Drawing Maps

The relational skeleton beneath mathematics, physics, and biology

For a long time I assumed I was just building mathematics. That seems like a reasonable conclusion when you spend years creating operators, equations, symbolic systems, and ways of describing relationships that recur across very different domains. The pattern looked mathematical, and even was, so I called it mathematics.

Recently, though, I caught myself doing something unusual. Instead of developing a framework outward, toward all the contexts it naturally touched, I deliberately narrowed one of them down to a single niche. The experience was disorienting in a way I didn't expect, and it left me with a question I hadn't thought to ask: what exactly am I building?

The obvious answer is still mathematics. Yet the longer I sit with it, the less certain I am that mathematics is the right place to begin. Most mathematical work starts with quantities, and mine rarely does. When I sit down to work, I am not usually thinking about mass, force, velocity, probability, optimization, or utility, and in many cases I find myself quietly disagreeing with the definitions, assumptions, and boundaries those concepts carry. My attention keeps drifting somewhere else, toward relationships, direction, transition, and threshold. Toward what gets preserved, and toward the transitory shift-movement between one state and another.

The work feels less like solving equations and more like drawing maps or following them. Sometimes those maps resemble cartography, locating positions, tracing pathways, marking boundaries and navigable terrain. Sometimes they resemble topology, asking what stays intact when everything superficial is allowed to change. And sometimes they seem to reach beneath both, toward something more fundamental, a way of describing how relationships themselves organize and transform.

This doesn't diminish the mathematics. If anything, it might explain why mathematics shows up at all. Perhaps the equations were never the foundation. Perhaps they are one representation of a deeper relational structure, a formal language that arrives after the fact, to describe relationships that were observed first.

Of course, I am not the first to look at the world and see structure where others see substance. Felix Klein reframed all of geometry this way in 1872: a geometry simply is the study of the properties preserved under a group of transformations. Henri Poincaré’s topology asks what stays intact as a shape is stretched and bent, why a coffee cup and a donut are, structurally, the same thing. And Emmy Noether showed something that should unsettle anyone who assumes quantity comes first: every conservation law, energy, momentum, the quantities physics treats as bedrock, follows from a prior, continuous symmetry. The preserved relationship is the source; the conserved quantity is what falls out of it. Seen this way, even invariant mass is downstream of a geometric structure, the Lorentz-invariant length of the energy–momentum four-vector, rather than an independent starting point.

Across several domains I have studied, neural connectomes, folding protein lattices, and crystalline atomic substrates, a similar relational spine appears to emerge. Traditional frameworks often assume that the pairing dynamics within these systems rely on outward alignment or similarity. But when we look at how these structures actually establish contacts, that assumption collapses under empirical weight. Instead, they couple through complementary geometric opposition.

We can formalize this relationship between Tertiary Substrate Delta (TSΔ) structures through a geometric opposition operator:

C_ij = A_i ⊗ B_j

Here, the coupling matrix C_ij doesn't measure a baseline similarity or a static quantity. The operator ⊗ maps the precise, opposing tension required for two distinct configurations, A_i and B_j, to establish a stable bond. Whether predicting structural hubs in the mouse cortex or tracking residue contacts in lysozyme, this invariant relationship holds across domains without parameter tuning. Structures couple when one sits in the open direction, away from where the other's neighbors already cluster, not when their geometries line up.

Like Klein, Poincaré, and Noether, I am hunting for the preservation under the transformation. Except here, the map tracks how a system holds itself together through opposition rather than alignment. I'm not offering a conclusion here. I'm recording an observation. I do not ask “What quantity am I measuring?" I find myself asking, "What relationship is being preserved or not?”

For now, that seems like the more interesting question.