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82-483: A palindrome is a word, number , phrase, or other sequence of symbols that reads the same backwards as forwards, such as madam or racecar , the date " 22/02/2022 " and the sentence: "A man, a plan, a canal – Panama ". The 19-letter Finnish word saippuakivikauppias (a soapstone vendor), is the longest single-word palindrome in everyday use, while the 12-letter term tattarrattat (from James Joyce in Ulysses )

164-460: A "Universal Palindrome Day". Other universal palindrome days include, almost a millennium previously, 11/11/1111 , the future 12/12/2121 , and in a millennium 03/03/3030 . A phonetic palindrome is a portion of speech that is identical or roughly identical when reversed. It can arise in context where language is played with, for example in slang dialects like verlan . In the French language , there

246-444: A "fudge"). This is generally considered the first English-language palindrome sentence and was long reputed, notably by the grammarian James "Hermes" Harris , to be the only one, 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

328-486: 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 ,

410-421: A certain property and are palindromic. For instance: It is obvious that in any base there are infinitely many palindromic numbers, since in any base the infinite sequence of numbers written (in that base) as 101, 1001, 10001, 100001, etc. consists solely of palindromic numbers. Although palindromic numbers are most often considered in the decimal system, the concept of palindromicity can be applied to

492-532: A factor of the primorial n #, where n ≥13 and is the largest prime factor in the number. Fuller called these numbers Scheherazade numbers because they must have a factor of 1001. Scheherazade is the storyteller of One Thousand and One Nights , telling a new story each night to delay her execution. Since n must be at least 13, the primorial must be at least 1·2·3·5·7·11·13, and 7×11×13 = 1001. Fuller also refers to powers of 1001 as Scheherazade numbers. The smallest primorial containing Scheherazade number

574-526: A form of Ionic meter called Sotadic or Sotadean verse, 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 at Pompeii . This palindrome, known as the Sator Square , consists of a sentence written in Latin: sator arepo tenet opera rotas 'The sower Arepo holds with effort

656-448: A palindrome even after 700,000,000 iterations. Any number that never becomes palindromic in this way is known as a Lychrel number . On January 24, 2017, the number 1,999,291,987,030,606,810 was published in OEIS as A281509 and announced "The Largest Known Most Delayed Palindrome". The sequence of 125 261-step most delayed palindromes preceding 1,999,291,987,030,606,810 and not reported before

738-531: A paper was published demonstrating that every positive integer can be written as the sum of three palindromic numbers in every number system with base 5 or greater. Abraham ibn Ezra Abraham ben Meir Ibn Ezra ( Hebrew : ר׳ אַבְרָהָם בֶּן מֵאִיר אִבְּן עֶזְרָא , romanized :  ʾAḇrāhām ben Mēʾīr ʾībən ʾEzrāʾ , often abbreviated as ראב"ע ; Arabic : إبراهيم المجيد ابن عزرا Ibrāhim al-Mājid ibn Ezra ; also known as Abenezra or simply Ibn Ezra , 1089 / 1092 – 27 January 1164 / 23 January 1167)

820-512: A place in his view of the world. He also wrote various works on mathematical and astronomical subjects. Ibn Ezra composed his first book on astrology in Italy, before his move to France: In seven books written in Béziers in 1147–1148 Ibn Ezra then composed a systematic presentation of astrology, starting with an introduction and a book on general principles, and then five books on particular branches of

902-578: A research genome sequencing project discovered that many of the bases on the Y-chromosome are 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 2008 that the prevalent existence of palindromes in peptides might be related to

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984-475: A series of operations. First, the non-palindromic number is reversed and the result is added to the original number. If the result is not a palindromic number, this is repeated until it gives a palindromic number. Such number is called "a delayed palindrome". It is not known whether all non-palindromic numbers can be paired with palindromic numbers in this way. While no number has been proven to be unpaired, many do not appear to be. For example, 196 does not yield

1066-444: 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. By contrast, Karel Goeyvaerts 's 1953 electronic composition, Nummer 5 (met zuivere tonen) is an exact palindrome: not only does each event in

1148-428: Is 13# = 30,030. Fuller pointed out that some of these numbers are palindromic by groups of digits. For instance 17# = 510,510 shows a symmetry of groups of three digits. Fuller called such numbers Scheherazade Sublimely Rememberable Comprehensive Dividends , or SSRCD numbers. Fuller notes that 1001 raised to a power not only produces sublimely rememberable numbers that are palindromic in three-digit groups, but also

1230-467: Is a 224-word long poem, "Dammit I'm Mad", written by Demetri Martin . "Weird Al" Yankovic 's song " Bob " is composed entirely of palindromes. Joseph Haydn 's Symphony No. 47 in G is nicknamed "the Palindrome". In the third movement, a minuet and 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

1312-504: Is a bed of iron [is it not in Rabbah of the children of Ammon?]' you will understand the truth." Spinoza concluded that Ibn Ezra's reference to "the truth", and other such references scattered throughout Ibn Ezra's commentary in reference to seemingly anachronistic verses, as "a clear indication that it was not Moses who wrote the Pentateuch but someone else who lived long after him, and that it

1394-468: Is a palindrome is 2201, and it is a conjecture the fourth root of all the palindrome fourth powers are a palindrome with 100000...000001 (10 + 1). Gustavus Simmons conjectured there are no palindromes of form n for k > 4 (and n > 1). Palindromic numbers can be considered in numeral systems other than decimal . For example, the binary palindromic numbers are those with the binary representations: or in decimal: The Fermat primes and

1476-529: 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 in 11/11/11 11:11 or long digits as in 2 February 2020 . A notable palindrome day is this century's 2 February 2020 because this date is a palindrome regardless of the date 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

1558-410: Is a palindrome, as is the slow movement of his String Quartet No. 1 . His hour-long 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

1640-502: Is achieved by the Fibonacci word , which has density 1/φ, where φ is the Golden ratio . A palstar is a concatenation of palindromic strings, excluding the trivial one-letter palindromes – otherwise all strings would be palstars. Palindromic number Palindromic numbers receive most attention in the realm of recreational mathematics . A typical problem asks for numbers that possess

1722-409: Is also palindromic by definition. All numbers with one digit are palindromic, so in base 10 there are ten palindromic numbers with one digit: There are 9 palindromic numbers with two digits: All palindromic numbers with an even number of digits are divisible by 11 . There are 90 palindromic numbers with three digits (Using the rule of product : 9 choices for the first digit - which determines

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1804-514: Is also written as an acrostic square. Indeed, composing palindromes was "a pastime of Roman landed gentry". Byzantine baptismal fonts were often inscribed with the 4th-century Greek palindrome, ΝΙΨΟΝ ΑΝΟΜΗΜΑΤΑ (or ΑΝΟΜΗΜΑ ) ΜΗ ΜΟΝΑΝ ΟΨΙΝ (" Nipson anomēmata mē monan opsin ") 'Wash [your] sin(s), not only [your] face', attributed to Gregory of Nazianzus ; most notably in the basilica of Hagia Sophia in Constantinople . The inscription

1886-477: 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 as A281509 , and described as "The Largest Known Most Delayed Palindrome", with a delay of 261. Several smaller 261-delay palindromes were published separately as A281508 . Every positive integer can be written as the sum of three palindromic numbers in every number system with base 5 or greater. A day or timestamp

1968-441: Is composite, then either n = a b {\displaystyle n=ab} for some 1 < a < b − 1 {\displaystyle 1<a<b-1} , in which case n is the palindrome "aa" in base b − 1 {\displaystyle b-1} , or else it is a perfect square n = a 2 {\displaystyle n=a^{2}} , in which case n

2050-457: Is equal to the original sequence in reverse complement. A palindromic DNA sequence may form a hairpin . Palindromic motifs are made by the order of the nucleotides that specify the complex chemicals ( proteins ) that, as a result of those genetic instructions, the cell is to produce. They have been specially researched in bacterial chromosomes and in the so-called Bacterial Interspersed Mosaic Elements (BIMEs) scattered over them. In 2003,

2132-547: 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 ); and Hadleigh (Suffolk) . A 12th-century palindrome with the same square property is the Hebrew palindrome, פרשנו רעבתן שבדבש נתבער ונשרף perashnu: ra`avtan shebad'vash nitba`er venisraf 'We explained

2214-432: Is from medieval rather than ancient times. The second word, borrowed from Greek, should properly be spelled gyrum . In English, there are many palindrome words such as eye , madam , and deified , but English writers generally cited Latin and Greek palindromic sentences in the early 19th century; though John Taylor had coined one in 1614: "Lewd did I live, & evil I did dwel" (with the ampersand being something of

2296-402: 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(k)-parsable or 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

2378-583: Is listed in dictionaries as being the longest single-word palindrome. The 9-letter term redivider is used by some writers, but appears to be an invented or derived term; only redivide and redivision appear in the Shorter Oxford English Dictionary; the 9-letter word Malayalam , a language of southern India, is also of equal length. According to Guinness World Records , the Finnish 19-letter word saippuakivikauppias (a soapstone vendor),

2460-494: Is never palindromic in base b {\displaystyle b} if n / 2 ≤ b ≤ n − 2 {\displaystyle n/2\leq b\leq n-2} . Moreover, a prime number p {\displaystyle p} is never palindromic in base b {\displaystyle b} if p < b < p − 1 {\displaystyle {\sqrt {p}}<b<p-1} . A number that

2542-406: Is non-palindromic in all bases b in the range 2 ≤  b  ≤  n  − 2 can be called a strictly non-palindromic number . For example, the number 6 is written as "110" in base 2, "20" in base 3, and "12" in base 4, none of which are palindromes. All strictly non-palindromic numbers larger than 6 are prime. Indeed, if n > 6 {\displaystyle n>6}

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2624-409: 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

2706-621: Is often written as NisiOisiN to emphasize this. Some people have changed their name in order to make it palindromic (including as the actor Robert Trebor and rock-vocalist Ola Salo ), while others were given a palindromic name at birth (such as the philologist Revilo P. Oliver , the flamenco dancer Sara Baras , the runner Anuța Cătună , the creator of the Eden Project Tim Smit , and the Mexican racing driver Noel León ). There are also palindromic names in fictional media. "Stanley Yelnats"

2788-419: Is possible to find the longest palindromic substring of a given input string in linear time . 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 k for k =1,2,... we define the density to be Among aperiodic words, the largest possible palindromic density

2870-554: 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. 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

2952-675: 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 composition Mixtur , originally written in 1964, consists of twenty sections, called "moments", which may be permuted in several different ways, including retrograde presentation, and two versions may be made in

3034-689: Is the longest in English. The word palindrome was introduced by English poet and writer Henry Peacham in 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 Latin acrostic word square , the Sator Square (which contains both word and sentence palindromes), and the 4th-century Greek Byzantine sentence palindrome nipson anomemata me monan opsin . Palindromes are also found in music (the table canon and crab canon ) and biological structures (most genomes include palindromic gene sequences ). In automata theory ,

3116-445: Is the name of the main character in Holes , a 1998 novel and 2003 film . Five of the fictional Pokémon species have palindromic names in English ( Eevee , Girafarig, Farigiraf, Ho-Oh, and Alomomola), as does the region Alola. The 1970s pop band ABBA is a palindrome using the starting letter of the first name of each of the four band members. The digits of a palindromic number are

3198-540: Is the palindrome "121" in base a − 1 {\displaystyle a-1} (except for the special case of n = 9 = 1001 2 {\displaystyle n=9=1001_{2}} ). The first few strictly non-palindromic numbers (sequence A016038 in the OEIS ) are: If the digits of a natural number don't only have to be reversed in order, but also subtracted from b − 1 {\displaystyle b-1} to yield

3280-695: Is the phrase une Slave valse nue ("a Slavic woman waltzes naked"), phonemically / yn slav vals ny / . John Oswald discussed his experience of phonetic palindromes while working on audio tape versions of the cut-up technique using recorded readings by William S. Burroughs . A list of phonetic palindromes discussed by word puzzle columnist 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". The longest single-word palindrome in

3362-645: Is the world's longest palindromic word in everyday use. English palindrome sentences of notable length include mathematician Peter Hilton 's "Doc, note: I dissent. A fast never prevents a fatness. I diet on cod", and Scottish poet Alastair 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: Veritas by David Stephens (1980, 58,795 letters), and Dr Awkward & Olson in Oslo by Lawrence Levine (1986, 31,954 words). Another palindromic English work

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3444-622: Is then 11 n − 1 {\displaystyle 11_{n-1}} ). Even excluding cases where the number is smaller than the base, most numbers are palindromic in more than one base. For example, 1221 4 = 151 8 = 77 14 = 55 20 = 33 34 = 11 104 {\displaystyle 1221_{4}=151_{8}=77_{14}=55_{20}=33_{34}=11_{104}} , 1991 10 = 7 C 7 16 {\displaystyle 1991_{10}=7C7_{16}} . A number n {\displaystyle n}

3526-658: Is troubled by the anomalous nature of referring to Moses as being "beyond the Jordan", as though the writer was oriented in the land of Cana'an (west of the Jordan River), although Moses and the Children of Israel had not yet crossed the Jordan at that point in the Biblical narrative. Relating this inconsistency to others in the Torah, Ibn Ezra famously stated, "If you can grasp the mystery behind

3608-481: The Oxford English Dictionary is the 12-letter onomatopoeic word tattarrattat , coined by James Joyce in Ulysses (1922) for a knock on the door. The Guinness Book of Records gives the title to the 11-letter detartrated , the preterite and past participle of detartrate , a chemical term meaning to remove tartrates . The 9-letter word Rotavator , a trademarked name for an agricultural machine,

3690-704: The Arabic poetic form of Muwashshah . The crater Abenezra on the Moon was named in honor of Ibn Ezra. Robert Browning 's poem " Rabbi ben Ezra ", beginning "Grow old along with me/The best is yet to be", is derived from a meditation on Ibn Ezra's life and work that appeared in Browning's 1864 poetry collection Dramatis Personæ . According to Jewish tradition, Abraham ibn Ezra was buried in Cabul , alongside Judah Halevi . [REDACTED]  This article incorporates text from

3772-445: 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 ACCTAGGT is palindromic because its complement is TGGATCCA , which

3854-552: The Mersenne primes form a subset of the binary palindromic primes. Any number n {\displaystyle n} is palindromic in all bases b {\displaystyle b} with b > n {\displaystyle b>n} (trivially so, because n {\displaystyle n} is then a single-digit number), and also in base n − 1 {\displaystyle n-1} (because n {\displaystyle n}

3936-401: The OEIS ). The number of palindromic numbers which have some other property are listed below: There are many palindromic perfect powers n , where n is a natural number and k is 2, 3 or 4. The first nine terms of the sequence 1 , 11 , 111 , 1111 , ... form the palindromes 1, 121, 12321, 1234321, ... (sequence A002477 in the OEIS ) The only known non-palindromic number whose cube

4018-426: The natural numbers in any numeral system . Consider a number n  > 0 in base b  ≥ 2, where it is written in standard notation with k +1 digits a i as: with, as usual, 0 ≤  a i  <  b for all i and a k  ≠ 0. Then n is palindromic if and only if a i  =  a k − i for all i . Zero is written 0 in any base and

4100-454: The Pussy Cat , is a palindrome. The first movement from Constant Lambert 's 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. British composer Robert Simpson also composed music in the palindrome or based on palindromic themes; the slow movement of his Symphony No. 2

4182-485: The first scholars who are known to have published a text about the division of the Book of Isaiah into at least two distinct parts. He remarked in his commentary to Isaiah that chapters 1-39 dealt with a different historical period (second half of the 8th century BCE) than chapters 40-66 (later than last third of the 6th century BCE). This division of the book into First Isaiah and Deutero Isaiah has been accepted nowadays by all but

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4264-468: The following problematic passages: 1) The final twelve verses of this book [i.e., Deuteronomy 34:1–12, describing the death of Moses], 2) 'Moshe wrote [this song on the same day, and taught it to the children of Israel]' [Deuteronomy 31:22]; 3) 'At that time, the Canaanites dwelt in the land' [Genesis 12:6]; 4) '... In the mountain of God, He will appear' [Genesis 22:14]; 5) 'behold, his [Og king of Bashan] bed

4346-413: The glutton who is in the honey was burned and incinerated', credited to Abraham ibn Ezra in 1924, and referring to the halachic question as to whether a fly landing in honey makes the honey treif (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

4428-519: 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's Messiah and Fauré's Requiem . 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

4510-447: 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, the Crucifixus movement of Bach's Mass in B minor . The purpose of such palindromic balancing is to focus

4592-506: The minuet is repeated. The interlude from Alban Berg 's opera Lulu is a palindrome, 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 Madrigals . Igor Stravinsky 's final composition, The Owl and

4674-474: The most conservative Jews and Christians. Ibn Ezra's commentaries, and especially some of the longer excursuses, contain numerous contributions to the philosophy of religion . One work in particular that belongs to this province, Yesod Mora ("Foundation of Awe"), on the division and the reasons for the Biblical commandments, he wrote in 1158 for a London friend, Joseph ben Jacob. In his philosophical thought neoplatonic ideas prevail; and astrology also had

4756-437: 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. Palindromic motifs are found in most genomes or sets of genetic instructions. The meaning of palindrome in the context of genetics is slightly different, from the definition used for words and sentences. Since

4838-522: The original sequence again, then the number is said to be antipalindromic . Formally, in the usual decomposition of a natural number into its digits a i {\displaystyle a_{i}} in base b {\displaystyle b} , a number is antipalindromic iff a i = b − 1 − a k − i {\displaystyle a_{i}=b-1-a_{k-i}} . Non-palindromic numbers can be paired with palindromic ones via

4920-467: The plain understanding. Indeed, Ibn Ezra is claimed by proponents of the higher biblical criticism of the Torah as one of its earliest pioneers. Baruch Spinoza , in concluding that Moses did not author the Torah, and that the Torah and other protocanonical books were written or redacted by somebody else, cites Ibn Ezra commentary on Deuteronomy. In his commentary, Ibn Ezra looks to Deuteronomy 1:1, and

5002-445: 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, or in formation of protein/protein complexes. In automata theory , a set of all palindromes in a given alphabet is a typical example of a language that is context-free , but not regular . This means that it

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5084-604: The publication of his expanded commentary on Genesis . Besides his commentaries on the Torah, Ibn Ezra would also publish a multitude of works on science in Hebrew. In doing so, he would continue his mission of spreading the knowledge he had gained in Spain to the Jews throughout the areas he visited and lived. This can be seen particularly in the works he published while living in France. Here, many of

5166-419: The same read backwards as forwards, for example, 91019; decimal representation is usually assumed. In recreational mathematics , palindromic numbers with special properties are sought. For example, 191 and 313 are palindromic primes . Whether Lychrel numbers exist is an unsolved problem in mathematics about whether all numbers become palindromes when they are continuously reversed and added. For example, 56

5248-402: 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. In classical music , a crab canon is a canon in which one line of

5330-509: The set of all palindromes over an alphabet is a context-free language, but it is not regular . The word palindrome was introduced by English poet and writer Henry Peacham in 1638. It is derived from the Greek roots πάλιν 'again' and δρóμος 'way, direction'; a different word is used in Greek, καρκινικός 'carcinic' (lit. crab-like ) to refer to letter-by-letter reversible writing. The ancient Greek poet Sotades (3rd-century BC) invented

5412-514: The spaces. Some names are palindromes, such as the given names Hannah, Ava, Aviva, Anna, Eve, Bob, and Otto, or the surnames Harrah, Renner, Salas, and Nenonen. Lon Nol (1913–1985) was Prime Minister of Cambodia. Nisio Isin is a Japanese novelist and manga writer, whose pseudonym (西尾 維新, Nishio Ishin ) is a palindrome when romanized using the Kunrei-shiki or the Nihon-shiki systems, and

5494-489: The subject. The presentation appears to have been planned as an integrated whole, with cross-references throughout, including references to subsequent books in the future tense. Each of the books is known in two versions, so it seems that at some point Ibn Ezra also created a revised edition of the series. There are a great many other poems by Ibn Ezra, some of them religious and some secular – about friendship, wine, didactic or satirical. Like his friend Yehuda Halevi , he used

5576-487: The techniques outlined by Hayyuj, Ibn Ezra would publish his first biblical commentary, a commentary on Kohelet in 1140. He would continue to publish such commentaries over mostly works from Ketuvim and Nevi'im throughout his journey, though he would manage to publish a short commentary over the entire Pentateuch while living in Lucca in 1145. This short commentary would be amended into longer portions beginning in 1155 with

5658-514: The third digit as well - multiplied by 10 choices for the second digit): There are likewise 90 palindromic numbers with four digits (again, 9 choices for the first digit multiplied by ten choices for the second digit. The other two digits are determined by the choice of the first two): so there are 199 palindromic numbers smaller than 10 . There are 1099 palindromic numbers smaller than 10 and for other exponents of 10 we have: 1999, 10999, 19999, 109999, 199999, 1099999, ... (sequence A070199 in

5740-602: 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 the recreational linguistics community by J. A. Lindon in the 1960s. Occasional examples in English were created in the 19th century. Several in French and Latin date to the Middle Ages . There are also line-unit palindromes, most often poems . These possess an initial set of lines which, precisely halfway through,

5822-460: The values of the groups are the binomial coefficients . For instance, This sequence fails at (1001) because there is a carry digit taken into the group to the left in some groups. Fuller suggests writing these spillovers on a separate line. If this is done, using more spillover lines as necessary, the symmetry is preserved indefinitely to any power. Many other Scheherazade numbers show similar symmetries when expressed in this way. In 2018,

5904-399: The wheels'. It is also an acrostic where 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 a word square that 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

5986-516: The work of Arab scholars he had studied in Spain. Beginning many of his writings in Italy, Ibn Ezra also worked extensively to translate the works of grammarian and biblical exegetist Judah ben David Hayyuj from their original Judeo-Arabic to Hebrew. Published as early as 1140, these translations became some of the first expositions of Hebrew grammar to be written in Hebrew. During his time of publishing translations, Ibn Ezra also began to publish his own biblical commentaries. Making use of many of

6068-522: The works published can be seen as relating to astrology, and use of the astrolabe . In his commentary, Ibn Ezra adhered to the literal sense of the texts, avoiding Rabbinic allegory and Kabbalistic interpretation. He exercised an independent criticism that, according to some writers, exhibits a marked tendency toward rationalism . In addition, he sharply criticized those who blended the simplistic and logical explanation with Midrash , maintaining that such interpretations were never intended to supplant

6150-652: Was (implausibly) attributed to Napoleon (alluding to his exile on Elba). Other well-known English palindromes are: "A man, a plan, a canal – Panama" (1948), "Madam, I'm Adam" (1861), and "Never odd or even" (1930). The most familiar palindromes in English are character-unit palindromes, where the characters read the same backward as forward. Examples are civic , radar , level , rotor , kayak , madam , and refer . The longest common ones are rotator, deified, racecar , and reviver ; longer examples such as redivider , kinnikinnik , and tattarrattat are orders of magnitude rarer. There are also word-unit palindromes in which

6232-412: Was Tudela. Little is known of ibn Ezra's family from outside sources; however, he wrote of a marriage to a wife that produced five children. While it is believed four died early, the last-born, Isaac, became an influential poet and later convert to Islam in 1140. The conversion of his son was deeply troubling for ibn Ezra, leading him to pen many poems reacting to the event for years afterward. Ibn Ezra

6314-405: Was a close friend of Judah Halevi , who was some 14 years older. When ibn Ezra moved to Córdoba as a young man, Halevi followed him. This trend continued when the two began their lives as wanderers in 1137. Halevi died in 1141, but Ibn Ezra continued travelling for three decades, reaching as far as Baghdad . During his travels, he began to compose secular poetry describing the lands through which he

6396-412: Was a different book that Moses wrote". Spinoza and later scholars were thus able to expand on several of Ibn Ezra's references as a means of providing stronger evidence for Non-Mosaic authorship. Orthodox writers, on the other hand, have stated that Ibn Ezra's commentary can be interpreted as consistent with Jewish tradition stating that the Torah was divinely dictated to Moses. Ibn Ezra is also among

6478-562: 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 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

6560-713: Was one of the most distinguished Jewish biblical commentators and philosophers of the Middle Ages . He was born in Tudela , Taifa of Zaragoza (present-day Navarre ). Abraham Ibn Ezra was born in Tudela, one of the oldest and most important Jewish communities in Navarre . At the time, the town was under the rule of the emirs of the Muslim Taifa of Zaragoza. However, when he later moved to Córdoba , he claimed it to be his place of birth. Ultimately, most scholars agree that his place of birth

6642-448: Was published separately as A281508 . The sum of the reciprocals of the palindromic numbers is a convergent series, whose value is approximately 3.37028... (sequence A118031 in the OEIS ). Scheherazade numbers are a set of numbers identified by Buckminster Fuller in his book Synergetics . Fuller does not give a formal definition for this term, but from the examples he gives, it can be understood to be those numbers that contain

6724-549: Was travelling as well as beginning to pen the deeply rational Torah commentaries he would be best remembered for. He appears to have been unrelated to the contemporary scholar Moses ibn Ezra . In Spain, Ibn Ezra had already gained the reputation of a distinguished poet and thinker. However, apart from his poems, the vast majority of his work was composed after 1140. Written in Hebrew , as opposed to earlier thinkers' use of Judeo-Arabic , these works covering Hebrew grammar, Biblical exegesis, and scientific theory were tinged with

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