Matematika, kalba, poezija: sąveikos ir traukos

GIEDRIUS ALKAUSKAS Vilnius University ORCID id: orcid.org/0000-0002-5747-363X Research areas: modular forms, functional equations, projective flows; natural-scientific aspects of musicology; the language and form of poetry. DOI: doi.org/10.35321/all86-10 MATHEMATICS, LANGUAGE, POETRY: INTERACTIONS AND ATTRACTIONS Mathematics, Language, Poetry: Interactions and Attractions ABSTRACT The article examines structures situated at the intersection of two of the fields of cognition (mathematics, language, and poetry) that imply nontrivial conclusions for the third. For example, it considers how to interpret creatively a Lithuanian text written in accordance with strict mathematical and grammatical rules. Further, drawing on mathematics, it reveals the core of the terza-rima form; at the same time, this poetic invention of Dante makes it possible to formulate a theoretical (and, in several cases, practical) problem. The article also discusses the assumptions determining when interactions among several disciplines are (or may be) fruitful. KEYWORDS: versification, fixed forms, tercets, quintains, Baranauskas, Dante, five-line stanza, bidisciplinarity, concrete poetry, synonym, metathesis, mnemonic, Luhn’s formula, tropes. ANNOTATION In this paper, we investigate the structures built in the intersection of two domains of knowledge (mathematics, language, poetry), which imply non-trivial consequences for the remaining domain. For example, we explore the possibilities of how to creatively interpret the Lithuanian text whose form is strictly limited by mathematical and grammatical rules. Further, mathematics helps us to reveal the core of terza rima by Dante and allows to pose a purely theoretical (and in few cases practical) problem about its generalizations. We also Straipsniai / Articles 227 GIEDRIUS ALKAUSKAS discuss the assumptions under which the interactions of several domains of creativity are (or can be) fertile. KEYWORDS: Versification, formes fixes, terza rima, quinta rima, Baranauskas, Dante, pentastich, bidisciplinary, concrete poetry, synonym, metathesis, mnemonic, Luhn algorithm, trope. 1. INTRODUCTION Improvisation in poetry is the most important thing, but before that there must be a substantive whole—the writer must understand both physics and mathematics. Sigitas Geda 1.1. Context. This article is the third in a cycle of four texts. The first was devoted to the language of poetry (2019), the second to its sound (2020). The final part will discuss the script of poetry: the reasons for nonstandard spelling (historical, political, technical, personal, intentional) and its potential artistic capacity; graphemes, syllabograms, logograms, hieroglyphs, pictograms, and graphics; petroglyphs, lithometry¹ and boustrophedon; the poetic dimensions of scripts (say, the Adlam and N’ko of West Africa, the language of Enoch) and the possibility of converting / extrapolating this into Lithuanian. Indeed, questions about the exolinguistic deficit / surplus in Lithuanian verse in graždanka², or in the books of the Lithuanian Tatars in Ruthenian written in Arabic script (for example, Ivan Luckevič’s kitab), are intriguing. And further: does the form of a stone dictate another spacetime for the texts carved into it?³ Do Luiso Kerolio’s Jabberwocky⁴ differ from a text that could not be understood but in fact has meaning, though lost forever? The ideas of the fourth part will orbit along such trajectories. The aim of this article is to distill several constructions based on the interaction of two fields (poetry, language, mathematics) that imply a nontrivial ¹ Measurement of stone (λῐ́θος μέτρον). This term is more appropriate here than measurement of the earth (geometry). ² Giedrius Subačius’s monograph (Subačius 2010) is devoted to another aspect of this topic. ³ Poetry carved in stone combines within itself both the art of time and the timeless geometry of space. A poem written in one of the quintina forms discussed later in this text (2.2.), namely, 13213 24324 31431 42142 (the standard quintina modulo 4), carved onto a stone stele shaped as a rectangular parallelepiped with proportions 1 × 1 × 2, conceals even the direction of time. ⁴ For the nontrivial linguistic structure of this poem, see https://en.wikipedia.org/wiki/Jabberwocky. Wonderful! Not surprising, since L. Kerolis (Lewis Carroll, real surname Charles L. Dodgson) was a professional mathematician. 228 Acta Linguistica Lithuanica LXXXVI Matematika, kalba, poezija: sąveikos ir traukos trečiosios srities struktūrą. Tai daryta pagal darbo apie muziką ir matematiką analogiją (Alkauskas 2018), kur bendro pažinimo ribos nagrinėjamos ne iš tiksliųjų mokslų (taip rašantieji – o tokių dauguma – retai išsilaisvina iš gamtamokslinio mąstymo schemų), o iš meno pusės. Išsamių pavyzdžių, išskyrus vieną (2.2.), nepateikiama. Taigi šis tekstas – tik idėjų daigynas. Bet net ir nedygus kankorėžis visą alk snio (taip, alk snio!) provaizdį savyje jau turi. 1.2. Counterstrategy. And yet: can the interaction of different creative forces be fruitful? The topic is nontrivial: although current scientific documents assign intrinsic value to multi- and interdisciplinarity, the intersection of several of both more often conceals the shortcomings of all the facets5 than opens a new portal. First, what does the paragraph title itself mean? Although it is merely a play, this word precisely reflects the spirit of the article and introduces the fruitful part at the intersection of mathematics, language, and poetry. Structural and mathematical linguistics will be touched upon only briefly in the text (somewhat more so computational linguistics), but it should be emphasized that the attention of the rational element of consciousness to the irrational has been and remains considerably greater in all cases of interaction between art and science than vice versa. Two characteristic quotations. The first is by Zelig S. Harris: “However, the debt of mathematical linguistics to mathematics is chiefly the attitude and the way of thinking rather than any particular result. It is perhaps no accident that most graduate students in this field come from mathematics or logic rather than from linguistics” (Harris 1969: 191).6 The second is by Oleg Grinbaum: «The main reason [for the low intensity of the use of mathematical methods in literary studies], we believe, is the absence, in the arsenal of scholars of versification who operate with the concept of artistic integrity, of such a formal analytical apparatus in which the integrity of verse could be represented and described in the language of mathematics» (Гринбаум 2002: 12).7 ⁵ Extending the ideas of Antanas Baranauskas’s mathematical terminology, and even more so those of Jonas Jablonskis (who, instead of the established term daugianaris, proposed daugmuo): how should each part in the union of several sets be called—jungmuo? And in the intersection of several sets—kertmuo? Here it is as if at the confluence of two rivers, each of which could be named sutekmenis (sutekmuo). Sutekmė (‘confluence’), tekmė, and įsrutis (‘inflow, river mouth’) are unsuitable. ⁶ “It should be said that mathematical linguistics most likely borrowed from mathematics an attitude and a way of thinking, but not any specific result. Therefore, it is unsurprising that most master’s students in this field have completed bachelor’s studies not in linguistics, but in mathematics and logic.” ⁷ “In our opinion, the main reason why mathematical methods are used so little in literary studies is that scholars who examine versification and use the concept of artistic integrity Straipsniai / Articles 229 GIEDRIUS ALKAUSKAS Therefore, those rare cases when the poles exchange places, when the relatively more irrational principle inspires the more rational one, are especially valuable. So what is this game with a title? Three Lithuanian words antskrydis, bergždžias, kontržvalgyba each have a cluster of five consonants. The word kontrstrategija (‘actions intended to eliminate the effect of an opponent’s strategy’), with a cluster of six consonants, used in specialized literature, contains within itself the germ of an algorithmically programmed8 optimization problem: to find all morphologically, phonetically, and practically possible Lithuanian words with the greatest possible value of a function of the sequence of letters constituting the word. Here, a word is identified with the sequence of letters constituting it. Let us denote it by ℓ. Let 𝔇 be the entire hypothetical dictionary of the Lithuanian language. Examples of such functions: 1. F(ℓ) = 0, if the word contains at least one vowel other than a; otherwise, the number of a vowels. 2. G(ℓ) = 0, if the word contains two adjacent vowels or two adjacent consonants; otherwise, the length of the word. 3. H(ℓ) is the length of the longest subsequence consisting solely of vowels. Let us examine the last example. The unusually fresh-sounding street name Paeismilgio in Kėdainiai (a stream Smilgaitis, archaic Eismilga) indicates a possible field of inquiry. Thus, in the word nuokalnė, replacing the stem kalnas with the semantically similar uolą yields nu uolė. This word was also used in the seventh canto of the poem “Šiauriniai reliktai”9. The optimization problem is formulated as follows: is it true that max H(ℓ) = 4? In that case, the set of words sought is H⁻¹(4).10 Clearly, the mathematical (logical) formulas written down merely formalize sentences that are even easier to express in words. Nevertheless, I deeply believe that only regular shifts back and forth between metalanguage, formulas, jargon as one extreme, and poetic language as the other, provide a basis for speaking about a genuine connection between mathematics and poetry. The continual devaluation of old values and revaluation of new ones. And the pendulum continues to swing. 1.3. Baranauskas and mathematics. Although it is gratifying that Jonas Kubilius collected the mathematical passages from Baranauskas’s diaries and letters, because of the violation of the pendulum principle his study Antanas Baranauskas ir matematika (Kubilius 2001) should be assessed with reservations. The book is intended for scholars to become acquainted with the main concepts, but they have not mastered the formal analytical apparatus by means of which the unity of versification can be presented and described in the language of mathematics.” ⁸ An algorithm is what a finite Turing machine can perform. ⁹ For more on the poem, see Alkauskas 2022. ¹⁰ For a function f: A → B and an element b ∈ B, the notation f⁻¹(b) denotes the set of preimages of b: all elements a ∈ A for which f(a) = b. 230 Acta Linguistica Lithuanica LXXXVI Matematika, kalba, poezija: sąveikos ir traukos with mathematical concepts: prime numbers, the sieve of Eratosthenes, the law of asymptotics, powers, cryptography, Archimedes’ constant π, the quadrature of the circle, infinity—and it achieves this goal. However, a reader seeking to learn about Baranauskas will be disappointed, because, while reading, they will find that quite a lot of the text has little to do with him. And another thing. For example, what the author writes about the formula discovered by the Poet, which Karl Hossfeld (Carl Hossfeld) published in 1890 in the renowned journal Zeitschrift für Mathematik und Physik: “[...] that result was important to the amateur Baranauskas. In fact, its value is not great” (Kubilius 2001: 30). The pendulum seems to be stuck at one extreme. And what will a school student think while reading? That Baranauskas is a failure? Another example: Kubilius discusses Baranauskas’s publication “O progresji transcendentalnej oraz o skali i sуłach umysłu ludzkiego,” printed in Warsaw in 1897. In it, the Žemaitian bishop-suffragan recursively constructs, for any a₁ ∈ N, the sequence aₛ₊₁ = aₛᵃˢ, which he calls transcendental. For example, if a₁ = 2, then a₂ = 4, a₃ = 256, while a₄ already has 617 digits. In modern mathematics, such numbers are not exotic. In the theory of Turing machines, there exist functions growing faster than any function describable in symbols, for example, one such as f(n) = aₙ. Kubilius’s subsequent comment (ibid., 84): “[Baranauskas’s work] examines the properties of [these] progressions and presents the ideas about infinity from the letters already cited by us. However, the main part of the work is devoted to questions of philosophy and theology.” The essential questions—precisely those questions—are omitted. It is good that two literary figures mention them: Sigitas Geda: “[Baranauskas’s] titanic nature is attested by yet another—utopian—idea. I said that a truly great poet must have utopian ideas. To calculate the volume of hell!”; Saulius Šaltenis: “by calculating it, you seem to drive all the evil in the world into a mathematical formula as into a trap. You ensnare Satan like a horrifying beetle in a piece of amber and impose God’s order!” (Geda 1998: 155–156). After all, the pendulum’s orbit forms a continuum of equivalent points. Any one of them is most likely highly unrepresentative11. Quoting Geda once again: “There is no essential difference between higher mathematics and poetry or music. This cosmic plan of Baranauskas testifies that everything he did in “The Forest of Anykščiai” burst forth in other forms in another activity” (ibid., 156). 11 This is not a universal law. For example, in the theory of functions of a complex variable, all local information (about the neighborhood of a single point) concerning an analytic function determines it globally as well. Straipsniai / Articles 231 GIEDRIUS ALKAUSKAS Very little—only half a page—is devoted in Kubilius’s book to Baranauskas as a creator of Lithuanian mathematical terminology. One would, of course, like more: accents, intonations, linguists’ comments, excerpts from the correspondence with Hugo Weber. Kubilius, who himself initiated the creation of the dictionary of mathematical terminology (MTŽ), should be regarded as a continuator of Baranauskas’s work. Vidmantas Pekarskas has written a fine overview of the history of these terms (Pekarskas 2003). 1.4. Poetry and visual art. According to Benediktas Januševičius, “visual poetry is a verbal text to which a new, additional dimension is imparted by a visual arrangement uncharacteristic of literature” (RP 17). But it seems that this addition is granted only on a few pages of the book Raidžių paveikslai. Some works are merely texts, formed not as meaningful strophoids but in the shapes of a rat, a horseshoe, and so on (why?! who does not know what some mill looks like?). Others, by contrast, are merely drawings made of graphemes. Yet, according to the editor, “a book of poetry is not a newspaper or an information publication; it has the right to be meaningless or misunderstood.” Misunderstood? Understandable! But meaningless? If there is no meaning, then is everything permissible? Like a vulgarized thought of Fyodor Dostoevsky. Yet not entirely vulgarized, because there is some connection between faith, meaning, and poetry, as the following quotation from T. S. Eliot shows: “For the knowledgeable poet, an allegory differs in no way from a pure visual image. And pure visual images become much stronger when meaningful—we do not need to know what that meaning is, but in becoming conscious of the image, we must know that it also has meaning” (Eliotas 1927: 397). However, Januševičius’s formula “graphic poetry is a poem pretending to be a drawing” is appealing because it is generally suitable as a definition of bidisciplinarity: by emphasizing the quality of the pure substance, it invites us to test the artifact (a sturdy pot is made only from good clay). “Theology pretending to be physics”—this applies to some of the articles in Česlovas Kavaliauskas’s work Teologijos žodynas (Kavaliauskas 1992). Admittedly, playing the character of poetry wearing a graphic mask is also the most difficult. There are two reasons for this: physical and physiological. Literature is an art of nonhomogeneous time, whereas graphic art is an art of homogeneous space. The distortion of space is possible either on very large scales or in illusory works. Film and music fit into homogeneous time¹². This is, of course, relative, but the orthogonality of graphic art and poetry is obvious. At the same time, it is extremely difficult for one ¹² The distinction between literature as an art of the past (although only just elapsed) tense and music as an art necessarily of the present tense, although often mentioned, is not entirely convincing, especially when considering the interval of artistic expression–meaning (i.e., syntax–semantics) as a continuum. But Rimvydas Stankevičius’s (Stankevičius 2015) idea that the past frequentative tense particularly favored by the poet Antanas Kalinauskas (applied to verbal derivatives formed with the suffix –inė­ti from verbs, for example, pareidinėju, užsegdinėjo) is an effort to see every action as occurring now, which possibly points to a synthesis of literature and music, is extraordinarily interesting. 232 Acta Linguistica Lithuanica LXXXVI Matematika, kalba, poezija: sąveikos ir traukos to transform one of these dimensions into another. Second, reactions known in brain science, called ERP (event-related potential) (HNL). Their studies have clarified how long it takes to perceive a word through the senses and to fit a word to its context. The moment when access to semantic information occurs is known. And everything happens—subconsciously. Without grasping the meaning, it is further refined at the level of consciousness. In the case of the interaction between graphic art and poetry, conscious coordination or a concept of meaninglessness is simply needed too often. Such activity overstimulates the brain. Thus, graphic poetry is a broad concept encompassing many techniques and ideas. Donaldas Apanavičius’s abstract (and mathematical) poems in this book are among the strongest. Here we find time, disappearance (RP 80–81), cryptolanguage, Kabbalah (ibid., 79), genesis ab ovo (ibid., 78). Particularly powerful are three reconstructions of OM (ibid., 75–77), in which language, mathematics, mandala, limit, phoneme, symbol, and mystery are intermingled; the subconscious reacts strongly to this (Fig. 1). FIG. 1. D. Apanavičius’s triptych The book also contains nominal mathematics: Rimas Vėžys mentions Euclid’s fifth postulate (see 4.1) (RP 23), while Vilma Fiokla Kiurė uses matrices (ibid., 51) (unfortunately, everything in this composition is in disarray; even the logical error13 does more harm than good). Among the good ideas, Mato Gimžauskas’s playful “Suahili dada” deserves mention (ibid., 86), in which “Tautiška giesmė,” written as though in pseudo-Swahili syllabograms, correlates strongly both with the topics of 4.2 (framing of memory) and with the aforementioned fourth part of this cycle of articles. Or S. Geda’s “A Drawing of Solitude during Joninės” (ibid., 73), which, it seems, has not yet been thoroughly examined by critics14. A brief and highly subjective analysis of this graphic text follows (Fig. 2). The mention of Joninės provides a firm point of reference: the solar calendar. The number of dots on the sides of the square varies from 1 to 7, so these piles divide the perimeter into 24 parts. Thus we have the phases of the moon, where an odd number, like the year 13 A=̅MATRIX AND 4R, when it should be 3R. 14 Although the poem itself has been mentioned in passing more than once (for example, Valentas 1997: 149), as has Gytis Norvila’s interview, which intuitively mentions the cosmos here (see https://www.bernardinai.lt/2012-02-22-gytis-norvilas-eilerastis-prieglobscio-zona/). Straipsniai / Articles 233 GIEDRIUS ALKAUSKAS and the beginning of the month (while young, while alone) marks the new moon. The drawing presented is the most popular former Sun–Moon calendar among various cultures. The Chinese, Hebrews, and Indians still use such calendars today. The ancient Scandinavians used one as well, for here is stanza 23 of “The Lay of Vafthrudnir”: Mundilfiori is the father of the Moon, the Sun his daughter. They are destined to travel across the sky every day—and thus we measure time (PE 101). Geda attached particular importance to Scandinavian mythology (as well as to Sumerian, Indian, Hebrew, and Greek mythology), and even more so to cosmoperception. In the cycle “Songs of Seven Summers” alone, Odin, Thor, and Loki are mentioned. Further, Jehovah’s order—to complete the creation of the world, the seventh Day of God, the balance of forces—is represented by the spring and autumn equinoxes, seven dice on the left and right. There the structure of the world is most stable, and change is slowest. The entire mass of the world, even visually, is concentrated around these stationary points, while the solstices are ruptures, transformations, upheavals, singularities. The upper unit (solitude, singularis) is Midsummer, the lower one—Christmas. In God’s calendar, the first day is the emergence of order from chaos. The inversion of chaos is Perfection. Order at the limit of the possible, at the abyss (Gr. χάος). Order inevitably becoming increasingly like chaos15. And one more thing: before the calendar system had been standardized throughout the planet, the winter solstice in the Southern Hemisphere coincided with the summer solstice in the Northern Hemisphere. But what need is there for those words—the entire idea is conveyed by a single graphic poem by Geda. FIG. 2. S. Geda’s “Drawing of Solitude at Midsummer” 15 The creators of structuralist, precisely mathematical music unexpectedly noticed that such compositions differ little from those created according to aleatory principles. 234 Acta Linguistica Lithuanica LXXXVI Matematika, kalba, poezija: sąveikos ir traukos 1.5. Notation and conventions. In mathematics, a number is an element belonging to a certain standard structure. Since no other kinds are considered (except in 4.1–4.3), number here means either a natural number from the set ℕ = {1, 2, 3, 4,...}, or an integer from the set ℤ = ℕ ∪ {0} ∪ −ℕ. The meaning of structure is precise, but only the concept of a group, to be defined later, will be needed. When M, N ∈ ℕ, the function M mod N (read as ‘M modulo N’) gives the unique integer L, 0 ≤ L ≤ N − 1, such that M − L is divisible by N without remainder. For example, 20 mod 7 = 6. Regarding versification, an uppercase letter will denote masculine, a lowercase letter—feminine, a lowercase letter emboldened and italicized—a dactylic rhyme (when the third syllable from the end is stressed). The index of the letter will indicate the number of feet (bisyllabic or trisyllabic) in the line. If all the lines of a stanza have the same number, one simply writes 4-iamb, 3-amphibrach, and so on. 2. POSMAI IR GRANDINĖS Modern languages tend to differentiate abstract thought (the only universal language remaining today is mathematics), but the universal properties of Latin are characteristic of Dante’s Florentine speech. Thomas Stearns Eliot 2.1. Versification and language. Part of the art–science of versification consists in adapting mathematics (in the broad sense) to a particular language, if we regard rhythm, phonics, instrumentation, form, and rhyme as manifestations of structure, symmetry, balance, and analogy16. Here are a couple of examples. In the classical syncopated line, intervals of 0, 1, 2, or 4 unstressed syllables alternate. Juozas Girdzijauskas studied the latter’s structure and established the following facts:17 “In Lithuanian poetry, two syncopation groups have emerged, with six18 principal meters and eighteen more detailed variants. [...] In almost all syncopation variants (except one), a bisyllabic interval of unstressed syllables predominates (from 51 percent to 75 percent). [...] This is also because its dominance 16 For more on poetic terms, see Matulaitienė 1997. 17 The term was introduced by Kęstutis Nastopka. 18 In type V and type VI syncopation, unstressed intervals of 3 and 5 syllables are permitted (Girdzijauskas 1978: 220–230). Straipsniai / Articles 235 GIEDRIUS ALKAUSKAS corresponds to the norms of standard Lithuanian—it also prevails in prose” (Girdžiauskas 1978: 225). A second example: the strange five-syllable iamb-dactyl19 metre is particularly characteristic of twentieth-century Lithuanian poetry. Bearing in mind this dependence of versification on language, new fixed forms20 pentastichs and their weaving methods will be examined below. The outlined construction, although largely theoretical, implies a nontrivial mathematical problem as well as several practically applicable, attractive forms of versification. It is hardly surprising that the greatest variety of syllabic-accentual versification was found in the work of precisely those poets who deeply explored the cultures of different traditions. Among them were Jonas Mačys-Kėkštas21, Edmundas Steponaitis22, Maironis, Vytautas Mačernis, S. Geda, and Tomas Venclova. Therefore, two octaves—Kėkštas’s and Alexander Pushkin’s—will serve as a suitable introduction to the main starting points of this chapter. The former ridicules the nationalists who discerned an almost Lithuanian dominance even in ancient Greece (“Homer’as esąs Omyras, Odisea – Adišius”), as well as the ordinary versifiers of Aušra (Mačys-Kėkštas 1910: 43). Yet Kėkštas’s stanza is strikingly similar to Pushkin’s: ...To tell the truth, it will be difficult for me to give up writing octaves: Here you arrange the lines in an even, smooth row, Like soldiers, wherever it is fitting and convenient; You drive your thoughts in, one after another, Without much order or rules, At the end, having stuck a sting in someone, You say: that is enough of war for now. The iambic tetrameter is becoming tiresome to me: Everyone writes in it. It is time to leave it to the boys for amusement. I wanted Long ago to take up the octave. And indeed: I could manage The triple rhyme. I shall set off gloriously! After all, rhymes live with me easily; Two come of their own accord, and bring the third. J. Mačys-Kėkštas, “Having Gained Freedom” (1902) A. C. Пушкинъ «Домикъ въ Коломнѣ» (1830) Form without imagery, vocabulary, and a Milašian “landscape of the soul” in itself still means nothing, but in exceptional cases formal inventions already carry an authorial charge. This is what happened with Onegin’s stanza (iambic tetrameter aBaB ccDD eFFe GG, which was also used by 19 The term was introduced by Juozas Girdžiauskas. 20 An allusion to the formes fixes: ballade, virelai, rondeaux of fourteenth- and fifteenth-century French poetry and music. 21 One of the first in Lithuania, together with Pranas Vaičaitis, to adopt exclusively syllabic-accentual versification (Girdžiauskas 2001). 22 Already the first stanza of the debut poem “The Dawn of Lithuania” (Steponaitis 1988: 24) has a nontrivial form a₅b₆b₆c₅a₅c₄ (iamb; the third is similar, while the second stanza is somewhat different). 236 Acta Linguistica Lithuanica LXXXVI Matematika, kalba, poezija: sąveikos ir traukos Jurgis Baltrušaitis’s work “Dust and Stars23), written in Lithuanian in 12 stanzas, Mikhail Lermontov’s «Бородино» (iamb a₄a₄B₃c₄c₄c₄B₃) and the poem «Сашка» (5-iamb AbAbAbccDDee), Edgar Allan Poe’s “Raven” (trochee a₄a₄.B₈.c₄c₄.c₄B₄.B₈.B₄; three lines have internal rhymes, and are therefore split in half here). As if authorial content without specific content. Herein lies the similarity with mathematics: the latter exists in a Platonic reality, while all practical reincarnations are merely imperfect models of the ideal. In search of such formal pre-poetry, quintains and septains will be examined below. 2.2. Weaving tercets. For convenience, rhymes in this paragraph will be denoted by numbers rather than letters. For example, “the scheme of a septain is 3142342.” If two stanzas have the same meter, with only the rhyme shifted, they are called stanzas of identical formula. Repeating the septain written above four times, but cyclically shifted by one, two, and, respectively, three rhymes, yields the stanzas 3142342 4253453 5364564 6475675. Rhyme 4 occurs here two, two, two, and one time. Tercets (terza rima) are a weave of tercets 121 232 323 and so on. This scheme was first used in Dante’s “Divine Comedy,” written in hendecasyllabic24 iambs. Reverse tercets are also possible, perhaps never yet used. Namely, 212 323 434 and so on. Since each of Dante’s cantos ends with one line, the so-called rime rilevate (in the 10th canto of “Inferno,” these would be rhymes 45–46–45 46), reverse tercets are obtained by reading each canto from the end. For more on the subtle architecture of Dante’s poem, see Hart 1987. The word weaving intuitively fits here. After all, a lock of hair is usually divided into three parts, and the subsequent formation of a braid resembles the creation of a tercet-based canto both symbolically and topologically. Moreover, Skirmantas Valentas (Valentas 1997: 19), continuing Rüdiger Schmitt’s research, supplements the reconstructed ancient Indo-European metaphor “to weave, braid” = “to create poetry” with Baltic-language facts, for example, Lithuanian pinti ‘to speak in a twisted, intricate manner’, austi ‘to chatter’ (LKŽe). If in Dante’s case the metaphor’s aptness is obvious, why did poetry also resemble weaving in antiquity? Valentas mentions the synonymy of dainius and poetas, but does not fully bring to light the commonality that explains everything. Namely, music. The alternating singing and recitation of choruses in Ancient Greek theaters already itself resembles weaving: the pitches of the voices change places like warp threads, while dynamics (louder – quieter) create a sense of changing distance (closer – farther). This claim may be supported by the meter of Scandinavian poetry, drotkvaett (dróttkvætt), which is “the only one of the skaldic meters characterized by sentences interwoven with one another” (Ruseckienė, trans. 2017: 17). This means that only when certain phrases in the stanza are underlined does their sequence, like that of the remaining phrases, become meaningful. Such a metrical construction is explained 23 Indeed, all of Baltrušaitis’s rhymes, except G, are feminine. 24 11 syllables. Straipsniai / Articles 237 GIEDRIUS ALKAUSKAS through the merging of the parts of two choruses, which over time were reduced to a single vocal line. Recalling the formes fixes mentioned in 2.1, as well as the synonymy of canto and poem up to the nineteenth century, the hypothesis of the musical origin of weaving as a euphemism for poetry gains support. In conclusion, the mathematical unravelling of this fabric. In topology, the so-called braid groups (in English, braid groups; for groups, see 3.1) are particularly important. A graphical example (Fig. 3) explains much without words. In this case, b₁ · b₂ = b₃ FIG. 3. Example of the product of braid-group elements 2.2. Generalization of tercets. The Dantean form has a cyclic symmetry, which can be described mathematically as follows. Extending the weave to −∞ and +∞, one obtains the sequence .... −2, −1, −2, −1, 0, −1, 0, 1, 0, 1̄, 2, 1, 2, 3, 2, 3, 4, 3, 4, 5,.... This sequence, together with two additional parameters, determines the form of the lines: the position at which the weave begins (marked by a dash), and the length of the stanza. In this case, the number is 3. And what is a sequence? It is any function f: ℤ → ℤ. The weave begins at the term f(1). If we imagined that Dante’s work began to be read from the second stanza, this would correspond to the sequence f(n + 3). The form of the new poem would remain identical to the old one, with only the rhymes shifted by one. Then the aforementioned cyclic symmetry is the equality f(n + 3) = f(n) + 1. There are many such sequences. For example, {…, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, …}, but this sequence is not interwoven. Let us examine another sequence {…, 1̄, 5, 1, 2, 6, 2, 3, 7, 3, 4, 8, 4, 5, 9, 5, …}. This one has the required cyclicity, but the gap between identical rhymes circled by a rectangle is too large. A construction such as the following emerges from a systematic search for the core of the tercet structure. Definition. Let Q ∈ ℕ. We shall call the function f: ℤ → ℤ a simple Q-tine if it satisfies the following four conditions: 1. f(n + Q) = f(n) + 1, for every n ∈ ℤ; 2. f(ℕ) = ℕ. Let f⁻¹(1) = {p₁, p₂, …, pQ}²⁵. For all r ∈ ℕ such that both indices at p belong to the set {1, 2, …, Q}, the inequalities are valid 3. pᵣ₊₁ − pᵣ < Q; 4. pᵣ₊₂ − pᵣ > Q. ²⁵ The fact that the set of preimages of each number consists of Q elements follows from condition 1. 238 Acta Linguistica Lithuanica LXXXVI Matematika, kalba, poezija: sąveikos ir traukos The simple Q-tuple that satisfies the additional condition 5. The quantity pQ − p1 is the smallest possible among all simple Q-tuples; we shall call it the minimal Q-tuple. The simple Q-tuple that satisfies another additional condition 6. | f⁻¹(1) ∩ ℕ | ≥ 226, we shall call the normalized Q-tuple. If a Q-tuple is both minimal and normalized, we shall call it standard. It follows from condition 4 that for every simple Q-tuple f and every natural number ℓ, the set f⁻¹(1) ∩ [−ℓQ + 1, −ℓQ + Q]27 has 0, 1, or 2 elements. Let T(f) be the largest number ℓ for which this set has exactly one element. The standard Q-tuple that satisfies the last additional condition 7. The quantity T(f) is the largest possible among all standard Q-tuples; we shall call it the canonical Q-tuple. If these conditions seem vague, the examples will clarify how they translate into the language of versification. Condition 1 has already been discussed. Condition 2 is needed to ensure that only positive rhymes, as well as rhyme 1, occur in the first stanza. Condition 3 requires that any two adjacent rhymes be separated by less than the length of a stanza. Condition 4 says that in any sequence of Q lines (for example, in one stanza), each rhyme is repeated no more than twice. When Q = 5, the first four conditions are satisfied by as many as 25 different simple quintuplets. Let us take two of them: 32132 and 32312. Now let us write three stanzas representing these schemes: In the left-hand example, the distance between the first and last occurrence of rhyme 3 is 12 lines, whereas in the right-hand example it is 13. Examining all 25 cases shows that 12 is the smallest possible distance. The calculations show that condition 5 is satisfied by five minimal quintuplets: 32132, 13213, 21321, 13243, 21324. On the other hand, condition 6 is satisfied by nine normalized quintuplets: 31231, 23121, 31213, 13231, 13213, 21321, 12341, 12314, 12134. 26 The symbol |𝒜| denotes the number of elements of the finite set 𝒜, while ∩ denotes the intersection of sets. 27 Square brackets [a, b] denote a closed interval. Straipsniai / Articles 239 GIEDRIUS ALKAUSKAS This condition requires that rhyme 1 in the first stanza (where it will occur only there) be represented by at least 2 lines; otherwise, one line in the poem would remain unrhymed. Two of the written sets of five and, respectively, nine elements have two common members. These are the standard quintinas: 21321 and 13213. For the first of them 𝒯(f) = 2, and for the second 𝒯(f) = 1. Thus the canonical quintina is unique: 21321. At the level of line arrangement, condition 7 means that the line that has no rhyme in the first stanza corresponds to the greatest possible rhyme. Obviously, 21321 (rhyme 3 has no pair) is a more natural form with regard to the flow of time than 13213 (rhyme 2 has no pair). With the help of mathematical reasoning, a more general result is also proved. Statement. For every odd Q = 2q − 1≥ 3, there exists a unique canonical Q-tina (q − 1) (q − 2) … 1q (q − 1) (q − 2) … 1. As mentioned, there are 25 simple quintinas. A computer program counts by exhaustive enumeration that there are 96 sextinas and as many as 427 septinas. Let the number b(Q) generally denote the number of simple Q-tinas. An interesting sequence is obtained 1, 2, 3, 8, 25, 96, 427, 2176, 12465, 79360, …. What is the pattern here? In such cases, one must check an online sequence database. The discovered sequence has already been recorded there; see OEIS A065619. Clearly, it was defined in an entirely different way, with no connection to terzinas. This, incidentally, is one of the fundamental principles of combinatorics: count in two ways. This very often makes it possible to discern the structure of an object. The exponential generating function of the sequence is also known: ∑ᵠ∞₍Q=1₎ b(Q) · xᑫ / Q! = x(1 + sin x) / cos x (here Q! = 1 · 2 · … · Q). Such equalities no longer surprise representatives of the physical sciences, because an abundance of miracles dulls the senses. A pity, though natural. Other questions arising here are: how many minimal; normalized; standard Q-like ones are there? And what is different if Q is an even number? In this case, the situation is more complicated. Thus, a construction inspired by poetry implies a mathematical problem involving a nontrivial structure. It may be left to pure mathematics. In the context of poetry, something else is more interesting: no specific language (for example, Lithuanian) figures in the definition of canonical Q-tinas. Once one begins arranging the lines, it immediately becomes clear that the quintina 21321 (which is developed further in the article) contains too little internal structure. A truly formal author must discuss both meter and language. Dante fixes the number three: 3 parts of the poem; 3 lines, 33 syllables per stanza; 33 cantos in each part. The Trinity, the tree of the three worlds, the triad, the trifunctional Indo-European pantheon²⁸. Meanwhile, five is the Pythagorean hugieia (ὑγίεια) ²⁸ The theory of the linguist and scholar of religion Georges Dumézil. 240 Acta Linguistica Lituanica LXXXVI Matematika, kalba, poezija: sąveikos ir traukos the pentagram and the Pythagorean number²⁹ 5 = 2 + 3, which is also equal to the hypotenuse of the smallest Pythagorean triple (3, 4, 5)³⁰. Thus, mathematics not only reveals the structure of the canonical quintina, but also acts upon it through symbols. 2.3. Pentastichs with triple rhyme. In Dantean tercets, ⌊3/2⌋=2 lines are rhymed out of three. To rhyme the same number in a five-line stanza, ⌊5/2⌋=3 lines³¹ must be selected. If the remaining two rhymes are identical, ten variants are obtained combinatorially: 22111, 21211, 21121, 21112, 12211, 12121, 12112, 11221, 11212, 11122. Four examples from Lithuanian poetry, corresponding to the formula 12112³², are presented below from different schemes. We, alive, break through the heavy [burden] in caves; Not days, but centuries fall upon us with hardships! But the world encircles us with darkness in vain, We shall swiftly rise to the heights on different winds— Already the sun shines brighter! I will sing out, it seems, the sorrow With an intoxicating song! How the lonely birch grew green— How the little wind raised the wave— How my beloved visited—– Juozas Mačys–Kėkštas “Bread and Toys” (1902) Amphibrach a₄b₄a₄a₄b₃ Balys Sruoga “Along the Paths of the Gods” XVI (1923) 4-chorēas aBaaB Tear the white sail from the mast, We shall sink the rudder into the depths! Only by gathering hot hearts together, We shall passionately steer the ship through the stormy sea To the shore. Vincas Mykolaitis–Putinas “The Fisherman’s Song” (1920) Iambodactyl a₄b₃a₄a₄b₃ We were blessed with sharp swords, our castles were healed with fire, black lilies blossomed in ponds, fearsome beasts pursued us roaring, perhaps for a week, perhaps for several ages. Aivaras Veiknys “Prieglobstis” (2014) 3-anapest aBaaB It would also be possible to add—and all the schemes are different!—A. Veiknys’s poems “Trys paveikslai apie vienatvę” (2021, 4-iamb AbAAb) and “Durys” (2014, 5-iamb aBaaB), Bernardo Brazdžionis’s “Tu man buvai” (1944, 4-iamb ²⁹ Pythagorean tuning of musical instruments is based on the interval of a pure fifth, that is, on the ratio of the frequencies of the fundamental tones ³⁄₂. According to the Encyclopaedia Britannica, the number 5 for the Pythagoreans is also the all-significant union of the feminine (2) and masculine (3) principles. ³⁰ A triple of natural numbers (a, b, c) is called Pythagorean if a² + b² = c². According to the Pythagorean theorem, segments of such lengths form a right triangle. ³¹ The brackets ⌊x⌋ denote the integer part of the real number x (in English, the floor function, hence the visual similarity of the symbol to a floor). ³² The five-line stanza 12112 is also the most popular form in the Spanish genre Quintilla of the 15th–16th centuries. Straipsniai / Articles 241 GIEDRIUS ALKAUSKAS Kornelijus Platelis’s “Moon Garden” (1984, 5-amphibrach aBaaB), V. Mykolaitis-Putinas’s “Hymns of Pessimism” V (1925, 4-amphibrach aBaaB), Albinas Bernotas’s “After a Long Silence” (1978, 4-amphibrach aBaaB), V. Mačernis’s “Man” (1943, iamb a₇B₇a₈a₅B₅) and “***” (dactyl a₄B₄a₄a₄B₃; in the poem, three lines have dactylic anacruses—one unstressed syllable at the beginning), and Juozas Erlickas’s “Ballad about a Man with a Pitchfork” (1987, the first stanza—iamb A₄b₃A₄A₄b₃; in the others, the number of feet varies somewhat, while maintaining the 12112 scheme). All these stanzas contain pauses: catalectic lines (one syllable is missing), or lines shortened by a foot. Extrapolating this to quintains, one may assert: the natural breathing of the text in such a five-line stanza requires pauses. The alternative of catalectic lines is ruled out by over-rhyming: it is difficult to imagine a natural flow of the text when each rhyme is matched by both 3 masculine and 2 feminine cadences (or vice versa). Only the second alternative remains. There is, indeed, one exception. Pure hendecasyllabic iamb also suits quintains. Here each line ends with a pause that completes it to six feet. One pause is short, but the anacrusis of the other line, acting as an upbeat, seemingly extends the pause to two syllables. And indeed, the first discovered poem with the 12112 rhyme scheme in which both the number of syllables in the lines and the types of their cadences coincide is Aidas Marčėnas’s “From the Outskirts to the Center” (2000, 5-iamb abaab). In the context of this section, it is simply necessary to quote the first stanza: Suiręs Dante, dantenos kraujuotos, senukas ten, kur vakar būta ber­no, peizažas iš fragmentų suklijuotas – dangus ir žemė, moteris ir luotas, šviesa – lyg čia prasideda inferno (ŠLPCh 142). Iambic tetrameter is less suitable, because, if the pause were counted, each line would consist of five feet, and the stanza of 25 feet. The oddness of this number is what determines the sudden turns in the flow of the text. In this way, three forms suitable for quintains—trochaic, dactylic, and iambic—emerge: As mentioned, the interval of unstressed syllables consisting of two syllables is most characteristic of the Lithuanian language. It is precisely for this reason that the dactylic example is used in this article 242 Acta Linguistica Lithuanica LXXXVI Matematika, kalba, poezija: sąveikos ir traukos the beginning canto “The First Battle without Ąsgard” of the author’s poem “Northern Relics.” The first two stanzas: Splits the deepening expanse the Cranes’ ploughshare... To the one who has won the sowing, Nothing is given. Only the Eternal Circle is given. The cries northward turn to stone, Crumble the frost, Anxiety trembles on the surface, The All-Opener opens A blue-hued Eye. 2.4. Septinas. Canonical septinas are already less practical. For, first, each rhyme would be repeated seven times, which threatens serious monorhyme. Second, the rhymes would shift cyclically only after seven lines. When examining various 7-cyclic sequences, it would be more practical to take every second stanza. Thus, several exceptionally attractive septinas are found. For example, this one, which will also be used in the second poem of the tetralogy (see Alkauskas 2022): 1st stanza 3 1 4 2 3 4 2 2nd stanza 5 3 6 4 5 6 4 3rd stanza 7 5 8 6 7 8 6. The odd rhymes are repeated three times with intervals of 3, 3, while the even ones are repeated four times with corresponding intervals of 2, 4, 2. 3. GROUPS Mathematics, alongside language and music, is one of the primary expressions of the free creative power of the human mind. Hermann Weyl 3.1. The structure of a group and metatheses. A binary operation is a kind of magic box with doors on the left, right, and top. After placing an element on each side, the box operates and generates something that can already be removed through the top. Mathematically, this is written as a □ b = c. For simplicity, the box is denoted not by the symbol “□” but by the standard “·”. A group (Γ, ·) is a set with a binary operation if the pair satisfies the following axioms: 1) Neutral element (identity): there exists an e ∈ Γ such that a · e = e · a = a for every a ∈ Γ; 2) Inverse element: for every a there exists a b such that a · b = e; Straipsniai / Articles 243 GIEDRIUS ALKAUSKAS 3) Associativity: a · (b · c) = (a · b) · c. Based on these axioms, it is proved that the identity element is unique, as is the inverse of each element. The latter is then denoted by a⁻¹. A binary operation may also satisfy the equality a · b = b · a. Such a group is called commutative. In the real world, this is atypical: after all, it is not the same whether one puts on a sock first or puts on a shoe. Nor is it the same what happens first: the bus stopping or the passenger jumping out of it³³. In a group, after the identity, the simplest are elements of order two (if they exist), for which a · a = e. Such elements are called reflections or involutions. The theme of mirror reflection in linguistics is certainly not new. Metathesis can also be interpreted in this way (Lith. kepti // Slav. nekмu), as can Algirdas Patackas’s developed law of transformation (Patackas 2014: 85), describing the semantic and symbolic connections between words: plat-us // talp-us; gyl-is // lyg-is³⁴ and so on. Despite the well-founded criticism from linguists, the latter phenomenon should not be dismissed as entirely unscientific (Saudargas 2014: 100). All the more so because, according to Dainius Razauskas, “In one respect, the works of A. Patackas and A. Žarskus remain relevant. Precisely in that they reveal an attempt to identify the linguistic phenomenon examined by S. Valentas at the highest philological level [in the book Mė(lynojo)nulio lingvistika] not in authorial poetry, but in the Lithuanian language itself” (Razauskas 2006: 59). The theme of mirror reflection can be extended, but according to an entirely different principle. Let us call this reverse text design (ATD). This is a form of creativity in which, because of predetermined rules, the means of constructing the text are limited, leaving complete freedom for its interpretation and for giving it meaning through context, allusions, and manipulations. Original examples are presented below. The aforementioned predetermined rule becomes clear after just a few of them. 3.2. Text. Metathetic fables—metamorphic silences, or linguistic autopsies on the theme of two-member words x(e) (ė) (ie)-l (ė) (ia) (iai) (e) *** About frosts and the inevitable otherworldliness of November VYSTA GĖLĖS – GYSTA VĖLĖS. *** About the life of the Ainu, growing clearer in the month of March, by the Išikaris River 33 In essence, this concerns the composition of functions, which is already associative by definition; however, its commutativity is an exceptionally rare property. 34 Particularly interesting mathematical cases: length – volume, vertical – horizontal. 244 Acta Linguistica Lithuanica LXXXVI Matematika, kalba, poezija: sąveikos ir traukos SAKUROS VĖLELĖ. VAKARO SĖLIAI. *** A novel about unhappy love for solo cello GĖLĖS RĖŽIA – ČĖLĖ GRIEŽIA. *** A fable about undying hope and whipped cream VIENOS PĖLĖS PIENĄ VELIA. *** The paradise of the Ojibwe living by the Great Lakes VALGO ŽELĖ ŽVALGO VĖLĖ. *** About boundless curiosity and toy robots in childhood RIEKIA – LĖLĖ. LIEKA RĖLĖ. *** The myth of Sovijus and the burial of the dead in water DĖLĖS VĖLUOJA. VĖLĖS DVĖLUOJA. *** As on earth, so in heaven (a story about the queen of birds, like a heavenly Marie Antoinette, who during a famine retorted to those asking her for food: “No bread? Eat cake!”) KAISTA MIELĖS – MAISTAS KIELEI. *** The movement of the Latgalians toward Deltuva and Nalšia in the 7th century SĖLINA VĖLEI. VĖLINA SĖLIAI. *** The Savior’s sermon about the loss of earthly nature and the heavenly nature thereby gained (Lk 9, 24, merely retold at the Skapiškis market) VIELŌS CELĖS – ČIELOS VĖLĖS. *** A story about an ancient Sedukian who, while digging and planting, discovered a brick fragment from the Second Temple of Jerusalem ŽABALAS KĖLĖ. GABALAS ŽĖLĖ. KABALA GĖLĖ. *** About the primeval community of gatherers and hunters (in which hunters, using traps, also become gatherers) PELĖS STRINGA – STRELĖS PINGA. *** A story about Attila the Hun, who won Ildikė’s hand without any chivalry GREITA STRĖLĖ – RAITA GĖLĖ. *** Straipsniai / Articles 245 GIEDRIUS ALKAUSKAS What is the insignificant but most essential part of a large object? DRELĖS – RATAS, RELĖS – DRATAS. *** Everything here is logical. Physicists will confirm that the essential mechanism of a drill (n. borrowed drelė) is a screw, while that of a relay is an electromagnet consisting of coils of wire (old. borrowed drato) and a ferromagnet. Other schemes for applying ATD can also be devised. Let us take characteristic phrases and proverbs, but transform them into oxymorons or puns. For example, you can’t escape anywhere – you’ll have to escape from here; it’s not burning yet to burn down the bridge; we’ll see whether we’ll see that; so I remember it from time immemorial; this cult is no longer cultivated; no one is getting ready to undress and so on. The most interesting thing, of course, is to interpret such a one-liner. But the idea that the semantic charge of a poem’s title can be decisive is certainly not new – some poems in S. Geda’s book Skrynelė dvasios pagauti (Geda 1994) can be deciphered only a posteriori by rereading the title. 3.3. Synonymy and roots from the unit. First, a definition: “True synonyms are words of similar or identical meaning that denote the same concept and are used to mark various semantic nuances or stylistic features. [...] Due to polysemy, a word may enter into synonymous relations with some words through certain meanings and with others through different meanings” (Lyberis 1981: 3). Let us now imagine that the words α₁ and α₂ are true synonyms. Ideographic, generic, emotional, derivational—all are suitable. Both those recorded in a dictionary and those newly constructed, provided that linguistic intuition does not contradict the synonymy assigned to them. Let us denote this relation by α₁~α₂. Now let each adjacent pair in the sequence α₁, α₂, α₃,..., αₙ be synonyms (in different pairs, the same word may be used homonymically). If we were to look for examples in which α₁~αₙ, we could hardly avoid a single cluster of synonyms. But the additional requirement ∃ p, 1 < p < n: α₁ ≁ αₚ already creates the necessary tension. Another question: when does α₁~−αₙ? In other words, can a chain of synonyms lead to an antonym? This question is where poetry lies. Precisely in the relation between the relative (metaphor) and the absolute. Quoting Lyberis again: “Winter, having covered it with white sheets, will bite your green bud (Maironis). Here the word sheets acquires the meaning ‘snow’ only in context and is a vivid metaphor” (ibid.). Indeed, one of the types of metaphor is contextualization, which lends different words or phrases features of synonyms. But contextualization can be replaced by the rhythm of an algorithm that leads the word α₁ through a chain of synonyms to the word αₙ. Thus, this chain acquires features of metaphor. The creation of pseudo-examples of synonyms that do not conflict with linguistic intuition can also be good poetry. 246 Acta Linguistica Lithuanica LXXXVI Matematika, kalba, poezija: sąveikos ir traukos Let n ∈ ℕ. A complex number α is called an nth-degree root of minus one if αⁿ = −1. For example, i² = −1, (¹⁄₂ + ³⁄₂i)³ = −1³⁵, the chain α₁~−αₙ is a linguistic analogue of a root of minus one. For example: heat, fever, ague, illness, chill, cooling, cold. The word džiuova is used here homonymically—as both ‘drought’ and ‘tuberculosis’. In Lyberis’s formulation, a synonym is a shade. Another color with double the frequency of the wave may also be a synonym of an optical color. Let us suppose that violet (800 terahertz) is synonymous with marginal red light (400 terahertz). If this is not sufficiently convincing in optics (both frequencies lie at the margins of visible light), acoustics helps, since a doubled frequency corresponds to a clearly synonymous (of the same name) interval of a pure octave (for example, the sol of the first and second octaves). Therefore, in the case of optics and acoustics, the antonym would be exactly halfway between them, where the ratio of wave frequencies is √2 ≈ 1.4142. Yet it is obvious that one can slowly move from one to the other, shifting exclusively through shades and echoes³⁶. In the context of language, such cases are singular. That makes them all the more interesting. The example of heat – cold given above may not be particularly interesting (nor was anything better sought), but the subject itself touches something deep in the linguistic network. All that remains is to keep winding the line³⁷. 4. MATHEMATICAL APPLICATIONS OF POETRY In Siberia I became interested in the exact sciences—physics and mathematics. At night I studied the theory of both and wrote poems. Rev. Česlovas Kavaliauskas 4.1. Archimedes’ constant. The number π is the most important of the so-called periods³⁸. The value of this constant is equal to the ratio of the circumference to the diameter, approximately ³⁵ The imaginary unit i, satisfying the equality i² = −1, is a real, existing object in mathematical Platonic reality, but such a misleading name has become established historically. ³⁶ Acoustic waves of close frequencies strongly dissonate because of the so-called beating effect. But if an echo is a sound (message) repeated from memory whose frequency may deviate somewhat from the original, then echo becomes a direct optical analogue of shade. ³⁷ The phraseological expression roll up one’s rods has the opposite meaning to this artless allegory; its two components are otherwise near-lexical synonyms of the components of the former expression. ³⁸ The precise definition of periods is rather technical (see Kontsevich, Zagier 2001). Straipsniai / Articles 247 GIEDRIUS ALKAUSKAS 3.1415926535897932384626433832795. This, of course, is not the primary definition. Euclidean geometry in the plane operates with lines and points, whose mutual relations are determined by the postulates of Euclid. When the words point and line are heard mathematically, real objects are imagined; however, one may operate only with concepts that obey predefined relations called axioms. These are givens, seeds that grow into shoots (theorems) and a harvest (theory). An example of an axiom: Through a point not lying on a given line passes exactly one line that does not intersect it. This is Euclid’s famous fifth postulate (axiom), which was long attempted to be derived from the others. Had this succeeded, the fifth would no longer have been needed. One may recall the work of the mathematician and poet Omar Khayyam, Comments on the First Postulates of Euclid’s Book (around 1100), in which he believed he had found such a derivation. Unfortunately, in the nineteenth century it was proved that Euclid’s fifth postulate is independent of the others. Its alternatives also provided the foundation for the so-called non-Euclidean geometries. Ivan Karamazov spoke about the latter to Alyosha: «Meanwhile there were and even now there are geometers and philosophers, and even some of the most remarkable among them, who doubt that the whole universe or, even more broadly, all existence was created according to Euclidean geometry alone, and who even dare to dream that the two parallel lines which, according to Euclid, can in no way meet on earth might perhaps meet somewhere in infinity» (Dostoevsky 1879: 396). Only the author is, incidentally, not quite precise here: in mentioning non-Euclidean geometry, he more likely describes the essence of projective geometry. In the latter, the following statement is always true: two noncoincident lines intersect at a single point. This geometry has another feature: in any of its theorems, exchanging the concepts of points and lines yields a true theorem. The dual assertion to the preceding one is: through two distinct points passes exactly one line. At the same time, this is an excellent illustration that line and point are merely abstractions. Imagining a fictitious Šėliai language in which the meanings of the words line and taskis are exchanged, all Lithuanian theorems would also hold in Šėliai. To emphasize how mathematical abstraction differs from an object in the imagination, it may be noted that one model of the projective plane is three-dimensional space with a fixed zero point O, in which a “point” is a line passing through O, while a “line through two points” is the plane determined by two such lines. Thus, alongside Dostoevsky, Justin Marcinkevičius is remarkably apt: All the time I seek a single point, All the time I seek a single word... from which all lines proceed. [...] Though I know I will never find it (DALP2 18). 248 Acta Linguistica Lithuanica LXXXVI Matematika, kalba, poezija: sąveikos ir traukos The lack of an intuitive definition of π can now be explained simply. Within the framework of Euclidean geometry, it is simply impossible to define the length of curves (for example, a circle). For this, another branch of mathematics, analysis, with its own axioms, is required. The primary definition of π may be surprising: This integral is an excellent springboard to the 45th tercet of Canto XXXIII of Dante’s “Paradiso”: What geometer is so wholly devoted to measuring the circle, yet cannot, through thought, discover that principle which he needs. You might say, a geometer who tried to square the circle, at last realizing that he strives and suffers in vain. Why, in this case, is measuring the circle and squaring the circle a fairly accurate translation? Dante is speaking of the ancient problem of squaring the circle. This asked whether it was possible, using only a compass and straightedge, to construct a square whose area equaled that of a unit circle. The negative answer was proved by Ferdinand von Lindemann in 1882. We shall not go into this further—this passage of the “Commedia,” a perfect tridisciplinarity of theology, poetry, and mathematics, has been fairly well studied (Herzman, Towsley 1994). On π and the geometric structure of the poem itself, precisely planned by the author (see Hart 1987). Dostoevsky mentions the problem of squaring the circle in “Notes from a Dead House,” comparing its insolubility with the equalization of punishments for identical crimes. This fact is omitted from the otherwise quite comprehensive compendium The Palgrave Handbook of Literature and Mathematics (PHLM). 4.2. Mnemonics. Dario Castellanos (Castellanos 1988: 152–153) discusses mnemonics in various languages that help one remember the digits of π in the decimal system. The encoding here is simple: the number of letters in a word corresponds to a digit. A well-known English version, and a somewhat less well-known Greek one: How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics! Ever the great God measures geometrically; in order to define the circumference of the circle by its diameter, he brought forth an infinite number... The author then provides rhymed examples in French and Spanish. It is obvious (and research confirms this) that poetry is better suited to this purpose than prose (Levitin 1999: 200–211). Indeed, human memory is rather weak when it comes to details: from an ordinary conversation, most people will remember only the emphasis, emotion, or mood. In light of this fact, the memory of performers of ancient ballads and epics appears almost phenomenal. Yet research has revealed that performers do not memorize every detail: rhythm, rhyme, alliteration, melody, grammar, and semantics constrain the freedom of the text, which also helps Straipsniai / Articles 249 GIEDRIUS ALKAUSKAS avoid mistakes. In much the same way, tactile sensations frame pianists’ memories. However, after reexamining eleven epic singers following a break, it became clear that the text is not always repeated identically, but obeys semantic and poetic constraints: the vocabulary varies synonymically, while pale or weak words are sometimes replaced by others without changing the meaning. In the context of memory records, this is where a potential danger lies. Sung poetry will encode many digits of π, but one unfortunate synonym can ruin everything. For example, Castellanos (Castellanos 1988: 153) gives a French poem from 1717 based on the calculation of 126 digits of π. Unfortunately, the 113th is incorrect. Once this was noticed, the word secteur (7 letters) had to be replaced with quadrant (8), and the rhyming antérieur (9) with précédent (9). There is another shortcoming of this encoding. It is undisputed (although not yet proved) that the number π is normal in every numeral system. That is, in its decimal representation, the digit 0 will occur asymptotically with probability 1/1039. Since there are no zero-letter words, zero can be encoded by ten-letter words. Second, among one-letter words in poetry there are o and į, several interjections, dialect words40, and the letters of the alphabet41 are used quite freely. Yet it is obvious that if every tenth word were one-letter long, the text would not be natural. All these words should be assigned to a separate class—let one-letter words encode nothing. 4.3. Memory control. A solution to the problem of memory’s unreliability with regard to details is offered by the algorithm implemented in barcodes and QR (quick response) codes of various purposes, credit-card numbers, and book identifiers in the ISBN and EAN systems. Specifically, the checksum method (Stakėnas 2007: 127). If one needs to remember the digits 3603735, the simplest check digit would be 3 + 6 + 0 + 3 + 7 + 3 + 5 mod 10 = 7. Thus, in order to check the word lengths in the line the books already laid out like burbot on a bench⁴², one can supplement it with a seven-letter word, for example, the books already laid out like burbot on a bench front (unfortunately, this ruins the trope). Some errors that obey the form of the text (already → and, like → as) will be noticed, but if the reciter accidentally swapped 39 Let us suppose that we need to prove that a set 𝒜 has a certain property not characteristic of a typical statistical set. Therefore, there exists some internal structure constraining 𝒜, and the set does not obey statistical laws. Often, the required property is proved precisely by grasping this structure. On the other hand, problems asking one to prove that a specific set (the sequence of digits of π) obeys statistical laws are among the most difficult. 40 The famous Ė of “The Forest of Anykščiai” (o, bet, ogi). 41 A stylistic device favored by Juozas Erlickas, for example, “Much more gives the soul / joy, and so on,” “I am like a train that ran / from A to B and then C disappeared,” and so on. 42 A line from S. Geda’s poem “In the Village of Karvaičiai, Which Is Buried by Sand,” only with the original already replaced by jau. 250 Acta Linguistica Lithuanica LXXXVI Matematika, kalba, poezija: sąveikos ir traukos the words books and laid out, the iamb’s s1 caesura would not be spoiled, but the check digit for this error would unfortunately not detect it. Mathematics can nevertheless help more significantly, because the chosen algorithm is not good enough. If Luhn’s formula were used, it would detect both a one-digit discrepancy and nearly all errors involving the transposition of two adjacent digits (it would notice only not the error 09 ↔ 90). 4.4. Algorithm. A finite sequence of digits {s₁, s₂, s₃, s₄, s₅, s₆,..., sᵣ} is given. 1. It is replaced by the sequence K = {2s₁, s₂, 2s₃, s₄, 2s₅, s₆,...}, i.e., every other term is doubled. 2. All two-digit numbers in K are replaced by the sums of their digits (for example, 12 → 3, 14 → 5, and so on), thereby yielding the sequence L. 3. All terms of L are added, and the desired check digit equals this sum modulo 10. For example, let us examine the same 3603735. Here K = {6, 6, 0, 3, 14, 3, 10}, L = {6, 6, 0, 3, 5, 3, 1}. The check digit 6 + 6 + 0 + 3 + 5 + 3 + 1 mod 10 = 4, so the word end will detect more possible errors than the word front. The algorithm for a mnemonic with checking will now be described only approximately, since a careful statistical analysis of the sample text and a discussion of possible complications must first be undertaken. It would then be better to use, instead of the Luhn method, the algorithm devised by the Dutch mathematician Jacobus Verhoeff, which detects even more errors, including 09 ↔ 90⁴³. The text for the mnemonic is written in classical terzinas, each line having 11 syllables. Some statistics. The graph shows the length in letters of all 45 tercets of Canto X of the “Inferno” translated by Sigitas Geda (punctuation marks are not counted). All lines, except one each in tercets 7, 11, 21, 23, 30, and 33 (where the syllable count is either 9 or 13), have 11 syllables. The mathematically “non-Dantesque” lines were adjusted. This now strictly hendecasyllabic iambic text is also used for the statistics. The average number of letters in a tercet is 84.44; on average, syllables consist of 2.558 letters. The black line represents the normalized distribution function, interpolated linearly⁴⁴. Algorithm. Let {pₛ, s ∈ N} be the sequence of digits of π. That is, p₁ = 3, p₂ = 1, p₃ = 4, p₄ = 1, p₅ = 5, and so on. Let us divide the digits into blocks of 10. We now complete each block with a check digit according to 43 In Dutch, numbers are usually read in pairs. J. Verhoeff, studying postal codes, calculated that phonetic errors, when the number 1x is confused with x1 (for example, 15, Vijftien and 50, Vijftig), account for 0.49 percent of all errors. This is a small but noticeable proportion. 44 The distribution function F(x) = P(Z < x) of a random variable Z indicates the probability with which it assumes a value smaller than x. The maximum of such a function is 1. In the graph, this value is multiplied by 5 so that all the data fit in a single illustration. Straipsniai / Articles 251 GIEDRIUS ALKAUSKAS the Luhn formula, which is inserted into the sequence. Let the resulting new sequence be {qₛ, s ∈ N}. Replace every 0 with 12, and replace every remaining digit s with s + 2. Let R = {rₛ, s ∈ N} be the resulting sequence. One- and two-letter words may be used at any time when creating the poem-mnemonic; they are ignored in decoding. Let us suppose that the first v terms of sequence R have already been distributed into stanzas. The average value of the terms of sequence R is (3+12)/2 = 7.5. Let j − 1 be the greatest natural number for which rᵥ₊₁ + rᵥ₊₂ + … + rᵥ₊ⱼ₋₁ < 80 (this regime was chosen so that, after adding the next term of the sequence, the average missing number of letters in a tercet is obtained, i.e. 84.5). The next stanza will consist of words of lengths rᵥ₊₁, rᵥ₊₂, …, rᵥ₊ⱼ. All that remains is to create the corresponding tercet. Incidentally, this encoding will also detect another type of error, namely, missing or extra words. For example, two words were recited when one word was required (in the same example with burbot, closely lies ↔ laid out). After such a mistake, nearly all check digits will detect an error, so it is not difficult to determine in which stanza the number of words changed. ⁴ FIG. The number of tercets in Canto X of the “Inferno” for a given number of letters A genuine algorithm requires all possible complications to be examined. The fact that Luhn’s formula does not detect the error 09 ↔ 90 means that the check number will not indicate it if, during recitation, 12- and 11-letter words are transposed. This is potentially possible in a disyllabic meter (for example, shakiabeard young man⁴⁵). This chapter did not aim to create the best algorithm, but it is obvious that the effort is worthwhile. A more sophisticated algorithm would reveal subtler statistical differences between spoken, literary, and poetic-metrical language. This is what J. Girdzijauskas also studied, though somewhat differently. ⁴⁵ The corrected line from Maironis’s ballad “Čičinskas,” “goat-bearded young man.” Sven Shakiabeard (Svein Tjugeskjegg 963–1014) was king of Denmark. 252 Acta Linguistica Lithuanica LXXXVI Matematika, kalba, poezija: sąveikos ir traukos The entire “Inferno” consists of about 25,000 Lithuanian words. Singers of epics have demonstrated that the human brain is capable of learning that many. After excluding the check digits, about 22,000 digits remain. Forty years ago, a similar number was also the world π memory record (today it will exceed 70,000). But what is the point of learning those bare numbers? It is another matter when there is a poem in which the interesting mathematics is merely a secondary result. 5. CONCLUSIONS The interaction and bidisciplinarity of two arts, two sciences, or art and science conceal more dangers than they offer new possibilities. The article sought to reveal the potential shortcomings of such combinations: profanation, when one art or science participates in the interaction at an extremely elementary level; artificial adaptation, when one art or science obeys the laws of the other, or when the creator approaches the alliance one-sidedly (such is the case with most texts by representatives of the exact sciences about music and mathematics); the external disintegration of the work and personality of a recognized genius (Baranauskas, Dante, Goethe, etc.), when his or her relatively peripheral activity is viewed in isolation from the primary one; and differing mechanisms of cognitive-emotional operation and impact. However, there are many examples in which such combinations prove successful. The article has shown that implicit and explicit interactions are possible in the cases of mathematics and poetry, and mathematics and language. The former include poetic forms in which mathematical instruments are needed only for their theoretical justification, but these instruments no longer remain in the poetry itself. The latter include various cases of reverse text design, in which mathematics and grammar limit the possibilities of constructing the text (this may be compared with various theories of musical harmony—diatonicism, serialism, proportional canons, and so on), as well as mathematical applications of poetry which, although not themselves particularly functional, reveal deeper phenomena concerning poetic language and form. LITERATŪRA Alkauskas Giedrius 2018: Music-Mathematics Interconnections: An Approach through Science (Works by Mathematicians), and an Approach through the Arts (Musica mathematica by Rima Povilionienė). – Lietuvos muzikologija 19, 331–350. Alkauskas Giedrius 2019: Kalba poezijoje: filologija vs. biofilija. – Acta Linguistica Lithuanica 81, 235–250. Straipsniai / Articles 253 GIEDRIUS ALKAUSKAS Alkauskas Giedrius 2020: Prasmė легируota garsu: poezija kaip muzika. – Acta Linguistica Lithuanica 83, 331–350. Alkauskas Giedrius 2022: Raštijos dėliȯnė kaip kūrybos kodas. – Kultūros barai 1/2, 35–43. Baranauskas Antanas 1994: Rinktinė: poezija, giesmės, dienoraštis, laiškai, sud. R. Mikšytė, Vilnius: Baltos lankos. Baranauskas Antanas 1995: Raštai 1 (Poezija), par. R. Mikšytė, M. Daškus, Vilnius: Baltos lankos. Castellanos Dario 1988: The Ubiquitous π. – Mathematics Magazine 61(3), 148–163. DALP2 – Dvidešimto amžiaus lietuvių poezija 2, sud. V. Kubilius, Vilnius: Vaga, 1995. Dantė 2007: Dieviškoji Komedija: Pragaras, vertė S. Geda, Vilnius: Lietuvos rašytojų sąjungos leidykla. Elliot Thomas S. 1921: Dantė, vertė K. Platelis. – Dieviškoji Komedija: Pragaras, vertė S. Geda, Vilnius: Lietuvos rašytojų sąjungos leidykla, 388–409. Geda Sigitas 1994: Skrynėlė dvasioms pagauti: eilėraščiai, Vilnius: Vaga. Geda Sigitas, 1998: Man gražiausias klebonas – var nėnas: interviu, Vilnius: Vaga. Girdzijauskas Juozas 1978: Tarp silabotonikos ir verl ibro: sinkopis lietuvių poezijoje. – Poezijos pavasaris, Vilnius: Vaga, 220–235. Girdzijauskas Juozas 2001: Eliedara. – Lietuvos literatūros enciklopedija, Vilnius: Lietuvos literatūros ir tautosakos institutas, 125–127. Grinbaum Oleg N. 2002: Гринбаум Олег Н. Строка, строфа и стих как ритмическая система [Stroka, strofa i stikh kak rifmicheskaja sistema]. – Материалы XXXI Всероссийской научно-методич. конф. преподав. и аспир. Филологического ф-та СПбГУ [Materialy XXXI Vserossiiskoi nauchno-metodich. konf. prepodav. i aspir. Filologicheskogo f-ta SPbGU] 4(2), 12–28. Harris Zellig 1969: Mathematical linguistics. – The Mathematical Sciences. A Collection of Essays, Cambridge, Mass.: M.I.T. Press, 190–196. Hart Thomas Elwood 1987: The Cristo–Rhymes and Polyvalence as a Principle of Structure in Dante’s “Commedia”. – Dante Studies 105, 1–42. Herzman Ronald B., Towsley Gary W. 1994: Squaring the Circle: “Paradiso” 33 and the Poetics of Geometry. – Traditio 49, 95–125. HNL – Handbook of the Neuroscience of Language, eds. B. Stemmer, H. Whitaker, USA: Academic Press Elsevier, 2008. 254 Acta Linguistica Lithuanica LXXXVI Matematika, kalba, poezija: sąveikos ir traukos Kontsevich Maxim, Zagier Don 2001: Periods. – Mathematics unlimited 2001 and beyond, Berlin, New York City: Springer, 771–808. Kubilius Jonas 2001: Antanas Baranauskas ir matematika, ser. Iš Lietuvos matematikos istorijos, kn. 1, Vilnius: Matematikos ir informatikos institutas. Levitin Daniel J. 1999: Mechanisms of Memory for Musical Attributes. – Music, Cognition, and Computerized Sound: An Introduction to Psychoacoustics 1, ed. P. R. Cook, Cambridge, Mass: M.I.T. Press. LKŽe – Lietuvių kalbos žodynas 1–20 (1941–2002), red. kolegija G. Naktinienė, J. Paulauskas, R. Petrokienė, V. Vitkauskas, J. Zabarskaitė, vyr. red. G. Naktinienė, e. variantas, Vilnius: Lietuvių kalbos institutas, 2008 (atnaujinta versija, 2017). Prieiga internete: https://ekalba.lt/lietuviu-kalbos-zodynas. Mačys-Kėkštas Jonas 1910: Eilės, Pittsburg: Leidimas ir spauda Brolių Baltrušaičių. Matulaitienė Stasė 1997: Poezijos gramatika, Vilnius: Gimtasis žodis. Pekarskas Vidmantas 2003: Pirmųjų lietuviškų matematikos terminų istorijos fragmentai. – α+ω 3, 76–86. Prieiga internete: http://web.vu.lt/mif/v.stakenas/a+o/2003-3/2003-3-74-86.pdf Povilionienė Rima 2013: Musica Mathematica: tradicijos ir inovacijos šiuolaikinėje muzikoje, Vilnius: Lietuvos muzikos ir teatro akademija. Prieiga internete: https://www.tornado-beta.lt/en/musica-mathematica/. Razauskas Dainius 2008: Mė(lynas)nulis ir dangaus kiaulių paslaptis: „žaidimas žodžiais“ lietuvių tradicijoje. – Liaudies kultūra 6, 58–60. Ruseckienė Rasa, vert. 2017: Edda. Snorri Sturluson, atsak. red. I. Steponavičiūtė-Aleksiejūnienė, Vilnius: VšĮ Akademinė leidyba. Stakėnas Vilius 2007: Kodai ir šifrai: informacijos kodavimo ir kriptografijos pagrindai, Vilnius: TEV, Vilniaus Universitetas. Stankevičius Rimvydas, sud., 2015: A. Kalanavičius. Dvi saujos laiko: eilėraščiai (kartu su P. Dumšiene), Vilnius: Lietuvos rašytojų sąjungos leidykla. Steponaitis Edmundas 1998: Rinktinė, par., įvadą ir komentarus parašė R. Karmalavičius, Poezijos biblioteka Versmės, Vilnius: Vaga. Subačius Giedrius 2010: Lietuvių kalbos ekspertai Rusijos imperijos tarnyboje, Vilnius: Lietuvių kalbos instituto leidykla. Valentas Skirmantas 1997: Lingvistinis pasaulis poezijoje, Vilnius: Mokslo ir enciklopedijų leidybos institutas. Straipsniai / Articles 255 GIEDRIUS ALKAUSKAS MTŽ – Matematikos terminų žodynas, red. J. Kubilius, Vilnius: Mokslo ir enciklopedijų leidykla, 1994. OEIS – The online encyclopedia of integer sequences, seka A065619. Prieiga internete: https://oeis.org/A065619. PE – Poetinė Eda, vertė A. Vijūnas, Vilnius: Aidai, 2009. PHLM – The Palgrave Handbook of Literature and Mathematics, eds. R. Tubbs, A. Jenkins, N. Engelhardt, London: Palgrave Macmillan, 2021. RP – Raidžių paveikslai: vizualioji poezija lietuvių kalba, sud. B. Januševičius, G. Skudžinskas, Vilnius: Nerutina, 2018. ŠLPCh – Šiuolaikinės lietuvių poezijos chrestomatija, sud. S. Parulskis, Vilnius: Alma Littera, 2002. Mathematics, Language, Poetry: Interactions and Attractions SUMMARY This text is the third in a series of four texts devoted to various aspects of poetic language. The aim of this part is to present a few constructions where the interaction of two domains of creativity occurs (two out of poetry, language or mathematics), which a posteriori has a nontrivial intersection with the remaining domain. In our opinion, this is the instance where interdisciplinarity is truly fertile. Generally, this is not the case. Dealing with questions of quality in bidisciplinary art or science, our main standpoints are as follows. First, both domains should participate in an interaction on the comparable level of sophistication, each of the subjects being not subordinate to the laws of the other. As a non-example, most books on music and mathematics are written by mathematicians with a natural–scientific attitude. To make our case more substantial, we examine the book devoted to concrete poetry, finding both fruitful and artificial examples. Second (this brings a human dimension to the subject), we postulate that the legacy of any great artist should be considered not in isolation but in its entirety. This means the following: the area of distinction and the area of (a conditional) amateurism should not be separated from one another. The examples include the mathematical writings of Daniil Kharms, or the treatise in optics Zur Farbenlehre by Goethe. We examine the book Antanas Baranauskas and mathematics by Jonas Kubilius more thoroughly. Baranauskas was a poet who also worked extensively in the fields of homiletics, dialectology, orthography, 256 Acta Linguistica Lithuanica LXXXVI Matematika, kalba, poezija: sąveikos ir traukos terminology, and mathematics. However, Baranauskas’ output in the latter is investigated by Kubilius purely from a professional point of view. Qualitative judgements are generally correct. However, if we take into account that a part of Baranauskas’ mathematical work is integrated into his philosophical and theological treatises, it is natural to hope for a much more ambitious immersion in the mathematical legacy of Baranauskas. We further propose several interactions of mathematics-language-poetry, which, hopefully, have the needed qualities. We explore terza rima, the poetic invention by Dante, and find its core. This leads to the definitions of simple, minimal, normalized, and, correspondingly, canonical Q-times. The latter is a straightforward generalization of Dante’s form. Now we can prove the following result: for every odd positive integer Q, canonical Q-time is unique. The construction, however, does not take into account any particular language. If we do so for Q = 5, it emerges that a smooth flow of the text requires such a pentastich to possess few incomplete lines (hendecasyllabic iamb is an exception). Since the interval of two unstressed syllables is the most frequent in spoken Lithuanian, it is natural to write 5-times in a dactylic meter. The example of two such strophes is presented. As a further example of fruitful bidisciplinary interaction, we propose the linguistic analogue of roots of unity. We finish the paper with presenting an improved method for a mnemonic technique. The standard procedure uses poetry, but the encoding itself employs only rudimentary arithmetics. Our method, however, uses also check digit algorithm. When developed in full, the method may reveal interesting facts in the field of computational linguistics. Įteikta 2022 m. vasario 28 d. GIEDRIUS ALKAUSKAS Vilniaus universitetas, Informatikos institutas Naugarduko g. 24, LT–03225 Vilnius, Lietuva [email protected] Straipsniai / Articles 257