISSN0043-7980.
• The Palindromist.Palindromist Press, 1996–.
• Howard W. Bergerson.Palindromes and Anagrams.Dover Publications, 1973.ISBN978-0486206646.
• Dmitri A.Borgman.Language on Vacation.Charles Scribner's Sons, 1965.ISBN978-0006523086
• Stephen J. Chism.From A to Zotamorf: The Dictionary of Palindromes.Word Ways Press, 1992.ISBN978-0963515209.
• Michael Donner.I Love Me, Vol. I: S. Wordrow's Palindrome Encyclopedia.Algonquin Books, 1996.ISBN978-1565121096.

External links

• Chisholm, Hugh,ed. (1911)."Palindrome".Encyclopædia Britannica.Vol. 20 (11th ed.). Cambridge University Press. p. 633.
• "Palindromes".Several languages.European Day of Languages(EDL).Celebrated Sep 26