“OUT of FIXED PROPORTION, BEAUTY RISES”: a REVIEW of MATHEMATICS in FORMAL POETIC CONSTRAINT by Tiffany Nielander

“OUT OF FIXED PROPORTION, BEAUTY RISES”: A REVIEW OF

MATHEMATICS IN FORMAL POETIC CONSTRAINT

by Tiffany Nielander

A Thesis Submitted to the Faculty of

The Harriet L. Wilkes Honors College

in Partial Fulfillment of the Requirements for the Degree of

Bachelor of Arts in Liberal Arts and Sciences

with a Concentration in Mathematics and English Literature

Wilkes Honors College of

Florida Atlantic University

Jupiter, Florida

May 2018

“OUT OF FIXED PROPORTION, BEAUTY RISES”: A REVIEW OF MATHEMATICS IN FORMAL POETIC CONSTRAINT by Tiffany Nielander

This thesis was prepared under the direction of the candidate’s thesis advisors, Dr. Meredith Blue and Dr. Gavin Sourgen, and has been approved by members of her supervisory committee. It was submitted to the faculty of The Honors College and was accepted in partial fulfillment of the requirements for the degree of Bachelor of Arts in Liberal Arts and Sciences.

SUPERVISORY COMMITTEE:

______Dr. Meredith Blue

______Dr. Gavin Sourgen

______Dean Ellen Goldey, Wilkes Honors College

______Date

ABSTRACT

Author: Tiffany Nielander

Title: “OUT OF FIXED PROPORTION, BEAUTY RISES”: A

REVIEW OF MATHEMATICS IN FORMAL POETIC

CONSTRAINT

Institution: Wilkes Honors College of Florida Atlantic University

Thesis Advisors: Dr. Meredith Blue and Dr. Gavin Sourgen

Degree: Bachelor of Arts in Liberal Arts and Sciences

Concentrations: Mathematics and English Literature

Year: 2018

As Bertrand Russel once said, “The true spirit of delight, the exaltation, the sense of being more than man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry” (Russel 60). Poetry and mathematics are recognizably linked through aesthetics and counting at the most fundamental level, but these basic connections can be further extended to formal constraints in poetry. The link between mathematics and poetry, as well as formal poetic constraint based on mathematical structures and principles is inherently organic. The sestina and the sonnet are traditional poetic forms that contain intrinsic mathematical structures. The Fib, the S+7 algorithm, and computer-generated poetry are modern forms which have been explicitly based on mathematical structures.

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ACKNOWLEDGEMENTS

Thank you to my advisors, Dr. Meredith Blue and Dr. Gavin Sourgen: Dr. Blue, you have been with me from my first days at the HC four years ago, and you have been there for every moment since. Dr. Sourgen, I regret that I only had the chance to take two of your classes, but Irony & Satire will always be one of the best classes that I have ever taken.

You have both done so much for me this year, and I am so grateful for your time and support through the ups and downs of the process that has culminated in this document.

Thank you to my family: Mom, Macy, and Dad, what do I even say to you? You have been there for me through everything-panic attacks, triumphs, and failures-and I could not have made it this far without you. I love you so much and I cannot imagine achieving anything without sharing it with you.

Thank you to my loving partner, Tripp: I cannot put into words how grateful I am that you are a part of my life. Thank you for understanding my anxiety and stress, and always being there for me. Your constant support has gotten me through most of the last three years and I look forward to our future together.

Thank you to my fantastic friends: Thank you for buying coffee, and snacks. Thank you for letting me cry in your laps. Thank you for the long nights of Netflix binging and Uno- playing, even though I almost died. Thank you for your unyielding support. Thank you for hugs, I-love-yous, cuddles. Thank you for late nights in the housing office. I love you guys so much. Kesh, you have been my best friend basically since day one, thank you for everything you have done for me these four years, and I will miss you so much. I love you. And to my roommates, past and present: I treasure the support and friendship of all of you, and I wish you all the very best in everything you do.

Thank you to Adam “Dad-am” Schwarz: I appreciate you and everything you have done and still do for me as a member of the Housing team and as an individual. I could not have asked for a more supportive, involved, and understanding boss. I love you. *fist- bumps*

Thank you to Dr. Warren McGovern: You have shaped so much of my experience at the

HC, always for the better. Thank you for an incredible four years.

Thank you to Professor Rachel Luria: Thank you for not only being an amazing professor, but also the most wonderful advisor we could have asked for in Cliché.

Thank you to my HC professors, Drs. Michael Harrawood, Yasmine Shamma, Mark

Tunick, Christopher Ely, William O’Brien, Terje Hoim, Christopher Strain, and

Jacqueline Fewkes, for making the past four years memorable and absolutely incredible.

Table of Contents

Chapter 1: Introduction: On Formal Poetic Constraint, the Organic, and Mathematics .... 1

Chapter 2: Mathematics in the Sestina and the Petrarchan Sonnet ...... 12

Section 2.1: The Sestina and the Permutation ...... 12

Section 2.2: The Petrarchan Sonnet and the Pythagorean Theorem ...... 20

Chapter 3: The Fib, the S+7 Algorithm, and Computer-Generated Poetry ...... 26

Section 3.1: The Fib ...... 26

Section 3.2: The S+7 Algorithm ...... 29

Section 3.3: Computer-Generated Poetry ...... 31

Section 3.4: Coding the Fib and the S+7 ...... 33

Conclusion ...... 38

Appendix A ...... 40

Appendix B ...... 43

Works Cited ...... 45

List of Figures

Figure 1 ...... 15

Figure 2 ...... 18

Figure 3 ...... 19

Figure 4 ...... 23

Figure 5 ...... 30

Figure 6 ...... 36

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Chapter 1: Introduction: On Formal Poetic Constraint, the Organic, and

Mathematics

In his book How to Read a Poem, Terry Eagleton defines ‘form’ as “all those aspects of a literary work… which are relevant to how the work presents its materials”

(Eagleton 166). Form in this definition pertains to the disjoint elements, such as rhyme and tone, posed by a poem to convey itself in the physical world; it is the characteristics of its shape as individual components. The ‘formal poetic constraint’ incorporates these elements into a predetermined external structure of patterned rhyme, meter, and more, such as a limerick, often termed a “form.” I will use the term “fixed form” in the cases that are in reference specifically to classified constraints.

The notion of formal constraint and its application to the structures of poetry has been debated for centuries. Some have criticized fixed forms of poetry as acrobatics, being ostentatious but providing little to nothing of real substance to the overall reading of the poem. Oliver Wendell Holmes criticized fixed form in his novel Over the Teacups when he declares “Rhythm alone is a tether, and not a very long one. But rhymes are iron fetters” (Holmes 79). More often, form is criticized if it does not reflect the sentiments or essence of the ‘content of the poem’; i.e., if form and content can be distinctly separated, then the form is superfluous. There are others, however, who see new possibilities in strict formal constraint, or in fact, fixed forms as literature in themselves, worthy of study. This chapter will discuss the perspectives of two literary ‘movements’ and three particular authors who have all theorized on the importance and proper usage of strict formal constraints in poetry. The perspectives that this chapter examines will inform the

function of the fixed forms that I will be examining in the following chapters: by tracing the evolution of the purpose of constraint, I hope to illustrate the significance that mathematics has in forming such constraints.

I will begin this discussion with British Romanticism because its writers were notorious for their extensive employment of formal constraints to contend with their untamed internal energies. Form and formal constraint in Romantic poetry, though they seem to be cognitively disnonant, are a necessity for the very reason that the Romantics themselves emphasized poetry as “the spontaneous overflow of powerful feelings,” as

Wordsworth theorizes. However, the Romantic fixation on self-creation and on internal self-definition, manifesting in poetry and art that is often completely self-reflexive and concerned entirely with its own dynamics, would logically require a set of rules within which to operate and define it from within as much as from without (Curran 216). As

Stuart Curran writes in Poetic Form and British Romanticism,

In an art that exists for its own sake, form is essential. Yet that very fact should underscore why form was so inescapable a necessity for Romantic subjectivity, a ground for either commitment or disengagement, but always a ground for self-mirroring and self-creation. Any simple logic would suggest the necessity within such a dialectical field of complementary mirroring and recreation of the predicated other as well. (Curran 216)

Formal constraint is thus a vehicle for redefinition and individualism that jump-starts the process for the Romantic obsession with the mirroring of the internal structures to the external structures of the poem so that they become a single commentary on poetry, art, and the form that they donned. If not for the constraints in which the poem dresses itself, there would be nothing for the poem to discuss, and therefore the naked Romantic poem could never exist.

The interdependence of form and internal dynamics in Romantic poetry is reflected in S. T. Coleridge’s theory of form, in which he takes the relationship a step further. He does not criticize formal constraint as artificial, but instead makes a distinction between poetries in which a form is superimposed artificially and those in which their form is a natural consequence of the content. He defines the terms mechanical and organic form in the following passage from “Shakespeare’s Judgement

Equal to His Genius” (1836):

The form is mechanic when on any given material we impress a predetermined form, not necessarily arising out of the properties of the material – as when to a mass of wet clay we give whatever shape we wish it to retain when hardened. The organic form, on the other hand, is innate; it shapes as it develops itself from within, and the fullness of its development is one and the same with the perfection of its outward form. Such is the life, such is the form. Nature, the prime genial artist, inexhaustible in diverse powers, is equally inexhaustible in forms: each exterior is the physiognomy of the being within, its true image reflected and thrown out from the concave mirror… (Coleridge 55)

The mechanical form arises from forcibly placing the material of the poem within formal constraints that are not inherent in the poem itself as necessitated by the ‘content’, while the organic form praises the use of formal elements that occur genuinely and naturally, not just as products of the material, but as an inextricable element of the poem as a unified object. This is not an opposition to formal constraint in all circumstances, but to the molding of a poem into a specific shape simply for the sake of doing so without any significance to the material of the poem itself. It draws on the assumption that form and content should be inseparable from one another. Coleridge discusses this in his 1818 lecture “On poesy or art”:

… There is a difference between form as proceeding, and shape as superinduced;—the latter is either the death or the imprisonment of the thing;— the former is its self-witnessing and self-effected sphere of agency. (Coleridge 336)

Here he makes the important distinction between form and shape: one being that which is an inextricable element of the essence of the poem itself, something that carries meaning and weight of its own and allows the poet to give voice to the poem. The latter being that which is a set of descriptions which are independent of the character of the poem and instead of promoting the voice of the poem, stifles and chokes it. This distinction is what allows the Romantic poet to write a sonnet about the sonnet but prevents him from forcing a poem strictly about the forest into a shape to which the forest bears no relation, such as a sonnet. The fixed form is permitted to inspire a poem about the form and the poem itself, but it is intolerable for a poem to wear a formal constraint as a cloak. The role of external constraint in Coleridge’s ideology is as a mirror to the content throughout the creative process: the content produces the form as the form simultaneously produces the content.

T. S. Eliot, in his “Reflections on Vers Libre” offers another important perspective on the use of constraint in poetry. Eliot believes that formal constraint is the most effective when the poet does not use it systematically, as a crutch. He argues that it is only effective when it is applied masterfully, conservatively, and precisely where it is necessary (Eliot “Reflections on Vers Libre”). He specifically discusses the use of rhyme and patterned meter in poetry as the elements that are commonly omitted in order to create the artificial freedom of vers libre.

He first argues that “There is no escape from metre; there is only mastery,” in order to use metrical patterns to accentuate “moments of the first intensity” (Eliot

“Reflections on Vers Libre”):

We may therefore formulate as follows: the ghost of some simple metre should lurk behind the arras in even the ‘freest’ verse; to advance menacingly as we doze, and withdraw as we rouse. Or, freedom is only truly freedom when it appears against the background of an artificial limitation. (Eliot “Reflections on Vers Libre”)

Eliot is proposing that poets should use patterned metrical constraint as a means to accentuate the moments in which the very same pattern is broken, always hinting at the comfortable familiarity of consistence, but never fully achieving it, creating an effect that is unique to each poem and that reflects the content as the poet sees fit.

Eliot then discusses the benefits of the removal of rhyme from vers libre, as it is the removal of a distraction and, at times, a prop:

When the comforting echo of rhyme is removed, success or failure in the choice of words, in the sentence structure, in the order, is at once more apparent. Rhyme removed, the poet is at once held up to the standards of prose. Rhyme removed, much ethereal music leaps up from the word, music which has hitherto chirped unnoticed in the expanse of prose… And this liberation from rhyme might be as well a liberation of rhyme. Freed from its exacting task of supporting lame verse, it could be applied with greater effect where it is most needed. (Eliot “Reflections on Vers Libre”)

The implications for constraint by rhyme are similar to that by metre: that such constraints should not dictate to the poet, but that the poet should have control over them to use them to their own wishes and to the advantage of their verse, but not to be controlled by them. He compares unrhymed verse to prose to emphasize the importance of technique and craftsmanship in poem when there are no sing-song words to draw attention away. Eliot then proposes the most effective way for a poet to incorporate form into his or her verse:

But the most interesting verse which has yet been written in our language has been done either by taking a very simple form, like the iambic pentameter, and constantly withdrawing from it, or taking no form at all, and constantly approximating to a very simple one. It is this contrast between fixity and flux, this

unperceived evasion of monotony, which is the very life of verse. (Eliot “Reflections on Vers Libre”)

The poet should neither succumb to the imposition of form on their verse, nor dispose of it entirely, but hold it in tension with the ‘free’ to avoid the monotony of repeated patterns while maintaining an approximation of them.

Eliot’s view on the role of form fits relatively well into Coleridge’s theory on the organic. Both Coleridge and Eliot reject form as an element with the power to control the poet as he/she writes the poem, but instead favor it as being dictated by the poem itself in the organic, or by the poet. Coleridge’s organic sees formal elements of constraint as necessary, only as the poem itself calls for them naturally. Eliot’s perspective, similarly, advocates the poet’s choice in their use of formal constraints as a way to transcend and totally control them, rather than their being forcibly held to conventions that do not fit the poem’s needs nor allow the poem to stand with its own strength. The major difference between Coleridge and Eliot is their view on the agent of control which should dictate the constraints being applied: Coleridge sees the poem as an interweaving of form and content, and so the poem itself would prescribe the constraints that are already inherent to its content; Eliot sees the poet as the mediator of form, and therefore as the one that would place formal elements in the poem where he or she thinks they are needed to create his or her intended effect.

W. H. Auden assimilates both the perspectives of Coleridge and the Romantics, and that of Eliot when he conceives of a theory on the significance of constraint in poetry. In his essay, “Writing,” he asserts that:

Rhymes, meters, stanza forms, etc., are like servants. If the master is fair enough to win their affection and firm enough to command their respect, the result is an

orderly happy household. … if he lacks authority, they become slovenly, impertinent, drunk and dishonest. The poet who writes “free” verse is like Robinson Crusoe on his desert island: he must do all his cooking, laundry and darning for himself. In a few exceptional cases, this manly independence produces something original and impressive, but more often the result is squalor… (Auden 22)

Auden is arguing here that both poetries of fixed form and those coming out of free verse require masterful technique and exceptional skill to be of any worth. The elements of formal structure necessitate a firm hand to mold them just the way the poet wishes, without stifling them and rendering them meaningless and superfluous. However, were a poet to completely discard those elements of form in favor of “freedom,” he must be able to do all the work that form does for him, which, Auden intimates, is an exacting and almost impossible task.

In Auden’s perspective of form, constraint plays the role of removing pressure from the writer to create a poem of words as they are, not as more than they are.

Constraints and formal elements allow the poet to imbue each word with more meaning than would be conveyed by simply their own denotations. He also sees constraint as removing an amount of control from the poet, as evidenced in this quote: "Blessed be all metrical rules that forbid automatic responses, force us to have second thoughts, free from the fetters of Self" (Haffenden 274). The control is not lost, however, it simply moves to the poem itself, allowing it the capacity for self-creation, recalling the formal theory of the Romantics, as well as anticipating the views of constraint of the Oulipo.

Auden sees these constraints as elevating poetry above what a poet can offer simply on his or her own; formal constraints ask of the poet to ruminate and find exactly what he means to say, remaining aware of all consequences of each word choice and placement.

His metaphor of the servants and master of the house is at odds with this is in strict terms, but it succeeds in relating to the reader the image of a joint effort between poem and poet.

The Ouvroir de Litterature Potentielle, “the workshop for potential literature”, or

Oulipo, founded by Francois Le Lionnais and Raymond Queneau in 1960, then take

Auden’s perspective and expand on it further to create their own theory of formal poetic constraints. The Oulipo was a group of French writers and mathematicians who believed that constraint could be a source of inspiration for their poetry, and so sought to create new poetic forms of constraint, to open “new possibilities previously unknown to authors” (Lionnais qtd. in Motte 38).

The Oulipo has endeavored to explore, to inventory, to analyze the intimate processes and resources of the language of words, of letters. This exploration is naturally based on the use of constraint, either through the use of ancient constraints pushed to the far limit of their possibilities, or through systematic research in new constraints… [The Oulipo] merely seeks to formulate problems and eventually to offer solutions that allow and everybody to construct, letter by letter, word by word, a text. To create a structure – Oulipian act par excellence – is thus to propose an as yet undiscovered mode of organization for linguistic objects. (Motte 43-46)

They have two main inclinations: anoulipism and synthoulipism. Anoulipism is the process of the analysis of past works in the cannon, or outside of the cannon, to

‘discover’ new possibilities of traditional fixed forms that were often unknown to the original poets that utilized or invented them (Motte 27-28). Synthoulipism is the principle, and more ambitious, goal of the workshop, however, of conceiving of new constraints and possibilities of forms that have never been seen before (Motte 27-28).

They not only push existing forms and groupings of constraints further than the original structures, such as the haiku to the Fib (a haiku with metrical line lengths based on the

Fibonacci sequence), but they have created numerous new fixed forms. They have begun

to formulate a collection of these potential forms but refuse to use them further than to compose a handful of examples of each to show how they would work and be used, as

‘proofs’. They believe in leaving them in a potential state: literature that is potential not through the reading, but through the writing, because “the only text of value is the one that formulates the constraint; all texts resulting therefrom… [belong] to the ‘applied

Oulipo’” (Motte 12). The application has little merit because it does not represent a potential state from which a study can be made of the potentialities of the constraint.

The Oulipo see constraint as functioning to awaken the writer to the rules already inherent in language and poetry: “to the extent that constraint goes beyond rules which seem natural only to those people who have barely questioned language, it forces the system out of its routine functioning, thereby compelling it to reveal its hidden resources”

(Motte 41). Constraint forces the writer to break their unconscious and undisputed patterns and become hyper aware of the language that they are using, thus bringing their writing under their control entirely, rather than unknowingly allowing innate rules to dictate their language for them. The limitations imposed by a fixed form force a writer that has not questioned language to do so, because the constraint may force them to break the natural patterns to which they are accustomed, and from which they know no other way. As JoAnne Growney writes “Formal constraints concerning the shape of a poem prod the poet to exceptional care in word choice. There is not space for an extra syllable.

Meaning must be condensed” (Growney 3). In addition, constraint is a way for the writer to consciously and proudly embrace a set of rules and bend it to their will. The Oulipo do not see constraints as controlling the poet or the poem, but as a way for the poet to show his or her skill and discover the poem he was meant to write, rather than the poem that he

intended. John Glenday writes, “The great thing about [fixed] form is that it hinders us from saying what we had originally intended to say” (Glenday 8). Jaques Roubaud hypothesizes that there are two rules of Oulipian constraint that should be followed when creating the ‘proof’ for a new form. The first rule is that “a text written according to a constraint must speak of this constraint,” a suggestion of the organic within the context of what Coleridge would have thought to be a completely inorganic process (Motte 12). The second rule is that “a text written according to a mathematizable constraint must contain the consequences of the mathematical theory it illustrates,” which follows logically from the first rule, but raises the level of difficulty, and therefore the level of mastery by the poet, to achieve a poem which satisfies the requirement (Motte 12).

Mathematical constraints, further than the Oulipian requirement toward the organic, fundamentally tend toward the organic because of the relationship between mathematics, beauty, and the laws of the natural world. In his article, “The

Phenomenology of Mathematical Beauty,” Gian-Carlo Rota argues, “The truth of a theorem does not differ from its beauty by a greater degree of objectivity; rather, the distinction between truth and beauty in mathematics is made on the basis of their properties of truth and beauty, when viewed as worldly phenomena in an objective world” (Rota 175). Since mathematics is a way to conceive and make sense of the truths of the natural world, and all poetry is necessarily preoccupied with its interaction with the world by the very fact of its own existence, then it follows that applying mathematics to poetic constraint and form allows the resultant poetry to enter the discourse from a new angle. The tensions of the mathematics as a way to interpret natural and spatial truth and

the poem’s concern with its own space and being, make poetry composed with mathematical constraint continuously organic.

The inheritance of the idea of form as liberation can be traced from the Romantics to the Oulipo. Coleridge and the Romantics saw constraint as giving control of the poem over to the poem itself, while Eliot saw it as a way for the poet to maintain control. Both stipulate, however, that when constraint is used incorrectly, it has the power to dictate the poem, hence creating a discontinuity between the form and the content. Auden then inherits both the concept of poetic self-creation and of the poet as instiller of meaning in his theory on form, combining them to create a theory of constraint that allows for the transcendence from ‘freedom’ to the true statement that a poet is trying to make in a poem. This perspective is then adopted by the Oulipo, who have taken it as a battle cry for their workshop to create new forms, imbued with mathematics and possibility.

Chapter 2: Mathematics in the Sestina and the Petrarchan Sonnet

Formal poetic constraints based on mathematics exist already in the canon of our traditional forms. Two such mathematical forms are the sestina and the Petrarchan, or

Italian, sonnet. The sestina, credited to Arnaut Daniel in the late twelfth century, was treasured by poets such as Dante and Petrarch, falling out of use for centuries until being widely revived by twentieth century American poets, such as Elizabeth Bishop and Ezra

Pound (Turco 251-252, Strand and Boland 24). The Italian sonnet was popularized by

Petrarch, though it was not invented by him, and has gone in and out of style and use since the fourteenth century (Strand and Boland 56-58). Both the sestina and the sonnet reveal organic links between the mathematical structures and the character of the forms.

Section 2.1: The Sestina and the Permutation

The sestina is a form of poetry that has six 6-line stanzas, or sexains, followed by a three-line tercet known as an envoy (Turco 251). The sestina is based on the repetition of end words: the lines of each sexain always end with the same end words, or teleutons

(Turco 251). However, each time, the order of the words changes, and each teleuton never belongs to the same line number twice, such that the teleuton of the third line of the first stanza will not be at the end of the third line in the other stanzas. In Elizabeth

Bishop’s “A Miracle For Breakfast,” the word ‘coffee’ ends the first line of the first stanza and does not do so again, until the envoy:

At six o'clock we were waiting for coffee, waiting for coffee and the charitable crumb that was going to be served from a certain balcony --like kings of old, or like a miracle.

It was still dark. One foot of the sun steadied itself on a long ripple in the river.

The first ferry of the day had just crossed the river. It was so cold we hoped that the coffee would be very hot, seeing that the sun was not going to warm us; and that the crumb would be a loaf each, buttered, by a miracle. At seven a man stepped out on the balcony. (l. 1-12)

Likewise, crumb, miracle, sun, and river never end the same line number as they do in the first stanza, while balcony is not the third teleuton until the envoy, similar to the repetition of coffee, and the order of the repetition if not random.

These changes follow a pattern known as the retrogradatio cruciata, or “the backward crossing”, wherein the words are reweaved in the same way for each sexain and are never in the same order (Krysl 9). The retrogradatio cruciata is the permutation

123456 휎 = ( ), or (163542), a 6-cycle on the symmetric group on 6 elements (S6). 615243

A permutation of a set A is a function 휑: 퐴 → 퐴 that is both one to one and onto… Let 휎 be a permutation of a set A. The equivalence classes in A determined by the equivalence relation [For a, 푏 ∈ 퐴 , let 푎 ~ 푏 iff 푏 = 휎푛 (푎) for some 푛 ∈ ℤ.] are the orbits of 휎… A permutation 휎 ∈ 푆푛 is a cycle if it has at most one orbit containing more than one element. The length of a cycle is the number of elements in its largest orbit. (Fraleigh 76-89)

By examining a sestina, and assigning the numbers one through six to the teleutons, one finds the following pattern:

123456 615243 364125 532614 451362 246531

By executing the permutation once more, the order returns to 123456, thus it is a 6-cycle.

This means that all of the teleutons permute at once and with one another, rather than in smaller groups that would result in repeated end words. The permutation (163542) is one of many 6-cycles on S6. All of the 6-cycles are constructed by retaining one as the first digit and permuting the last 5 digits, meaning that are 6! = 5! = 120 unique permutations 6 that could have been chosen for the sestina. However, the retrogradatio cruciata has another interesting property: it has two domains of imprimitivity, (134) and (652).

An imprimitive group is a group G of one-to-one mappings (permutations) of a set S onto itself, for which there exists a partition of S into a union of disjoint subsets S1,…, Sm, 푚 ≥ 2, with the following properties: the number of elements in at least one of the sets Si is greater than 1; for any permutation 푔 ∈ 퐺 and any i, 1 ≤ 푖 ≤ 푚, there exists a j, 1 ≤ 푗 ≤ 푚, such that g maps Si onto Sj. The subsets Si themselves are called domains of imprimitivity of the group G. (“Imprimitive Group”)

This means that of the 120 permutations of size six, there are only 11 other permutations with the same two properties as (163542): that of being a 6-cycle with the domains of imprimitivity (134) and (652). The other permutations are constructed by finding the combinations of the two domains, holding 1 constant as the first digit, and choosing one digit at a time and alternating between the two domains for the other five.

Other than the cycle (163542), you can construct the following 11 permutations:

(163245), (164532), (164235), (153642), (153246), (154632), (154236), (123645),

(123546), (124635), (124536).

If one were to visualize the movement of the retrogradatio cruciata it would like this:

Figure 1: The Retrogradatio Cruciata

It makes a spiral pattern moving in from 6 to 3, creating compressions and expansions as it makes and breaks word pairings through circumstance, as well as varying the number of lines between the words and themselves (Krysl 7). The fluctuations of the word combinations created by the coiled shaped of the permutation changes the meaning of each teleuton as it evolves with every stanza and every new relationship it makes with the words around it. Marilyn Krysl corroborates this in her article, “Sacred and Profane: The

Sestina as Rite”: “We can say that previous utterances of the teleuton inform each present instance with the knowledge of its own history, at the same time this history is transformed and enlarged by each present instance” (Krysl 12). “A Miracle For

Breakfast” displays the dynamism of connotation in a sestina just in the relationships surrounding the word ‘coffee’ in the first two stanzas. In the first stanza, the narrator is

“waiting for coffee” (1), while in the second stanza, the narrator has a hope pertaining to the coffee. Both the waiting and hoping are horizontally juxtaposed with the coffee to imply a looking forward to the point in time wherein the coffee becomes a reality. The

vertical associations that ‘coffee’ makes in the first and second stanzas, however, are very different. In the first stanza, ‘coffee’ is only adjacent to ‘crumb,’ both of which are types of food, emphasizing the narrator’s desire for sustenance, coffee being especially substantial as it brings energy to the drinker. On the other hand, the second stanza sees

‘coffee’ caught between ‘river’ and ‘sun,’ two elements of the environment and natural world, which had previously been adjoining. The insertion of ‘coffee’ between ‘river’ and

‘sun’ situates her want for nourishment as more than just internal to her, but as a natural fixture like the river is to the landscape and the sun is to the day and the perception of time. These fluctuations of meaning are not the only consequence of the shape of the permutation.

The curled shape also serves to partially alienate the readers, while the recurrence of the same words produces a sense of recognition within them, preventing complete estrangement. The first teleuton of every stanza is the last word of the previous stanza, and the immediate repetition of the last words allows the poem to proceed permuting the words and break with the reader’s expectation for the perpetuation of the previous ordering without alienating them completely. It introduces the idea of the change in word order with the most recently used teleuton in order to create a relationship between each stanza and a feeling of comfort for the reader. It also moves the first end word of the previous stanza only to the second line of the next stanza, maintaining a closeness to the previous order to continue the ease of the change. This type of repetition evokes a sense of familiarity while also executing a drastic change in which those teleutons that were previously furthest apart become partners. Algernon Charles Swinburne may have recognized this and implemented it in his pattern for rhymed sestina as utilized in the

sestina, “Sestina,” which does not utilize a permutation pattern for constructing the ordering of each stanza, but maintains the property of carrying down the last teleuton followed by the first, in order to preserve the movement of the traditional sestina despite removing the stipulation that every teleuton can only belong to a line number once

(Swinburne 46-48). The last three teleutons of each stanza then soothe the discomfort created by the pairing of the first and last teleutons, by permitting a similar grouping to the previous stanza in the 2-4-3 order. The perpetuation of the waiting and hoping for

‘coffee’ in “A Miracle For Breakfast” evokes the property of recalling the familiar that the 2-4-3 cluster does, correlating with ‘coffee’ as the second teleuton in the second stanza. The evolution of the ‘coffee’ as sustenance lacking in the narrator and as natural truth similarly reflects the changes in the teleuton order because of the permutation resulting in the disaffection of the reader.

Of the other 11 permutations that have the same properties as the retrogradatio cruciata, only two have an ordered aesthetic quality, the other 9 being more chaotic than systematic:

Figure 2: (154236) and (153642)

(154236) has a similar spiral pattern to that of the sestina’s permutation, but when the line crosses over itself to reach the fourth word, it places the first word of the previous stanza at the end of the stanza following it, alienating both the word and the reader. The

(154236) permutation would further problematize the sestina form as the first line of each stanza ends with the second-to-last teleuton of the previous stanza, which does not preserve the connection between each stanza as (163542), the retrogradatio cruciata, does. Additionally, it sets the last four teleutons in each stanza as the last, second, fourth, and first, teleutons of the previous sexain, the order of which does not evoke the groupings of the preceding stanzas. Overall, the properties of this permutation and its shape are more chaotic than is ideal for the feelings that the sestina is meant to evoke.

The permutation (153642) likewise sets the first teleutons of each stanza as the fifth of the former, while it does utilize the property of the retrogradatio cruciata of positioning the first as the latter’s second. It also maintains a similar property to the retrogradatio cruciata in that its ending teleutons were originally close together, but

instead of shuffling them, it leaves them in the same 2-3-4 order, which does not create as much difference between the stanzas as is necessitated by the sestina. This ordering pattern would keep the reader fluctuating between two extremes: the stanza beginning with the fifth teleuton begins an extreme level of discomfort, exacerbated by the procession of the first and final teleutons, and then thrown entirely to the opposite of over familiarity by the replication of the 2-3-4 ordering. The fluctuation between these two extremes aggravates the reader’s uneasiness, pushing it toward anxiety, rather than the soothing character of the (163542) permutation.

Another possible construction that would satisfy the aesthetic property of the spiral belonging to the retrogradatio cruciata would be the permutation (142356) which represents the expanding spiral starting at the fourth teleuton.

Figure 3: (142356)

The only other spirals, those beginning with 3 and expanding, or beginning with 1 and contracting, represent 5-cycles leaving one teleuton with a constant position. The

(142356) permutation does the opposite of the ordering of the sestina as it opens each

sexain with teleutons that were close together previously, while ending with those were separated:

123456 435261 256314 361542 514623 642135.

Rather than creating a slight discomfort with new pairings and soothing it with something recognizable, (142356) opens with the similar grouping and ends with the polar grouping, providing no place for an ease of changes for the reader.

Jacques Roubaud also devised of an alternative to the sestina that permutes through 7 teleutons in seven sexains (Glaz and Growney 45). It keeps the retrogradatio permutation but imposes an additional mathematical constraint on the teleuton pattern.

After performing the retrogradatio cruciata, the new form requires that the first teleuton

1234567 be changed according to another permutation 휎 = ( ). The additional 2465371 permutation, however, removes the bond between each stanza that the repeated end teleuton creates, further disrupting the reader’s expectations to the point of emotional havoc. However, he preserves the recognizable end grouping property of the traditional sestina for its ability to alleviate the excess stress created by the insertion of the new word where an established one should be.

The retrogradatio cruciata, as a 6 cycle, is a full permutation cycle. The completion of the cycle reflects the alienating-soothing dynamic of the sestina pattern as well. The closure of the cycle is a circular closure, which requires a return to the origin point to finish, paralleling the quality of the soothing repetitions that characterize the sestina. However, since the original order of the first stanza is never repeated in full, the

circular closure is imperfect, a discontinuity comparable to the disruptions within each stanza that alienate the reader of the sonnet. Thus, the character of the permutation cycle reflects the character of the sestina, and this shows that the composition of the sestina with a constraint based in the 6-cycle is organic.

Section 2.2: The Petrarchan Sonnet and the Pythagorean Theorem

A traditional sonnet has 14 lines of rhymed iambic pentameter, and there are two common variations of the sonnet: the Petrarchan and the Shakespearean. The

Shakespearean, or English sonnet, is based on three quatrains, or 4-line stanzas, followed by two lines, or a couplet. It often follows the rhyme scheme is ABABCDCDEFEFGG

(Turco 264). The Petrarchan, or Italian, sonnet is separated into an octave, or 8-line stanza, and sestet made up of six lines, often following the rhyme pattern

ABBACDDCEFGEFG, though the pattern of the sestet can vary (Turco 263). The format of ideas within the Petrarchan sonnet is based on this octave-sestet division: the octave is the place for the poet to introduce and address their subject, while the final sestet allows the author to complicate or resolve it, following the volta, or “turn” representing the change in the direction of the ideas (Birken and Coon 76). Many have also argued that the division of the octave and sestet for their own purposes resembles the way in which humans think and imposes a “holism of identity” as Matthew Chiasson and Janine Rogers argue in their article “Beauty Bare: The Sonnet Form, Geometry and Aesthetics” (57).

Chiasson and Rogers then claim that Oppenheimer takes the idea of the whole identity to contend that the form “reconcile[s] the split between the poet and his conventional courtly-lover personae that had been the dominant model for lyric poetry into the 13th

century” (Chiasson and Rogers 57). The psychological distinction between the octave and sestet as individual units that make up a single whole lends itself to a 6:8:14 proportion.

However, Oppenheimer argues in his article, “The Origin of the Sonnet,” that:

[The final two lines] seem always, and despite appearances and rhyme scheme, to stand off by themselves, and so in a sense to leave the previous twelve lines as a rhetorically and numerically separate unit within the poem... The numbers to be considered, therefore, are not simply eight and six, or six:eight, but six:eight:twelve. …The proportions of 6:8 and 6:8:12 did play exceedingly interesting roles in history of ideas, …and most particularly in Renaissance architecture, where they describe the "harmonic" proportions of rooms. (Oppenheimer 303)

The proportion of 6:8:12 also lends itself to an approximation of the Golden Ratio, 휑 ≈

1.618 ….

8 + 6 8 12 + 8 12 = 1.75, = 1. 3̅, = 1. 6̅, = 1.5 8 6 12 8

However, the 6:8:12 proportion does not estimate the irrational number 휑 closely enough to embody the Golden Ratio, nor does it represent the sonnet form holistically as it excludes the last two lines.

The Petrarchan sonnet is, however, characterized by another geometric property:

The Pythagorean triangle.

퐴2 + 퐵2 = 퐶2

The Petrarchan has three important numerical elements: 8 for the octave, 6 for the sestet, and the 10 syllables making up each line. The ratio, 8:6:10 satisfies the Pythagorean

Theorem. It is also a multiple of the primitive Pythagorean triple, 3:4:5, the smallest ratio that satisfies the Theorem (Weisstein). In fact, as the octave can be divided into quatrains and the sestet into tercets by the rhyme, while each line can be divided into two sections

of pentameter, the 3:4:5 triple is not only alluded to but contained in the sonnet form.

From this, Chiasson and Rogers conclude that:

The sestet (6) and the octave (8) are like the two perpendicular legs of the right- angled triangle, representing the distinct poetic split or fork. The thing that structurally binds them and reconciles the split is the iambic pentameter (10), which persists through the entire poem and would represent the hypotenuse of a right-angled triangle; ultimately, the hypotenuse completes the triangle and makes it a closed geometrical figure. (Chiasson and Rogers 57)

The psychology of the sonnet is not only symbolized by its triangular shape, it is entirely intrinsic to it: the closure of identity is embedded within the geometrical closure of the

Pythagorean triangle, just as the opposing identities are rooted in the right angle of the triangle, exhibiting its structural solidarity.

Figure 4: The Petrarchan Sonnet as a Triangle

The Pythagorean triangle has other significant geometrical and aesthetic implications to the sonnet:

… there are two aesthetic qualities of the Pythagorean Theorem and Pythagorean Triples that should be of special interest to readers of the sonnet: both theorems are generative and spatial in nature: they are generative in the sense that they are procreative and prolific, and they are spatial in the sense that they interrogate the meaning of physical and intellectual space… (Chaisson and Rogers 55).

The Theorem is generative in its infinite applications beyond Euclidian geometry, as illustrated by its incorporation into the composition of the sonnet as a constraint, and the primitive Triple is generative in that it is the most basic proportion from which all other

Pythagorean ratios are composed. The sonnet is likewise procreative as it inspires so many other forms and motivates experimentation and innovation similar to the adaptability of the Pythagorean Theorem, as well as its being the formula that has generated every sonnet ever written and other poems as well, akin to the primitive Triple

(Chiasson and Rogers 55).

This is exemplified by the poetry collection, Cent mille milliards de poèmes, by

Raymond Queneau, of the Oulipo. It is a collection of ten sonnets wherein each line of each poem is printed on a separate strip of card stock, so that you can open the book to a poem composed of a line from each of the ten sonnets, and the poems are all written to make sense grammatically and by rhyme with one another (Motte 3). The ten sonnets of fourteen individual lines each, then produce 1014, or 100,000,000,000,000 unique sonnets, and so it is likely that by choosing lines randomly, the reader in collaboration with the book itself would create a new sonnet that has never been read before. Cent mille milliards de poèmes is therefore potentially generative to an infinite degree as it offers a new poem to be read every time it is opened. If someone were to read a new sonnet every five minutes, for ten hours a day, it would take approximately 2.3 billion years to read them all, a relative infinity compared to the length of a human life.

60 100000000000000 833333333333 → 12 ∗ 10 = 120, → = 2283105022.83 5 120 365

The Pythagorean Theorem is also a significant element of the theory of Euclidean and other spaces, and therefore offers insight on the relationships of geometric proportions to wholes and the unity of them in space (Chiasson and Rogers 56). The

Pythagorean Triple is a representation of one such spatial proportion, as well as being representative of an entire form of geometric aesthetics and unity. The spatial relationship of parts to the whole exhibited by the two theorems relates back to the union of identity within the sonnet, as well as to the physical shape of the poem and how it relates to itself and everything around it.

The generative and spatial properties that are common to both the Pythagorean

Theorem and the sonnet prove that the sonnet exhibits the “consequences of the mathematical theory that it illustrates,” as the Oulipians require in their theory of the organic. The spatial consequence also serves to relate the sonnet to the organic basis for mathematical constraint in the way that it contributes another concept to the discourse of the spatial component of poetry and mathematical truth. Furthermore, the closure of the triangle of the sonnet shape that correlates with the closure of identity and thought that characterizes the Petrarchan sonnet not only satisfies the Oulipian definition of the organic, but Coleridge’s as well. The holism of thought and identity in the sonnet simultaneously composes the triangular shape while the closed geometrical figure engenders the holism of thought and identity.

Chapter 3: The Fib, the S+7 Algorithm, and Computer-Generated Poetry

This chapter concerns mathematical constraints in poetry as they have arisen after

1900 and with documented mathematical intension. It is common for poets to experiment with formal poetic constraints based on easily recognizable patterns, and it not surprising that many patterns are based on arithmetic and geometric sequences, since counting is an inherent process to all poetic metre. The Fib is based on one of the most well-known sequences, the Fibonacci Sequence. The S+7 Algorithm, on the other hand, is method of transforming works of literature in order to create original images and phrases. Computer generated poetry is still in the early stages of possibility as programmers try to replicate creativity in A.I.

Section 3.1: The Fib

A Fib is a type of poem that is based on a formal constraint constructed from the

Fibonacci sequence (Pincus). Its popularity is generally attributed to Gregory Pincus who was the first to name the form a “Fib,” in 2006, though it is unlikely he was the first to ever write a poem with a Fibonacci form. The first example he gives of the constraint is the following poem:

One Small, Precise, Poetic, Spiraling mixture: Math plus poetry yields the Fib. (Pincus)

Each line contains the number of syllables as the sum of the syllables of the two previous lines. The first two lines would contain one syllable each; the third line would contain

two syllables; the fourth, three; the fifth, five; and the sixth, eight. This pattern can be continued further by following the recursive definition for the Fibonacci sequence:

푓푛 = 푓푛−1 + 푓푛−2 , 푓표푟 푛 ≥ 3 where n is the line number and fi is the number of syllables in line i. However, eight is the smallest number of syllables Pincus suggests a poet use, and no more are necessary. A similar form can be constructed by following the Fib structure by counting words instead of syllables, but this form would be more reducible into individual parts and syllable sounds and would therefore lose the aesthetics of the Fibonacci sequence. For example, the first two lines could be “Forget/me,” which visually would seem to follow the ‘1,1’ sequence (as in 1,1,2,3,5,…), but audibly the listener would detect the ‘2,1’ of the syllables, and would not detect the Fibonacci pattern.

The constrained quality of this form allows the poet no space for anything superfluous: he or she must be precise in his/her diction and grammar in order to achieve what they wish in just the number of syllables allowed per line.

The Fibonacci sequence brings with it aesthetic consequences as well: it approximates the Golden Ratio, or 휑. Luca Pacioli composed De Divina Proportione, or

On the Divine Proportion published 1509, on the proportion

(푎 + 푏) 푏 = , 휑 = 1.618 … 푏 푎

In it, Pacioli introduces the ideas that laid the ground work for the study of the classical aesthetics implied by the Golden Ratio. He claims that “… many of the qualities which I find in our proportion, are also those which in this our very useful discourse, we attribute to God” (Pacioli 12). The first is the unity of the proportion being self-contained, as God is. The second is Trinity: as God is the union of the Holy Trinity, so is the Golden ratio

the creation of the three quantities, 푎 + 푏, a, and b. The third quality relates the irrationality of 휑 to the incomprehensible and undefinable nature of God in language.

The fourth quality is that both God and the proportion are continuous and discrete. The relationship of 휑 to God brings an impression of the divine to the poems created with the

Fib constraint. The properties of the Ratio that are comparable to the divine are internalized by the poem as soon as it formed, each line relating to those around it in the proportion approaching 휑 more closely the longer the poem goes on. The poem’s contribution to the discourse of universal truth is therefore furthered by the Fib’s relationship to God and the divine. The application of the Fib constraint to a discourse that the poetry already participates in makes it an organic form.

Adrian Bejan and J. Peder Zane offer a cognitive explanation for the human preference for shapes containing or approximating the golden ratio in their book Design in Nature: How the Constructal Law Governs Evolution in Biology, Physics, Technology, and Social Organization (2013). They conclude that “Shapes that resemble the golden ratio facilitate the scanning of images and their transmission through vision organs to the brain. The speeding up of this flow goes hand in hand with the architectures of the nervous system in the eye and brain” (Bejan and Zane 233). The proportions of shapes that resemble the golden ratio are easier for humans’ eyes to scan and transmit because the speed at which they can interpret horizontally is faster than the speed that they can interpret the shape vertically. The golden ratio imposes a shape that can be scanned both vertically and horizontally in the same amount of time based on the speed discrepancy between the two directions. The consistency of the time taken in both directions makes the process of transmitting the image to the brain for it to be interpreted far more efficient

and so humans unconsciously prefer to look at these shapes. The ‘divinity’ of 휑 can then be extended to the poem as it relates to the Golden Ratio: the cognitive implications of the ratio provide a level of intrinsic beauty to the shape that the form takes, especially if the writer extends it past eight syllables, as the sequence approximates closer to 휑, the larger the numbers become. The relationship of the Fibonacci sequence and the Golden

Ratio to the natural world, such as in the nautilus, and to the innate cognitive and biological processes of perception, further establishes the Fib as an organic form.

Section 3.2: The S+7 Algorithm

The S+7 algorithm is a process by which new poems are constructed from existing ones (Motte 61). The algorithm entails replacing each substantive, or noun, in a chosen poem with the seventh noun following it in a prechosen dictionary, for example

‘rainbow’ would become ‘rake’ using the dictionary I use below. The poet can make the constraint their own by changing the number of words they must shift in the dictionary, or by using adjectives instead of nouns if they want to write about a particular subject, thereby adding their own creativity into the process. An algorithmic pattern is necessary to achieve a semblance of order over the resultant text, because it maintains a level of objectivity within the subjectivity of the rules that the poets chooses to follow in their individual variations of the algorithm. Imposing an ordered pattern over the fragmentation that the algorithm produces also recalls the Romantic necessity for form to bring shape to the “untamed internal energies” as I said above.

Likewise, the algorithm can be seen as a mapping of the set of words you wish to exchange in a poem, defined by a property such as the quality of being nouns, to the set

of words you wish to exchange them with, defined by the property of ‘adding 7’ to each word. Using a dictionary with the algorithm, would be an injective map, unless the poem originally used every single word in the dictionary. Using a ‘word-bank’ other than a dictionary that is the same size as the set of initial words, or by forming the subset of just the words you would be exchanging them with from the dictionary, the author would be forming a bijective map. It would only be surjective, however, if two of the initial words were the same word, because they are unique by their location in the poem, but map to the same word in the image of the set.

Figure 5: The S+7 as a Bijection

The S+7 algorithm was invented by the Oulipian mathematician Jean Lescure, as a way for an author to relieve themselves of the limitations imposed by our limited experiences and thoughts at the moment of critical conception (Motte 58). The resulting poem will almost certainly be nonsensical, but the arbitrary word combinations can create unique phrases and images that an author may not have been able to conceive of on their own (Growney 4). The subjective constraints can create objective ideas that the author

can then shift into a new context, and reinject with their own experiences, emotions, and connotations.

The ‘patient’ poem, as it is receiving the transplanted phrases/ideas/images, would be written in a new set of constraints as well. The organic, as an antecedent to the

Oulipian theory of constraint, would suggest that a poem utilizing the phrases adopted by a poem in this way should allude to both the algorithm as well as the poem that gave birth to them. It might be written in the same form as the ‘donor’ poem or address the same subject. It might even discuss the transplantation itself, the way in which the number 7 was ‘added’ and how this addition affected the change on the donor. Moreover, as the

S+7 is producing a new poem from the ‘donor’ poem, the ‘donor’ poem is simultaneously, and therefore, organically producing the process of the change by providing itself as material to transform.

Section 3.3: Computer-Generated Poetry

Computer generated poetry also falls under the category of a mathematically constructed formal constraint as the computer must follow a strict program of randomizing, syntactical structures, and extensive grammar rules to write lines of verse that are recognizable as such and that make sense grammatically. A poem composed of complete ideas and sentences would require a program to understand grammatical conditionals, as well as the conditions that lead to exceptions to these rules. At the most basic levels it would require a vast string of conditional and recursive statements that determine the circumstances of the sentences that it is writing in order to match pluralization, maintain verb tenses, and use accurate punctuation. It is yet more

complicated to write poetry under formal constraints, because the code must at least be able to determine words that rhyme and how many syllables are in a word to account for metrical constraints.

Although the poems that result will usually be categorically abstract, the best programs can produce poems that pass the Turing Test. A Turing Test is “a test to establish the existence of artificial intelligence… with the understanding that if the interrogator is unable to correctly identify which responder is human the computer has demonstrated thinking ability comparable to a human's” (“Turing Test”). Many programmers that have written poetry-generating programs have submitted their poems to literature magazines, and the magazines have accepted them. The poems passed the

Turing Test by fooling experienced and selective poetry readers and bringing them enough enjoyment that they thought others should read them too.

The most significant element of poetry writing that is lost when a computer parses is that of creativity. Creativity is approximated by randomizing and chance, but true creativity is difficult to replicate in A.I, because, as Alexander Pope opens his poem,

“Sound and Sense,” “True ease in writing comes from art, not chance” (Pope 72). Sarah

Harmon is developing a learning program that creates similes in an attempt to begin to replicate the sort of creativity required for an A.I to write a poem comparable to

Wordsworth or Shakespeare. She has defined parameters for producing figurative language that approximate the human creative process. Her program, named FIGURE8, considers clarity, novelty, aptness, unpredictability, and prosody in its process to develop imaginative comparisons (Harmon, 72-73). FIGURE8 assesses its own similes by data mining and comparing its language to that of others, it tries to avoid clichés to seem more

unpredictable, but also checks that the things its comparing are contextually related, i.e. certainly few people have compared a fork to a couch, but it also makes no sense to do so in most situations. By scoring itself in the areas of clarity and likeability it attempts to learn from its mistakes and write better comparisons in the future.

Extending estimated creativity toward an entire poem requires even more learning by the program. It should know how and when to use all poetic devices, including figurative language, as well as having to understand enough about human emotion to know where to emphasize certain words and sounds in the verse. Syntax also becomes far more meaningful, and the A.I. program must be more dynamic in its decision making on sentence structure than would be a basic generator.

Section 3.4: Coding the Fib and the S+7

I wrote very basic programs meant to ‘write’ a Fib poem (see Appendix A), as well as an altered version of the S+7 (see Appendix B), to exhibit the problems of computer-generated poetry and to propose ways that the problem of creativity can be lightened. Neither program contains conditionals for grammar.

The Fib program took words of varying syllable lengths in multiple inputs by the user and randomly chose words from these banks fill in the lines of the poem. It also uses both if-statements and recursion to randomly decide the syllable lengths of the words in each line, following the syllable sums as intended by the Fibonacci sequence. The resultant poems make very little sense, as no grammar was imposed on them.

One-syllable words: a, I, me, them, this, it, that, an, is, if, not, yes, no, cat, dog,

door, muse, run, look, up, down, book, tree, sit, stand

Two-syllable words: music, apple, lover, sadden, crying, starlight, orange, carries,

pocket, pizza

Three-syllable words: depression, beauty, telephone, circumstance, Tiffany

Four-syllable words: beautiful, celebration, television, worrying, ordinary

Five-syllable words: personality, anniversary, extraordinary

Resultant poems: cat me yes door no stand carries yes I lover music dog Tiffany crying run worrying I beautiful cat extraordinary depression beautiful no depression worrying a telephone anniversary

This program takes user inputs for all of the words, addressing the inclusion of a kind of creativity through the user’s imposition of a concept prior to the poetic synthesis.

The user can choose the tone and subject of the poem by inputting words that suggest the ideas that they would like depicted and conveyed through the poem. However, there is no way to force the program to choose the words that the user would prefer, so the poem still remains objective and random, even though its themes or ideas may be limited.

I chose to input words with no specific theme or subject in mind. I also tried to include a variety of nouns, verbs, and adjectives so that the poems would be more than simple lists of objects, but could possibly form full thoughts, even if they were disjointed ones. The most obvious drawback to the outputs of this program is the lack of grammatical structure, which seems to almost always prevent its production of useful or though-provoking images, except by accident. For example, the first two poems all used the words “worrying,” “beautiful,” and “depression,” centering on a theme which is not entirely dissonant from the erratic diction making up the whole poem, as such chaotic

behavior can be a worrying symptom of depression. However, it was I, the user, that chose those words for the program to select and since there were relatively few words in the word-bank that I created it is not surprising that the three words came up together. On the other hand, the last poem engenders the very clear though unpredictable image of an

“extraordinary telephone anniversary.” The phrase “extraordinary telephone anniversary” exhibits the characteristics of the S+7 in that it is an original and possibly innovative image that a writer could transplant into another work or take as inspiration for a new work altogether; it borders on the whimsical but could provide brilliant imagery in just the right context. A grammar module would be the first and most significant improvement I would make to the code for this program in order for it to construct sensible phrases and sentences.

The S+7 program replaces every fourth word of the poem with the seventh word following it in a dictionary (“Index of /~Swei/CSC550”), since I chose not to code grammar rules into it, so the name of the algorithm that the program follows would be

W4+7, rather than S+7. The program takes in each line of a poem as a separate line of input from the user, then it uses array slicing to copy every fourth word into a new array so that it can use recursion and nested if-statements, or if-statements within if-statements, to replace the words and then it replaces the original words with the new words in a new array. It then prints the new poem for the user’s viewing pleasure. I chose Mark Twain’s limerick, “A man hired by John Smith and Co.” and the poem does not make complete sense due to the exchanges made, but it is quantifiably less absurd than the poems that were produced by the Fib generator.

Figure 6: “A man hired by John Smith and Co.” Transformation

The input poem fulfills a minimal definition of ‘creativity’ because it was written by a human, implying the author’s purpose and style, rather than a computer’s attempt at imitating it. The program internalizes the creativity of the poem and produces a poem that is more dynamic and comes closer to replicating a poem written creatively by a person.

The program is, however, only writing about a fourth of the poem, rather than the whole poem, and the poem still makes little sense.

I chose to input a limerick as a short example, but a longer poem might provide more opportunities for the algorithm to achieve more interesting phrases and images as supporters of the form have claimed. In the case of the limerick above, only one of the changes is a logical statement, being line four, “dumping dirt neath his door,” which might easily be exchanged in the original poem, without consideration for Twain’s purpose in using the word “near,” rather than ‘neath.’ The change of the words across classes is the greatest flaw in this program, resulting in most of the absurd phrases that the resultant poem contains. The only two that remain the same across the change are the prepositions “near” and “neath,” and the nouns “drivers” and “drool.” The first

improvements I would make to the code for this program is to include a module to categorize the words into their parts of speech and to include a larger dictionary of words.

The Fib as composed by the mechanical constructor of a program will always result in a poem with a mechanical form since the process of randomly choosing words that fit the syllable requirements creates an utter and complete disconnect between the

‘content’ and the shape that it takes. However, it is worth examining the code of the program itself as it is a product of the Fib constraint as well. The code that generates each line gets longer and more complicated as the number of syllables gets larger and the lines themselves get longer. The embodying of the Fibonacci sequence in both the generated poetry and the code the produces the poetry establishes the code as an organic text.

The S+7 algorithm, whether executed directly by a user or by a computer will result in the same poem either way, which can be organic if applied to a ‘patient’ poem in an organic way. However, again the program itself has poetic and organic merit. There is symmetry in the process for decomposing the poem into the set of words for exchange and the process for recomposing the poem with the set’s image in place of the original set. The symmetry also creates a layered effect in the visual of the code that is analogous to the layers of subjectivity that separate the user, as the ‘poet,’ by the program, from the outputted poem. The coded S+7 is therefore not only organic if applied organically, but it is so in the way that it embodies the distance between the poet and the reader as the distance between the user and the output.

Conclusion:

The notion of the organic can be traced from Coleridge in the nineteenth century all the way to the Oulipo’s present theory on the role that formal poetic constraint plays in the poet’s writing process. Coleridge requirement that form and content emerge simultaneously seems to conflict with the Oulipian philosophy that often prefers to regard constraints as literature of their own merit, but both value the relationship between a constraint and the content of the poem that takes on said constraint.

The tension between discomfort and relief that characterizes the sestina is reflected organically by the incomplete circularity of the permutation cycle that defines the variations of the teleutons as it approaches perfection but never truly realizes it. The marriage of the closure of the Pythagorean triangle with the closure of the poet’s identity and thought process is, likewise, an organic relationship based on the mirroring of the mathematical principle in the nature of the poetic form it generates.

The inherent cognitive and divine beauty of the Fibonacci sequence brings the organic to the Fib through its relationship to the natural world and its complementarity as an additional source of enjoyment for the reader. The simultaneous creation enacted by the S+7 Algorithm as it creates a new poem while the poet is bringing the algorithm, itself, to life, displays the organicism of the constraint. The programs that bring these constraints to life are similarly organic in the way that they visually reflect the additive and layered properties of the constraints, respectively.

The ways that each of these constraints are individually organic supports the idea that mathematical constraint in poetry is innately organic based on the spatial properties

and interactions with the natural world that they bring to the discourse surrounding poetry’s place and space in the world.

Appendix A

The Fib Program import random single=input('Enter 25 one-syllable words to begin your Fib. Separate them with spaces only.') double=input('Enter 10 two-syllable words.') triple=input('Enter 5 three-syllable words.') quad=input('Enter 5 four-syllable words.') quint=input('Enter 3 five-syllable words.') l_1=single.split() l_2=double.split() l_3=triple.split() l_4=quad.split() l_5=quint.split() line1=random.choice(l_1) l_1.remove(line1) line2=random.choice(l_1) l_1.remove(line2) length_var_3=random.choice((True,False)) if length_var_3: line3=random.choice(l_2) l_2.remove(line3) else: line3_1=random.choice(l_1) l_1.remove(line3_1) line3_2=random.choice(l_1) l_1.remove(line3_2) line3=line3_1+(' ')+line3_2 length_var_4=random.randint(1,3) if length_var_4==1: line4_1=random.choice(l_1) l_1.remove(line4_1) lv42=random.choice((True,False)) if lv42: line4_2=random.choice(l_2) l_2.remove(line4_2) line4=line4_1+(' ')+line4_2 else: line4_2=random.choice(l_1) l_1.remove(line4_2)

line4_3=random.choice(l_1) l_1.remove(line4_3) line4=line4_1+(' ')+line4_2+(' ')+line4_3 elif length_var_4==2: line4_1=random.choice(l_2) l_2.remove(line4_1) line4_2=random.choice(l_1) l_1.remove(line4_2) line4=line4_1+(' ')+line4_2 else: line4=random.choice(l_3) l_3.remove(line4) sylls5=[] line5=[] sylls5.append(random.randint(1,5)) rem1=5-sum(sylls5) while rem1>0: sylls5.append(random.randint(1,rem1)) rem1=5-sum(sylls5) for i in sylls5: if i==1: word=random.choice(l_1) l_1.remove(word) line5.append(word) elif i==2: word=random.choice(l_2) l_2.remove(word) line5.append(word) elif i==3: word=random.choice(l_3) l_3.remove(word) line5.append(word) elif i==4: word=random.choice(l_4) l_4.remove(word) line5.append(word) elif i==5: word=random.choice(l_5) l_5.remove(word) line5.append(word) sylls6=[] line6=[] sylls6.append(random.randint(1,5)) rem=8-sum(sylls6) while rem>0:

sylls6.append(random.randint(1,5)) rem=8-sum(sylls6) if sum(sylls6)>8: sylls6=sylls6[:-1] rem=8-sum(sylls6) while rem>0: sylls6.append(random.randint(1,rem)) rem=8-sum(sylls6) for i in sylls6: if i==1: word=random.choice(l_1) l_1.remove(word) line6.append(word) elif i==2: word=random.choice(l_2) l_2.remove(word) line6.append(word) elif i==3: word=random.choice(l_3) l_3.remove(word) line6.append(word) elif i==4: word=random.choice(l_4) l_4.remove(word) line6.append(word) elif i==5: word=random.choice(l_5) l_5.remove(word) line6.append(word) print(line1) print(line2) print(line3) print(line4) print(" ".join(line5)) print(" ".join(line6))

Appendix B

The W4+7 Program dictionary=open("dictionary.txt","r") allwords=dictionary.readlines() poem=[] i=input('Please enter a line of the poem. When finished enter \'eNd\'.') poem.append(i) while i!="eND": i=input('Please enter a line of the poem. When finished enter \'eNd\'.') poem.append(i) poem=poem[:-1] poem=[phrase.split(' ') for phrase in poem] poemstr=[] for line in poem: for word in line: poemstr.append(word) #print(poemstr) wordschng=poemstr[3::4] #print(wordschng) newwords=[] for i in wordschng: if i in allwords: pos=allwords.index(i) newwords.append(allwords[pos+7][:-1]) else: allwords.append(i) allwords.sort() pos=allwords.index(i) newwords.append(allwords[pos+7][:-1]) #print(newwords) newpoem=poem #print(newpoem) def find_in_list_of_list(mylist, char): for sub_list in mylist: if char in sub_list: return (mylist.index(sub_list), sub_list.index(char)) raise ValueError("'{char}' is not in list".format(char = char)) for word in wordschng:

ind=find_in_list_of_list(poem,word) newpoem[ind[0]][ind[1]]=newwords[wordschng.index(word)] for line in newpoem: print(' '.join(line))

Works Cited

Auden, W. H. The Dyer's Hand and Other Essays. Vintage Books, 1989.

Bejan, Adrian, and J. Peder Zane. Design in Nature: How the Constructal Law Governs

Evolution in Biology, Physics, Technology, and Social Organization. Anchor

Books, 2013.

Birken, Marcia, and Anne Christine Coon. Discovering Patterns in Mathematics and

Poetry. Rodopi, 2008.

Bishop, Elizabeth. “A Miracle For Breakfast.” North & South. Houghton Mifflin, 1946,

pp. 18-20.

Chiasson, Matthew, and Janine Rogers. “Beauty Bare: The Sonnet Form, Geometry and

Aesthetics.” The Journal of Literature and Science, vol. 2, no. 1, 2009, pp. 48–

64., doi:10.12929/jls.02.1.03.

Coleridge, Samuel Taylor. The Complete Works of Samuel Taylor Coleridge. Edited by

W. G. T. Shedd, vol. 4, Harper & Brothers, Franklin Square, 1884.

Curran, Stuart. Poetic Form and British Romanticism. Oxford University Press, 1986.

Eagleton, Terry. How to Read a Poem. Blackwell, 2007.

Eliot, T S. “Reflections on Vers Libre.” T. S. Eliot, tseliot.com/essays/reflections-on-

vers-libre.

Fraleigh, John B. A First Course in Abstract Algebra. 7th ed., Pearson Education Inc.,

2003.

Glaz, Sarah, and JoAnne Simpson Growney. Strange Attractors: Poems of Love and

Mathematics. CRC Press, 2010.

Glenday, John. “The Creative Process – Poetry & Prose.” Northwords Now, no. 27, 2014,

pp. 8–9., northwordsnow.co.uk/userfiles/file/issues/issue27/NNow_27_web.pdf.

Growney, Joanne. “Mathematics Influences Poetry.” Journal of Mathematics and the

Arts, vol. 2, no. 1, Mar. 2008, pp. 1–7. Taylor & Francis,

doi:10.1080/17513470801975615.

Haffenden, John. W.H. Auden: The Critical Heritage. Routledge, 1997.

Harmon, Sarah. “FIGURE8: A Novel System for Generating and Evaluating Figurative

Language.” Computational Creativity, Park City, UT, June29-July 2, 2015, edited

by Hannu Toivonen, Simon Colton, Michael Cook, Dan Ventura, Brigham Young

University, 2015.

Holmes, Oliver Wendell. Over the Teacups. Houghton, Mifflin and Company, 1896.

“Imprimitive Group.” Encyclopedia of Mathematics, Springer,

www.encyclopediaofmath.org/index.php/Imprimitive_group.

“Index of /~Swei/CSC550.” People.sju.edu, People.sju.edu,

people.sju.edu/~swei/CSC550/dictionary.txt.

Krysl, Marilyn. “Sacred and Profane: The Sestina as Rite.” The American Poetry Review,

vol. 33, no. 2, 1 Mar. 2004, pp. 7–12. JSTOR,

www.jstor.org/stable/10.2307/20682474?refreqid=search-

gateway:779a01de0ada56d3c2cc848103f07fe9.

Motte, Warren. Oulipo: A Primer of Potential Literature. Dalkey Archive Press, 2008.

Oppenheimer, Paul. “The Origin of the Sonnet.” Comparative Literature, vol. 34, no. 4,

1982, p. 289. JSTOR, doi:10.2307/1771151.

Pacioli, Luca. On the Divine Proportion. Translated by Dr. Jonathan Tennenbaum, John

P. Scialdone, Richard Sanders, 2005.

Pincus, Gregory. “The Fib.” GottaBook, Blogspot, 1 Apr. 2006,

gottabook.blogspot.com/2006/04/fib.html.

Pope, Alexander. The Complete Poetical Works of Alexander Pope. Student's Cambridge

Edition ed., Houghton Mifflin Company, 1903.

Rota, Gian-Carlo. “The Phenomenology of Mathematical Beauty.” Synthese, vol. 111,

no. 2, 1997, pp. 171–182. JSTOR, JSTOR, www.jstor.org/stable/20117626.

Russell, Bertrand. Mysticism and Logic: and Other Essays. Longmans, Green and Co.,

1919.

Strand, Mark, and Eavan Boland. The Making of a Poem: A Norton Anthology of Poetic

Forms. W.W. Norton & Company, 2000.

Swinburne, Algernon C. “Sestina.” Poems & Ballads: Second & Third Series. Chatto and

Windus, 1908, pp. 46-48.

Turco, Lewis. The Book of Forms: A Handbook of Poetics. Third ed., University Press of

New England, 2000.

“Turing Test.” Merriam-Webster, Merriam-Webster, www.merriam-

webster.com/dictionary/Turing%20test.

Twain, Mark. “A Man Hired By John Smith And Co.” PoemHunter.com, Keltis,

www.poemhunter.com/poem/a-man-hired-by-john-smith-and-co/.

Weisstein, Eric W. “Pythagorean Triple.” Wolfram MathWorld, Wolfram MathWorld--A

Wolfram Web Resource, mathworld.wolfram.com/PythagoreanTriple.html.