Tobler’s Law in Latent Space

There’s an idea in geography called Tobler’s First Law of Geography: “everything is related to everything else, but near things are more related than distant things.” It sounds almost obvious when you first hear it. Of course nearby things are similar. But recently I’ve been wondering whether AI is quietly breaking this intuition, or revealing that “near” was always more complicated than we thought.

Deeper into Tobler

Tobler was not the first to notice this pattern, but he was the one who popularized it. Earlier work by R. A. Fisher in 1935 described similar ideas in the context of crops, where nearby plants tended to be more alike than those farther apart. Tobler first introduced his formulation publicly in 1969, and it was published in 1970 in A Computer Movie Simulating Urban Growth in the Detroit Region.

That paper focused on modeling population growth in Detroit, simulating how cities expand over time based on a handful of variables. There is some evidence that Tobler himself was not entirely serious about the phrasing when he introduced it. He likely did not expect it to be quoted over and over again as a “law.”

And yet, it stuck.

The idea comes up constantly in geography because it works. Things that are near each other are often more similar than things that are far apart. You can see it in forests, in neighborhoods, in groups of people. It’s one of those statements that feels immediately obvious, almost tautological, and that obviousness is part of its power.

What are vector embeddings?

What got me questioning Tobler’s Law is the rise of vector embeddings, a technique that has become central to how AI systems understand the world. These embeddings are used in large language models as well as in earth observation foundation models. To see why this matters, I need to explain what embeddings actually are.

At a high level, embeddings are a way of turning things into points in a space, where distance represents similarity. Instead of organizing information only by physical location, embeddings organize it by learned relationships. In that sense, they are better thought of as spaces rather than networks, even though we often visualize them as clusters or constellations.

This idea first became popular through language models. Words, sentences, or entire documents can be represented as points in an embedding space, where things with similar meanings end up close together. “King” and “queen” are near each other. So are “Paris” and “France.” “Near” no longer means physically adjacent; it means semantically similar.

The same approach can be applied to images and geography. In remote sensing, satellite imagery is often divided into small square tiles. Each tile can be transformed into a point in an embedding space. These embeddings are not just based on what it looks like to us, but on infrared and other data humans cannot see. When you do this, patterns emerge that are not constrained by physical distance.

Tiles containing runways cluster together. Cornfields cluster with other cornfields. Chicken farms group with chicken farms. These tiles may be scattered across the globe, but in embedding space, they are near each other because they are similar.

What does this all mean?

This brings us back to Tobler. What does Tobler’s First Law mean in this virtual or latent space? If airports across continents are “near,” what kind of nearness are we talking about? And as we are able to sense more and more information about the world, does physical distance remain the dominant way we should understand relatedness?

One way to think about this is that Tobler’s Law still holds, but “near” now has many dimensions. Geographic distance is one of them. Embedding space is another. Time can be another still, since things close together in time are often more similar than things far apart in time.

In that sense, Tobler’s Law may be better understood not as a strict rule, but as a heuristic: a useful way of reasoning about similarity within a given space. Embeddings don’t break it. They extend it into new dimensions.

Or put differently, every model creates its own notion of distance. As George Box famously said, “All models are wrong, but some are useful.” Tobler’s model was useful for the space we could see. Embeddings invite us to ask what other spaces might matter.

-Kate