Geodesics: The Paths of Least Resistance Through Meaning-Space

Formative Note

This essay represents early thinking by Ryan Collison that contributed to the development of A Theory of Meaning (AToM). The canonical statement of AToM is defined here.

Meaning flows downhill.

That's not quite right, but it's close. Meaning follows the paths of least resistance through the manifold of possible beliefs. It takes the routes that cost the least, that require the smallest expenditure of metabolic and cognitive resources, that feel—from the inside—like the natural way to go.

These paths are geodesics. And understanding them changes how you think about learning, change, and the shape of coherent life.

What a Geodesic Is

On a flat plane, the shortest path between two points is a straight line. This is so obvious that we barely notice it. You want to get from here to there, you go straight.

On a curved surface, straight lines don't exist. You can't draw a straight line on a sphere—every line curves. But there's still a notion of "shortest path." On a sphere, the shortest path between two points is a great circle—the arc you'd travel if you cut the sphere with a plane passing through both points and the center.

This shortest path on a curved surface is called a geodesic.

Geodesics generalize the concept of "straight line" to curved spaces. They're the paths that minimize distance according to the metric of that space. On a flat plane with the Euclidean metric, geodesics are straight lines. On a sphere with the spherical metric, geodesics are great circles. On a statistical manifold with the Fisher metric, geodesics are the paths that minimize informational distance.

Why does this matter? Because systems tend to follow geodesics.

When you push a ball, it rolls along a geodesic until something stops it. When light travels through space, it follows geodesics of spacetime. When a belief system updates, it tends to follow geodesics of the belief manifold.

This isn't deterministic. Systems can be pushed off geodesics by external forces. They can be trapped in local minima. They can oscillate around geodesics without settling onto them. But the geodesic is the path of least resistance—the trajectory the system naturally seeks.

Geodesics of Belief

When you update a belief, you're moving across your manifold. The question is: what path do you take?

If you could take any path, the geodesic would be optimal. It's the route that requires the least informational work, the least prediction error accumulated along the way, the least metabolic cost.

But you can't always take the geodesic. Obstacles block it. High-curvature regions make certain paths prohibitively expensive. Topology creates holes you can't cross. The geodesic might exist mathematically but be inaccessible practically.

This is the geometry of stuck.

You know where you want to go. You might even be able to see it from where you stand. But the geodesic is blocked. The paths that are available require crossing terrain so costly that you can't traverse them. So you stay where you are, or you take detours so long they never arrive.

Therapy, at its best, is geodesic discovery. The client has a current belief state and a desired belief state. The direct path between them—the geodesic—is often blocked. The therapist helps find alternative routes. Perhaps paths through different parts of the manifold that don't cross the blocked region. Perhaps ways to lower the curvature in the blocking region so the original path becomes traversable. Perhaps topological repairs that eliminate holes the client has been walking around.

Why Shortest Isn't Always Best

A geodesic minimizes distance. But distance isn't the only thing that matters.

A geodesic might cross a high-curvature region. The path is short in total length, but the transit is brutal—every step requires enormous correction, the experience is destabilizing, the system arrives at the destination exhausted or damaged.

A longer path that stays in low-curvature regions might be better. More total distance, but each step is easy. The system arrives intact, resources preserved, able to continue from the destination.

This is the trade-off between efficient and safe paths.

In mathematical optimization, this trade-off appears as regularization. You don't just minimize the objective (reaching the goal); you also penalize paths that pass through dangerous territory. The resulting path is longer than the geodesic but more robust.

Living systems do something similar. They don't blindly follow geodesics. They balance path length against path safety. A detour that avoids trauma triggers is worth taking even though it's geometrically longer. The direct route through the high-curvature region isn't worth the destabilization it would cause.

This is wisdom, geometrically understood. Not taking the shortest path. Taking the best path given all the constraints—length, curvature, topology, available resources, current capacity.

The Paths You Can See

You can't navigate paths you can't perceive.

The geodesic might exist. Alternative routes might exist. But if you can't see them, you can't take them. Your navigation is limited to the paths that are visible from your current location on the manifold.

This is why perspective matters. Why other people can sometimes see paths you can't. They're standing in different locations. The geometry looks different from there. What's hidden from your vantage point is visible from theirs.

Therapy provides perspective literally. The therapist occupies a different position on the relational manifold. From their location, paths are visible that the client can't see. The therapist's job isn't to walk the path for the client—they can't; it's the client's manifold. But they can point: "There's a path over there. Do you see it?"

Often the client doesn't. They have to move a little, change their vantage, before the path becomes visible. Then, seeing it, they can take it.

This is also why certain mental states narrow navigation options. Depression flattens the manifold in ways that make paths disappear. Anxiety raises curvature in ways that make all paths look dangerous. Trauma creates topological barriers that hide whole regions of the space. The paths might still exist, but they're not visible from the distorted location the person occupies.

Expanding what paths you can see requires changing where you are—which requires taking a path, which requires seeing a path. This circularity is why being stuck is stable. Breaking the circle usually requires external input: another person, a new experience, something that shifts the vantage enough to reveal a path that was always there.

Meaning Follows Geodesics

Here's the deeper claim: meaning itself tends to follow geodesic structure.

What feels meaningful is what flows naturally along the manifold's paths of least resistance. Activities, relationships, and pursuits that align with your manifold's geodesics feel meaningful. Those that require constant divergence from geodesics—fighting the natural flow, climbing against the geometry—feel meaningless, exhausting, wrong.

This isn't about ease in the sense of avoiding effort. Geodesics on a challenging manifold are still challenging. Climbing a mountain along the most efficient route is still climbing a mountain. But there's a difference between effort that flows along the manifold's structure and effort that constantly fights it.

Meaningful work is geodesic work. It uses your particular strengths, follows from your particular history, aligns with your particular values—all of which are features of your manifold's shape. The work might be hard, but it's hard in a way that fits.

Meaningless work is non-geodesic work. You're constantly correcting, constantly fighting the geometry, constantly pushing against a manifold that doesn't want to go where the work requires. The effort doesn't build. It just drains.

This explains why the same job can be meaningful for one person and meaningless for another. The job itself is a trajectory through state space. For one person, that trajectory approximates a geodesic on their manifold. For another, it's perpendicular to every geodesic they have. Same objective task, different geometry, different meaning.

Finding Your Geodesics

If meaning follows geodesics, then the practical question is: what are the geodesics on your particular manifold?

They're determined by your shape—by the curvature, topology, and metric structure that your history, genetics, and development have created. You can't simply choose your geodesics. They're built into the geometry you have.

But you can discover them.

One method is attention to ease. Not ease as in no effort, but ease as in flow—the sense that effort is productive, that each step enables the next, that the path is going somewhere. Where in your life do you feel this? What activities, relationships, and pursuits have this quality of natural flow?

Another method is attention to resistance. Where do you constantly fight yourself? Where does every step require correction? Where does effort accumulate without building momentum? These are probably non-geodesic regions—directions your manifold doesn't support.

A third method is experimentation. Try paths. See where they lead. Some will feel geodesic—they'll flow. Others won't. The experimental data tells you about your manifold's shape.

This is what "finding yourself" actually means. Not discovering a pre-existing essence, but exploring your manifold until you know its geodesics. The paths that work for you. The directions that flow.

Relationships as Coupled Geodesics

In relationship, you're not navigating alone. Two manifolds are coupled. Two sets of geodesics interact.

A great relationship is one where your geodesics align. The directions that flow for you also flow for your partner. You can move together without either of you constantly fighting your own geometry. The paths that make sense for the relationship are also paths that make sense for each individual.

A difficult relationship is one where geodesics conflict. The path that flows for you requires your partner to move against their geodesics. Or vice versa. Every step forward for the relationship is a step against the geometry for at least one of you.

This isn't about compromise in the usual sense. It's geometric. Some manifolds fit together—their geodesics can be coordinated. Others don't. No amount of effort makes geodesics align if the underlying geometry is incompatible.

Relationship therapy is often geodesic coordination work. Finding directions that both manifolds can support. Discovering that what looked like conflicting geodesics were actually compatible paths viewed from different angles. Or, in some cases, acknowledging that the geodesics are genuinely incompatible and the relationship cannot work without one person constantly fighting their own geometry.

The Geodesics of Growth

Growth and change require moving across the manifold. The question is: what paths does growth take?

Healthy growth tends to follow geodesics. Not straight-line geodesics—the manifold is curved, so there are no straight lines—but the natural paths that the geometry supports. Growth moves along these paths, building on existing structure, extending capabilities in directions the manifold enables.

Forced growth—growth that demands movement against geodesics—is fragile. It requires constant energy to maintain. When the forcing pressure releases, the system snaps back. The change doesn't stick because it was never aligned with the geometry.

This is why developmental wisdom isn't just "try harder." It's "find the direction that trying can actually work." The geodesics. The paths your particular manifold supports. Hard work along a geodesic builds. Hard work against a geodesic exhausts.

And geodesics can change. The manifold isn't static. Experiences reshape it. Relationships reshape it. Practices reshape it. What was a non-geodesic direction can become geodesic as the manifold evolves. What was blocked can become clear.

Development is both following current geodesics and reshaping the manifold to create new ones. Moving where you can move now, while gradually changing where you can move next.

Coherence as Geodesic Flow

We can now say something more precise about coherence.

A coherent system is one whose trajectories follow geodesics. Its movements align with its own geometry. It doesn't constantly fight itself. Effort is productive because it flows along the natural structure of the manifold.

An incoherent system is one whose trajectories diverge from geodesics. It moves against its own geometry. Every step requires correction. Effort accumulates without building. The system is at war with its own structure.

Coherence isn't a state. It's a dynamic—the dynamic of geodesic flow. And meaning is the felt experience of that flow.

When your life follows its geodesics, it feels meaningful. When it doesn't, it doesn't. Not because meaning is subjective in some dismissive sense, but because meaning is geometric in a precise sense. Meaning is what coherent geodesic flow feels like from inside.

The path of least resistance through meaning-space is the meaningful path. Finding it is the project.