On Morphospace

An essay accompanying the Vivarium catalogue.

A body is not what an animal is. A body is what an animal does with a small handful of numbers.

This is the conceit of the Vivarium. Each creature in the catalogue is generated from between three and nine parameters — a leg length, a bell curvature, a tentacle count, a wing aspect ratio. The parameters are dialed by sliders; the body appears; another adjustment and the body becomes a sibling, a cousin, a phylum, a thing that cannot exist. The space of all possible bodies under a given formula is the morphospace of that formula.

The word is old. David M. Raup coined it in 1966 to describe the parameter space of mollusc shells — three parameters (whorl expansion, translation rate, generating curve) that together described every shell shape, real and imagined. Raup plotted the occupied regions and noticed something striking: most of the morphospace was empty. Real shells clustered in a few small zones. The rest of the space described shells that no animal had ever grown, that no fossil recorded, that violated no obvious law of physics. They were simply absent.

He called them missing.

The Vivarium is an exercise in the same question. The fireeel writhes through its corner. The jellyfish pulses through hers. The quadruped strikes through its. The wing lifts through its own. Each creature is a path through a small zone of its respective formula. The slider on the left moves the path. The slider on the right moves it further. Move far enough and the creature becomes implausible. Move further and it ceases to resemble anything biological. Move further still and it ceases to resemble anything at all.

But at every step, the formula is unchanged. The morphospace is the formula. The creature is a point.

This way of thinking dissolves a confusion that biology often produces: that the species is the fundamental unit. The morphospace says no. The fundamental unit is the function. Species are dense clusters; genera are wider clusters; phyla are entire regions. The animals are the topology; the morphospace is the manifold.

The questions this enables are different from the questions taxonomy enables.

Taxonomy asks: what is this animal? — and answers with a name. Morphospace asks: where in parameter space does this animal sit, and what does the rest of the space look like? The answer is not a name but a coordinate, and a neighborhood.

Some of those neighborhoods are populated. Selection has visited them; they work; the animals living there can swim, walk, fly, eat, reproduce. Other neighborhoods are empty. Sometimes the emptiness is informative — a region forbidden by physics. Sometimes it is curious — a region apparently viable, but no creature has stumbled into it, or evolution has driven its visitors to extinction. Sometimes it is provisional — the region is empty yet, and a perturbation, a mass extinction, a new ecological pressure could send some lineage spiraling into it next.

The Empty Quadrant preset in each creature is a deliberate visit to one of these unoccupied regions. The point is not to claim the parameters yield a viable organism. The point is to make the emptiness visible. The morphospace is larger than its tenants.

There is a further move, which the Vivarium does not yet make but which it sets up: treating the morphospace not just as a parameter list but as a coordinate system.

In a coordinate system, two points have a distance. The distance is meaningful — points close in the coordinate system are close for some reason: they have similar properties, similar derivatives, similar responses to perturbation. The system has a metric.

For morphospace, the natural metric is not Euclidean distance in parameter space. Two animals can have nearly identical parameters and entirely different fitnesses, or wildly different parameters and indistinguishable fitness. The natural metric is induced by the fitness landscape — by the function that maps parameters to viability under some criterion (locomotion, predation, mating, survival).

Treated this way, morphospace becomes a Riemannian manifold. The metric tensor at each point describes how sensitive fitness is to small parameter changes. Gradients in this metric — not in raw parameters — are what evolution actually climbs. Two animals are close if they share a fitness gradient direction; far if their gradients diverge.

This is the framing that turns morphospace from a descriptive convenience into a tool. Once you have a metric, you have geodesics — shortest paths between two organisms. You have curvature — measures of how the fitness landscape bends. You have isometries — symmetries that move morphologies into other morphologies of equivalent fitness. You have the apparatus of differential geometry, applied to bodies.

The Vivarium creatures, as currently rendered, are samples from the parameter manifold. They are not yet samples from the fitness manifold. The next move — and it is a real one — is to compute, for each parameter setting, the actual physical fitness in its domain (hydrodynamic for the jellyfish, biomechanical for the quadruped, aerodynamic for the wing) and to render that as the surface being explored.

When that's done, the empty quadrants will become legible in a new way: not just as unoccupied parameter regions, but as fitness valleys — bodies that exist in parameter space but that selection actively drains away from.

A last thought.

If morphospace is a coordinate system, then so are many things. There is a morphospace of molecules. There is a morphospace of algorithms. There is a morphospace of policies. There is, almost certainly, a morphospace of intelligences — a parameter manifold over the space of all possible cognitive architectures, with humanity occupying a tiny cluster and the rest of the space unsampled, populated by minds nature has not built and we have only begun to.

The Vivarium catalogues four creatures. But the way of seeing extends. Anything that can be described by a formula can be morphospaced. The catalogue is small because the work is hand-done. The catalogue is open because the way of seeing is what matters, not the four examples that demonstrate it.

Open one. Move the sliders. Make a body that has never existed.

The morphospace was always there. The animals are visitors.

Vivarium · Volume I · MMXXVI