àmanifold fly-ways that always end
in petals and pith, the numinous paths.
û David Roderick,
"Sonnet for a Novice Beekeeper"

Poet and mathematician Paul ValTry begins his essay on poetic form, "Les Coquillages", by noting that "the mollusk exudes its shell," letting the building material "seep through" the mantle to give it protective and coherent shape. Poems are not all that different and indeed malacology and the arts in general have much in common. "As we say, a 'sonnet,' an 'ode,' a 'sonata,' or a 'fugue,' to designate well-defined forms, so we say a 'conch,' a 'helmet,' a 'cameo'àall of them names of shells."
But poets in particular, unlike mollusks and snails, do not as ValTry says "derive the material of their works from their own substance" but rather from "a specified application of their minds." In other words, the poet as a divided being must strive artifically to be "natural," duplicating at a higher or conscious level of aspiration the very artificiality of the natural and effortless phenomenon itself. The poet works with numbers just as the mollusk, for example, develops its architecture according to the Fibonacci sequence in which each number, each whorl, is the sum of the preceding two. Poetic forms and genres are not adventitious: though they grow artifically, they grow like natural objects, arraying words in elegant and memorable order to accomplish a meaningful task and to ensure survivability. This is what ValTry denominates as perfection in art, the desire manifest in human production for "the sureness of execution, the inner necessity, the indissoluble bond between form and material that are revealed to us in the humblest of shells."
And in the humblest of poems too. Consider the terse, psalm-like lyric by Gerard Manley Hopkins, "Pied Beauty."

Glory be to God for dappled thingsù
For skies of couple-colour as a brinded cow;
For rose-moles all in stipple upon trout that swim;
Fresh-firecoal chestnut-falls; finches' wings;
Landscape plotted and piecedùfold, fallow, and plough; And ßll trßdes, their gear and tackle and trim.

All things counter, original, spare, strange;
Whatever is fickle, freckled (who knows how?)
With swift, slow; sweet, sour; adazzle, dim;
He fathers-forth whose beauty is past change:
Praise him.

Perhaps the first thing to remark is that "Pied Beauty" is a rather short paean of ten lines tagged by a hemistich and arranged as two grammatical sentences. If we examine it technically, we see that it falls into two syntactical units of thought of 6 and 41+2 lines, the first part formulated as a string of concrete particulars and the second as a general or abstract statement commenting on the nature of these particulars and ending with a hortative. Considered mathematically, in consolidating a ratio of 6 to 41+2, it mimes exactly the relation of octave to sestet, or 8 to 6, of the staple Petrarchan sonnet. Further, it conforms to the Petrarchan model not only structurally but also thematically insofar as the second section develops and qualifies the longer, antecedent passageùthe Petrarchan mode of thinking. The template Hopkins has followed is plainly the canonical 14-line "Italian" sonnet but the result is what he calls a "curtal sonnet", an abbreviated version of the original paradigm yet wholly consistent with it.
All this is obvious enough but the real question is: why? A little reflection on the subject he is addressing will make his purpose clear. Hopkins begins by giving glory to God for the mutable and variegated nature of the Creation, a world in constant motion swarming with intricate specifics in both its natural and human manifestations. But in the codaù"coda", deriving from the Latin word for "tail", suggests another reason why the poet called this kind of sonnet curtal or "curtailed", and why Milton, for that matter, called his extended sonnets "caudated"ùwe are now exhorted to praise God not only for the iridescent diversity of the Creation but for the paradox that the Author of change is Himself "past change." As in Spenser's "Cantos of Mutability" from The Faery Queene, which fathers-forth the very poem we are reading, we come to recognize that the beauty of the changeable is both caused and transcended by the beauty of the permanent, that the temporal is subsumed in the eternal, and that we can appreciate or construe the latter only through the medium of the former. The formal structure of the poem is consequently seen to exist in strict analogical symmetry with its theme. Hopkins has renovated, modified, changed the Petrarchan sonnet, giving us a poem which at first reading appears to have no affiliation with it, but he has simultaneously preserved the mathematical ratio of octave to sestet, establishing the permanence of the poetic tradition in which individual poems may vary, complexify and proliferate. While altering the gear and tackle and trim of the sonnet, he has succeeded in retaining its formal unity and changelessness. Hopkins is playing with a theologicopoetic version of Group Theory, pursuing a topological study of the Creation in which invariants are conserved under transformations.
Both the form and the theme of the poem would seem to spring from Augustine's gloss on the Creation in Epistles, 5, which Hopkins no doubt had clearly in mind when he set about composing it: "God isàthe unchanged Creator of all things that changeà[Who] adds, abolishes, curtails, increases or diminishes." But the poem also gestures toward Aquinas's Summa Theologica I, Q. 1, Art. 9 where we read "It is natural to man to attain to intellectual truths through sensible things, because all our knowledge originates in senseàFor what God is not is clearer to us than what He is. Therefore similitudes drawn from things farthest away from God form within us a truer estimate that God is above whatever we may think of Him," and to the conceptual fount of these reflections in St. Paul, Romans 1: 20: "For the invisible things of him from the creation of the world are àunderstood by the things that are made, even his eternal power and Godhead." This latter passage itself stems materiallyùor so it seems to meùfrom Psalms 148 whose paratactic language "Pied Beauty" recruits and modulates: "Fire, and hail; snow, and vapours; stormy windà praise the name of the Lord."
Thus even the way in which Hopkins assimilated his sources and allowed them to "seep through" parallels the dialectic of the one and the many which informs the poem. As a theologian relying on a variety of literary, doctrinal and scriptural texts, Hopkins attempted to clarify the singular relation between the temporal and the eternal, between the plurality of the world and its sole, unchanging Creator. But as a poet, he played with formal computations to construct a quasi-mathematical artifice, a poem abiding faithfully by the law of numbers which generates identity in difference but also difference in identity. The poem seems "natural," like an unpremeditated meditation, but, like ValTry's mollusk, it unfolds according to an inner law of precedent commutations which confer stability upon it. Stability, however, is not predictability. The only thing predictable about a good poem is the surprise we feel as the law of numbers produces something entirely individual within its generic envelope. Poetic form may be said to carry a mathematical signatureùsupremely evident in the sonnet but in other modes as well, such as the villanelle or the sestinaùand may be regarded as "natural" only in the degree to which it is internally structured. Yet it is precisely such internal structuration that permits both the insoluble mysteriousness and the bounded uniqueness of the thing made. ò