the square triangle

The sonnet form contains within it a mathematical theorem that is very beautiful – and quite literal: the Petrarchan English sonnet, with the octave/sestet/iambic pentameter stanza, embodies two geometrical constructs exactly: the Pythagorean Theorem and the Primitive Pythagorean Triple.

We can construct the Pythagorean Theorem out of the three primary numeric components of the sonnet; 8 (the octave), 6 (the sestet) and 10 (the number of syllables in each line): 82+62=102; 64+36=100. In this respect, the sonnet form does not merely represent the Pythagorean Theorem, it both is it and it does it. The form is symbolic but it also enacts the elegant mathematical form. But not only does the Italian sonnet form embody – or perform – one of the classically beautiful mathematical theorems, but it divides down to another essential mathematical beauty: the Primitive Pythagorean Triple.

If we take any triangle and multiply all of its sides by 2, we arrive at a scaledup version of the original triangle; it has all of the same angles as the original triangle and looks exactly the same (except enlarged). In mathematics, these triangles are called “similar triangles.” The 6, 8, and 10 triangle of the Italian sonnet is simply a scaled up version of 3, 4 and 5; this is also true for 9, 12, and 15; 12, 16 and 20, as well as 15, 20, and 25. In this sense, 3, 4, and 5 is the parent to all of these other triples and it is the most primitive of all of them since it cannot be subdivided any further, as long as we want to work in integers.

This idea of the Primitive Pythagorean Triple is analogous to irreducible fractions; mathematicians always write fractions in their lowest terms (instead of writing 12/8 or 6/4 they would write 3/2). Therefore, the sequence 3, 4, and 5 has special significance in this respect, since it characterizes an entire family of solutions to the Pythagorean Theorem. While the 3/4/5 triple isn’t the only Primitive Pythagorean Triple,21 it is significant that all Primitive Pythagorean Triples can be generated from the 3/4/5 triangle by use of three relatively simple algorithms. This means that 3, 4, and 5 is the most primitive of all Primitive Pythagorean Triples; it can be used to generate all of the others. The 3/4/5/ triple may be regarded, therefore, as the mother of all solutions, which captures perfectly both the centrality and the generative function of the sequence. Furthermore, in addition to being the smallest Primitive Pythagorean Triple that can generate all other Primitive Pythagorean Triples by a simple application, it also has the important feature that 3, 4, and 5 are consecutive numbers. For these reasons the 3/4/5 Primitive Pythagorean Triple holds much mathematical fascination, and is considered especially elegant. The Italian sonnet in English possesses this same reduction: sestet, octet and iambic pentameter can be subdivided into tercet, quatrain and pentameter...

A keystone form like a Primitive Pythagorean Triple, or a sonnet, can support whole new systems of knowledge, and contain a generative energy that ripples outward from its core, seeking “release / From dusty bondage into luminous air”. Of course, not all sonnets discuss mathematical or scientific subjects directly: very few of them do, in fact. But certainly all sonnets engage aesthetics generally and an aesthetics of form specifically, and on both those levels they are connected historically and structurally to principles of mathematical and scientific beauty, and so on those levels it is fair to say that the meaning of the sonnet form can be connected to scientific aesthetics. In an era where science is profoundly mathematical, and mathematics is a language that most non-scientists don’t speak, it is a beautiful idea that poetics may have the capacity to silently and covertly “speak” beauty mathematically, bringing us back to the shared intellectual heritage of science and literature.

- from "Beauty Bare: The Sonnet Form, Geometry and Aesthetics" by Matthew Chiasson and Janine Rogers, in the Journal of Literature and Science, Volume 2, No. 1 (2009). Thanks to Zach Wells for pointing it out.

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