Taxonomy of structures and techniques for mathematical poetry

Verbogeometry

This one comes first because the rest are read inside it. The forms that follow put concepts into an equation; verbogeometry puts them into a space. Take a pair of antonyms and draw a line between them, the way a number line runs between its extremes, and you have an axis whose gradations are levels of meaning rather than levels of quantity. I call that a word-axis. Set two of them perpendicular and they define a word-plane, every point of which is some blend of the words at the ends of the two axes.

The words are not only labels for the axes — they hold conceptual points along them. So a location in that space can be written as a coordinate word-pair, exactly the way (x, y) is written, and once points have coordinates every construction in analytic geometry becomes available. The midpoint formula finds the exact point between two ideas. The distance formula measures how far apart they stand. An inverse tangent returns the angle between one difference of meaning and another.

Each antonym pair adds a spatial dimension, and there is no limit to how many you use: word-cubes in three, word-hypercubes in four, and onward. The Muses names the eight corners of a word-cube whose axes are sensory / conceptual, utility / non-utility and design / craft. A God Path runs the distance formula through seven, one for each deity, every dimension a gradation running out of confusion.

The antonyms themselves come in kinds, and the kinds have a geometry. A simple antonym is a direct negation — fertile / infertile. A complex antonym is not — fertile / barren. The line drawn by a simple antonym stands normal to a plane holding the complex antonyms related to it, and a group of such lines run parallel builds a prism between two synonym-planes. That is the structure of The Transformation of Poison.

What the space will not give you is a number. Meaning is relative to the context each reader brings, so no word can be pinned to an exact location on its axis, and I would not want it pinned. What matters is not the value but the spatial relationship of the values to one another — the reader supplies the rest.

The Similar Triangles Poem

Also called the proportional poem. Two similar triangles hold four terms in a fixed relation. Nothing in that relation requires the terms to be numbers, so four concepts will sit in it just as well — and the moment they do, the geometry starts making claims about them.

The form can be solved for any of its four terms, and each solution is a different poem saying the same thing from a different vantage. That is the pleasure of it: the meaning does not sit in one arrangement, it survives all of them.

The Expanded Similar Triangles Poem

The similar triangles poem pushed past two ratios. Three similar triangles give six terms and three equal ratios — and with more ratios in hand, a new operation becomes available.

Take any two of the ratios and add or subtract their numerators together with their matching denominators. The result still holds the proportion. Each choice of pair and operation is a different flavour of the poem, and each flavour can then be solved for any of its terms.

What the expansion buys is a poem about differences rather than quantities — the gap between two terms standing in fixed ratio to a third.

The Orthogonal Space Poem

Three concepts on perpendicular axes: two multiplied to yield the third. I usually display it solved for one of the factors, because that arrangement lets you push the denominator toward nothing and watch the value climb without bound.

Multiplication matters here, and it is not interchangeable with addition. Add two concepts and they stay separable — you can still see the seam. Multiply them and they integrate, and what comes out has a magnitude neither had alone.

Twelve of these arranged contiguously in one Cartesian system make what I call a Dodecaorthogonal Space Poem, which is the structure of A Spectrum of Jewels.

The Paradigm Poem

Rather than an empty structure, an entire existing model — a law of motion, a physical relationship — is borrowed intact and used as the vehicle. The model arrives carrying its own semantics, and every relation inside it stays binding once the words go in.

In the language of embodied cognitive linguistics, the borrowed model is the source domain and the words placed in its variables are the target domain. The concrete maps onto the abstract, and it is that mapping — not the arithmetic — that does the poetic work.

The first one I made was Psychronometrics in 1983, which borrows Einstein’s time dilation to describe why a summer was longer when you were four.

Because the borrowed model brings its own variables, some of them turn out to be spaces. Karmic Influence on the Double Helix and The Power of Confidence both put the distance formula inside the physics, and in Congenital Wisdom the r of Newton’s law of gravitation is a five-dimensional separation — divisiveness from concurrence, chaos from control, the path of sin from the path of projectiles, descension from ascension, attachment from autonomy. The paradigm supplies the equation; verbogeometry supplies what one of its terms is made of.

The Double Derivative Poem

A derivative is the rate at which one thing changes as measured by the change in another. This form addresses the first and second derivatives of a phantom function — a function you are meant to feel your way into rather than plot on a coordinate system.

The idea came out of physics. Position, velocity and acceleration stand in exactly that relation — velocity is the first derivative of position, acceleration the second — and position is to velocity as velocity is to acceleration. Which is the similar triangles structure already, arriving from a different direction.

Substitution

Two finished poems that share a term can be joined. Solve both for the term they have in common, set them equal to one another, and a third poem falls out — one neither of the originals contained.

The new equation can then be solved for any variable in it, so a single shared word can open a whole family of poems. When the two source poems come from two different poets, the result is a collaborative substitution poem — a genuine collaboration, since neither author can predict what the join will say.

Worked through

Cedar Swamp Romance came out of two poems of mine that both contained the phrase my memories. Here is the first, as it was written:

Solve it for the phrase the two have in common:

The second poem was already in that form — it had been written solved for the same phrase:

Two different expressions, both equal to the same thing — so they are equal to each other. Setting them side by side is the whole move, and watch what the symmetry does to the shared phrase:

The phrase that joined them is no longer in the equation. It did the work and then dropped out, which is why the third poem can say something neither original contained. Solve what is left for Delaware River:

A second join: Money Gone

My friend Tim Gilbert — stage name TIMO, an actor, musician and songwriter living in London — writes what some would call quite dark songs, and I have always enjoyed his singular look at life. One day, in a miff of disgust, he said to me: God’s blood is money. The thought registered immediately in my mind as a similar triangles poem. Solved for Man, it became The Epistle of Timo.

The Epistle of Timo2024 · Digital See the artwork →

On another topic entirely, a different similar triangles poem had appeared in my mind: the value of The Victor is to Honor in War as Money is to Righteous Effort. This came nearly two decades before the recent war in Ukraine, but that war is a perfect example of this dark and somewhat cynical expression.

The Commensurability of Honor2005–2026 · Digital See the artwork →

What is interesting about these two seemingly unrelated expressions is that they both carry the term money. So here is another situation where we can substitute. Solve each of them for that shared term:

Now set them equal to one another:

And solve for Honor in War:

Delusions of Justification2005–2026 · Digital See the artwork →

This chain of events is the most remarkable part. By solving both for money, the context of money drops out entirely, and the equation that remains can be read with gusto: honor in war is equal to man times righteous effort times the victor, divided by God times blood.

Which shows how much fun these substitution poems can be, with others getting in on it. I call this little exercise Money Gone; the piece it became is Delusions of Justification.

The Axiomatic Poem

Not an equation but an axiom set. Take a formal system, substitute concepts for its primitive terms, and let the axioms carry the metaphor by force of their own logic. Nothing is asserted directly; everything follows.

In Peano’s String the system is Peano’s axioms for the natural numbers. Where Peano writes number, I write cat; where he writes successor, I write the God of; where he writes zero, I write Abraham. The axioms are otherwise untouched.

1. Abraham is a cat.
2. If x is a cat, the God of x is also a cat.
3. Abraham is not the God of any cat.
4. If two cats have equal Gods, they are equal.
5. If a set of cats contains the cat Abraham and it contains all the Gods of its members, then the set contains all the cats.

The fifth is induction. Read the five together and a theology assembles itself out of arithmetic — an endless succession of gods, each the god of the last, none of them first.

The same five axioms will sit in a different setting and read differently. In Cojoined they are laid over a Star of David built from scattered drops of blood, with a staff bearing the star and crescent standing beside it — the two traditions that share the cat Abraham as their zero.

Conceptual Blending

Not a form of mine. Blending is a description of how minds already work, from the cognitive linguists Gilles Fauconnier and Mark Turner. It belongs here because it is what is happening inside all eight of the structures above — and because some of my pieces run on it with no structure underneath them at all.

The blend is not a compromise between its inputs and it is not their sum. Projection into it is selective: not everything crosses over, and what does arrives stripped of its old context. That partiality is the condition under which a reader has to think.

An equation is an unusually efficient vehicle for it. The equal sign hands the reader a cross-space mapping already made — it asserts the correspondence rather than asking whether one is possible — and leaves them to find the generic space that would justify it. Multiplication goes further: added concepts stay separable and you can see the seam, while multiplied concepts integrate into a product neither input had alone.

Once you notice it running underneath, the taxonomy stops being a list of eight separate things.