Editing Palindrome (section)

==Other occurrences==

=== Classical music ===
[[File:Berg lulu palindrome mirror point.png|thumb|Centre part of palindrome in Alban Berg's opera ''Lulu'']]
[[Joseph Haydn]]'s [[Symphony No. 47 (Haydn)|Symphony No. 47]] in G is nicknamed "the Palindrome". In the third movement, a [[minuet]] and [[trio (music)|trio]], 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 from [[Alban Berg]]'s opera ''[[Lulu (opera)|Lulu]]'' is a palindrome,<ref name="Lulu">{{Cite web|title=Lulu|url=https://www.bl.uk/works/lulu|access-date=2021-08-07|website=[[British Library]]|language=en|archive-date=25 September 2021|archive-url=https://web.archive.org/web/20210925030200/https://www.bl.uk/works/lulu|url-status=live}}</ref> as are sections and pieces, in [[arch form]], by many other composers, including [[James Tenney]], and most famously [[Béla Bartók]]. [[George Crumb]] also used musical palindrome to text paint the [[Federico García Lorca]] poem "¿Por qué nací?", the first movement of three in his fourth book of [[Madrigal (music)|Madrigals]]. [[Igor Stravinsky]]'s final composition, ''The Owl and the Pussy Cat'', is a palindrome.<ref>A helpful list is at http://deconstructing-jim.blogspot.com/2010/03/musical-palindromes.html {{Webarchive|url=https://web.archive.org/web/20200806040510/http://deconstructing-jim.blogspot.com/2010/03/musical-palindromes.html |date=6 August 2020 }}</ref>{{Unreliable source?|date=June 2021|reason=Blogger is a blog hosting service that owns the blogspot.com domain. As a self-published source, it is considered generally unreliable and should be avoided unless the author is a subject-matter expert or the blog is used for uncontroversial self-descriptions.}}

The first movement from [[Constant Lambert]]'s [[ballet]] ''[[Horoscope (ballet)|Horoscope]]'' (1938) is entitled "Palindromic Prelude". Lambert claimed that the theme was dictated to him by the ghost of [[Bernard van Dieren]], who had died in 1936.<ref>Lloyd, Stephen. ''[https://books.google.com/books?id=hMzCAwAAQBAJ&dq=%22van+dieren%22+%22lambert%22+%22palindromic%22&pg=PA258 Constant Lambert: Beyond the Rio Grande]'' (2014), p. 258</ref>

British composer [[Robert Simpson (composer)|Robert Simpson]] also composed music in the palindrome or based on palindromic themes; the slow movement of his [[Symphony No. 2 (Simpson)|Symphony No. 2]] is a palindrome, as is the slow movement of his [[String Quartet No. 1 (Simpson)|String Quartet No. 1]]. His hour-long [[String Quartet No. 9 (Simpson)|String Quartet No. 9]] consists 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 of [[Anton Webern]] is often palindromic. Webern, who had studied the music of the Renaissance composer [[Heinrich 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 the [[symphony]], Op. 21. A striking example of vertical symmetry is the second movement of the [[Variations for piano (Webern)|Piano Variations]], Op. 27, in which Webern arranges every pitch of this [[dodecaphonic]] work around the central pitch axis of A4. From this, each downward reaching interval is replicated exactly in the opposite direction. For example, a G{{Music|sharp}}3—13 half-steps down from A4 is replicated as a B{{Music|flat}}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 composition ''[[Mixtur]]'', originally written in 1964, consists of twenty sections, called "moments", which may be [[Permutation|permuted]] in 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 is a complex musical unit and is played in the same direction in each half of the program.<ref>Rudolf Frisius, ''Karlheinz Stockhausen II: Die Werke 1950–1977; Gespräch mit Karlheinz Stockhausen, "Es geht aufwärts"'' (Mainz, London, Berlin, Madrid, New York, Paris, Prague, Tokyo, Toronto: Schott Musik International, 2008): 164–65. {{ISBN|978-3-7957-0249-6}}.</ref> By contrast, [[Karel Goeyvaerts]]'s 1953 electronic composition, ''[[Nummer 5]] (met zuivere tonen)'' is an ''exact'' palindrome: 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.<ref>M[orag] J[osephine] Grant, ''Serial Music, Serial Aesthetics: Compositional Theory in Post-war Europe'' (Cambridge, U.K.; New York: Cambridge University Press, 2001): 64–65.</ref>

In [[classical music]], a [[crab canon]] is a [[canon (music)|canon]] in 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 "[[Chi (letter)|χ]]" (pronounced /ˈkaɪ/.)  This is usually a form of reference to the crucifixion; for example, the ''{{lang|la|[[Mass in B minor structure#Crucifixus|Crucifixus]]}}'' movement of Bach's [[Mass 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, BWV 4|Christ lag in Todes Banden]]'', Handel's ''[[Messiah (Handel)|Messiah]]'' and Fauré's [[Requiem (Fauré)|Requiem]].<ref>{{cite book
  | author = Charton, Shawn E.
  | title  = Jennens vs. Handel: Decoding the Mysteries of Messiah
}}</ref>

A [[table canon]] is 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 the [[Sator Square]] from opposite sides, except that it is typically in two-part polyphony rather than in unison.<ref>{{cite book |title=The Craft of Tonal Counterpoint |last=Benjamin |first=Thomas |year=2003 |publisher=Routledge |location=New York |isbn=0-415-94391-4 |page=120 |url=https://books.google.com/books?id=Nkka1FYg2YYC&pg=PA120 |access-date=14 April 2011}}</ref>

=== Biological structures ===
{{Main|Palindromic sequence}}
[[File:DNA palindrome.svg|thumb|600px|right|Palindrome of [[DNA structure]]<br />A: Palindrome, B: Loop, C: Stem]]
Palindromic motifs are found in most [[genome]]s or sets of [[gene]]tic instructions. The meaning of palindrome in the context of genetics is slightly different, from the definition used for words and sentences. Since the [[DNA]] is formed by two paired strands of [[nucleotides]], and the nucleotides always pair in the same way ([[Adenine]] (A) with [[Thymine]] (T), [[Cytosine]] (C) with [[Guanine]] (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 sequence {{mono|ACCTAGGT}} is palindromic because its complement is {{mono|TGGATCCA}}, which is equal to the original sequence in reverse complement.

A palindromic [[DNA]] sequence may form a [[stem-loop|hairpin]]. Palindromic motifs are made by the order of the [[nucleotide]]s that specify the complex chemicals ([[protein]]s) that, as a result of those [[genetics|genetic]] instructions, the [[cell (biology)|cell]] is to produce. They have been specially researched in [[bacteria]]l chromosomes 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 the [[Y chromosome|Y-chromosome]] are arranged as palindromes.<ref>{{Cite web
 |url          = https://www.genome.gov/11007628/2003-release-mechanism-preserves-y-chromosome-gene/
 |title        = 2003 Release: Mechanism Preserves Y Chromosome Gene
 |website      = National Human Genome Research Institute (NHGRI)
 |language     = en-US
 |access-date  = 21 November 2017
 |archive-date = 1 December 2017
 |archive-url  = https://web.archive.org/web/20171201081105/https://www.genome.gov/11007628/2003-release-mechanism-preserves-y-chromosome-gene/
 |url-status   = live
}}</ref> 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,<ref name="aac">{{cite journal
  | author  = Ohno S
  | title   = Intrinsic evolution of proteins. The role of peptidic palindromes
  | journal = [[Riv. Biol.]]
  | volume  = 83
  | issue   = 2–3
  | pages   = 287–91, 405–10
  | year    = 1990
  | pmid    = 2128128
}}</ref><ref name="ac">{{cite journal
 |doi          = 10.1023/A:1023454111924
 |vauthors     = Giel-Pietraszuk M, Hoffmann M, Dolecka S, Rychlewski J, Barciszewski J
 |title        = Palindromes in proteins
 |journal      = J. Protein Chem.
 |volume       = 22
 |issue        = 2
 |pages        = 109–13
 |date         = February 2003
 |pmid         = 12760415
 |s2cid        = 28294669
 |url          = http://www.kluweronline.com/art.pdf?issn=0277-8033&volume=22&page=109
 |access-date  = 2011-02-17
 |archive-date = 2019-12-14
 |archive-url  = https://web.archive.org/web/20191214234759/https://www.wolterskluwer.nl/
 |url-status   = dead
}}</ref> but their role in the protein function is not clearly known. It has  been suggested in 2008<ref name="am">{{cite journal |vauthors=Sheari A, Kargar M, Katanforoush A, etal |year=2008 |title=A tale of two symmetrical tails: structural and functional characteristics of palindromes in proteins |url= |journal=BMC Bioinformatics |volume=9 |page=274 |doi=10.1186/1471-2105-9-274 |pmc=2474621 |pmid=18547401 |doi-access=free }}</ref> that 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 an [[alpha helical]] formation propensity of these sequences,<ref name="am" /> or in formation of protein/protein complexes.<ref name="X">{{cite journal
  | vauthors = Pinotsis N, Wilmanns M
  | title    = Protein assemblies with palindromic structure motifs
  | journal  = Cell. Mol. Life Sci.
  | volume   = 65
  | issue    = 19
  | pages    = 2953–6
  | date     = October 2008
  | pmid     = 18791850
  | doi      = 10.1007/s00018-008-8265-1
| s2cid = 29569626
 | pmc= 11131741
  }}</ref>

=== Computation theory ===
In [[automata theory]], a [[set (mathematics)|set]] of all palindromes in a given [[alphabet]] is a typical example of a [[formal language|language]] that is [[context-free language|context-free]], but not [[Regular language|regular]]. This means that it is impossible for a [[finite automaton]] to reliably test for palindromes.

In addition, the set of palindromes may not be reliably tested by a [[deterministic pushdown automaton]] which also means that they are not [[LR parser|LR(k)-parsable]] or [[LL parser|LL(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 the [[longest palindromic substring]] of a given input string in [[linear time]].<ref name=Jewels>{{citation
  | last1        = Crochemore
  | first1       = Maxime
  | last2        = Rytter
  | first2       = Wojciech
  | author2-link = Wojciech Rytter
  | title        = Jewels of Stringology: Text Algorithms
  | title-link   = Jewels of Stringology
  | publisher    = World Scientific
  | year         = 2003
  | isbn         = 978-981-02-4897-0
  | contribution = 8.1 Searching for symmetric words
  | pages        = 111–114
}}</ref><ref>{{citation
  | last         = Gusfield
  | first        = Dan
  | contribution = 9.2 Finding all maximal palindromes in linear time
  | doi          = 10.1017/CBO9780511574931
  | isbn         = 978-0-521-58519-4
  | location     = Cambridge
  | mr           = 1460730
  | pages        = 197–199
  | publisher    = Cambridge University Press
  | title        = Algorithms on Strings, Trees, and Sequences
  | year         = 1997
| s2cid = 61800864
 }}</ref>

The '''palindromic density''' of an infinite word ''w'' over an alphabet ''A'' is defined to be zero if only finitely many prefixes are palindromes; otherwise, letting the palindromic prefixes be of lengths ''n''<sub>''k''</sub> for ''k''=1,2,... we define the density to be

:<math> d_P(w) = \left( { \limsup_{k \rightarrow \infty} \frac{n_{k+1}}{n_k} } \right)^{-1} \ . </math>

Among aperiodic words, the largest possible palindromic density is achieved by the [[Fibonacci word]], which has density 1/φ, where φ is the [[Golden ratio]].<ref name=AB443>{{citation
  | last1         = Adamczewski
  | first1        = Boris
  | last2         = Bugeaud
  | first2        = Yann
  | chapter       = 8. Transcendence and diophantine approximation
  | editor1-last  = Berthé
  | editor1-link  = Valérie Berthé
  | editor1-first = Valérie
  | editor2-last  = Rigo
  | editor2-first = Michael
  | title         = Combinatorics, automata, and number theory
  | location      = Cambridge
  | publisher     = [[Cambridge University Press]]
  | series        = Encyclopedia of Mathematics and its Applications
  | volume        = 135
  | page          = 443
  | year          = 2010
  | isbn          = 978-0-521-51597-9
  | zbl           = 1271.11073
}}</ref>

A '''palstar''' is a [[concatenation]] of palindromic strings, excluding the trivial one-letter palindromes – otherwise all strings would be palstars.<ref name=Jewels />

===Calendar date===

February 2, 2020, was the most recent palindromic date which was can perfectly fit to any date formats in 8-digit FIGURES. And it happens very rare in any Millennium. The next of it will occur on December 12, 2121, which will be the last in this 3rd millennium.

3rd Millennium: February 2, 2020, and December 12, 2121.
* MM/DD/YYYY = 02/02/2020, 12/12/2121
* DD/MM/YYYY = 02/02/2020, 12/12/2121
* YYYY/MM/DD = 2020/02/02, 2121/12/12

4th Millennium: March 3, 3030
* MM/DD/YYYY = 03/03/3030
* DD/MM/YYYY = 03/03/3030
* YYYY/MM/DD = 3030/03/03