Geometry is inevitable in development

 I keep noticing that the more seriously I think about development, the more geometry sneaks into the picture.

At first, this feels a little strange. Development is about genes, molecules, signaling pathways, and transcription factors. None of these look geometric in any obvious way. Yet the moment we stop staring at single genes and start looking at populations of cells across time, ideas like distance, direction, and cost become hard to avoid.

The shift happens when we change what we think the object of interest is. If we focus on individual cells, we naturally ask what a cell is. What markers it expresses. What label it should be given. But development is not really about isolated cells. It is about how large populations reorganize themselves over time. The object is no longer a point. It is a distribution.

Once you are dealing with distributions, geometry arrives almost automatically. A distribution lives on a space, and that space has structure. In single cell data, that space is gene expression space, where each axis corresponds to a gene and each cell is a point with coordinates given by RNA counts. This space is high dimensional, but it is not arbitrary. Some directions correspond to coherent biological programs. Others are noisy or effectively inaccessible. The geometry is shaped by regulatory constraints.

Now think about development in this space. At an early time, the population occupies one region. Later, it occupies another. Development is not just that new regions appear. It is that mass moves. Cells drift, split, and concentrate along certain directions. Asking how this happens without invoking geometry starts to feel artificial.

This is why optimal transport feels less like a clever mathematical trick and more like a natural language for the problem. The moment you ask how one distribution becomes another, you are implicitly asking about paths and costs. How far does mass have to move. Which directions are cheap. Which are expensive. What rearrangements preserve global structure while allowing local change.

What matters here is not the optimization itself. Cells are not solving equations. The optimization is a lens we use to reveal constraints that are already there. When the minimal cost flow between two developmental stages has a particular structure, that structure is telling us something about the underlying landscape. It is telling us which futures were easy to reach and which required significant coordinated change.

Seen this way, fate is no longer a switch that flips at a precise moment. It is a direction in space that becomes progressively cheaper to move along. Early cells do not contain a fate label in any sharp sense. They sit in regions where some futures are nearby and others are far away. Geometry replaces decision.

This perspective also explains why development is so robust. If outcomes depended on brittle, local decisions, small perturbations would derail the process. But if development follows broad, low cost corridors in a structured space, then noise can be absorbed. Many microscopic paths can lead to the same macroscopic arrangement. Geometry makes room for flexibility without losing form.

So when geometry appears in biology, I no longer see it as metaphor. I see it as a consequence of asking the right kind of question. The moment we ask how something becomes something else, rather than what it is at a single moment, we are forced to care about space, distance, and flow.

For me, this has been the quiet lesson behind optimal transport. Not that it gives better answers in every case, but that it nudges me toward a different way of seeing. Development stops looking like a sequence of states and starts looking like a continuous, constrained motion. And once you see it that way, geometry is not optional. It is already there, waiting to be named.