The presence of the ground is equivalent to the absence of a third dimension. Thus irrelevant.

I see no reason to expect additional dimensions to make a fundamental difference here. This isn't a complex topological problem involving the constraints of graph connectivity. It comes down to nothing more than immediately available space, maintaining it as a group and making use of it as an individual.

Consider a tightly packed group of runners. If someone in the center goes down they'll likely be trampled and it's also likely to cause a mass pileup.

If birds were prone to that you'd expect the occasional collision at the center of a dense flock and a bunch of birds falling out of the sky as a direct result. Yet I've never once heard of that. The simple and obvious hypothesis is that they are hardwired for group navigation in a way that the vast majority of other animals aren't.

If you're having trouble seeing the point I'm trying to make here then try swapping out the example with runners for one with a fleet of planes. A swarm of hundreds of modern fighter jets in a dense 3D formation actively maneuvering in randomized arcs without any sort of pre-planning or rehearsal. If something went wrong can you really imagine all of the pilots successfully avoiding collision every single time? The birds around here put on shows like that multiple times every day.

The presence of the ground is very important because it's a solid body you can get crushed against! The fear of falling down and getting trampled is a major source of panic in bad crowd conditions. That alone makes a big difference.

This isn't about complex topological problems, it's about the square-cube law. In a 3d environment, the direction of movement of any individual covers a much smaller portion of their possible movement directions. This means it's much easier to form a "wave" in 2d, and this wave covers a much larger portion of their potential movement options.

Doesn't this obviously mean a 3d crowd is much less likely to crush an individual? And even if we're only talking about bird-on-bird collisions, head-on collisions make up a much smaller portion of all possible movement vectors.

Regarding your plane example - birds don't explode upon most collisions, and I'd assume they can easily recover from many angles a plane couldn't.

> the direction of movement of any individual covers a much smaller portion of their possible movement directions

Number of adjacent bins versus fraction of unoccupied bins. The latter is what matters (ie density). Plus the physical and mental ability to rapidly and accurately select and maneuver into one of them. Plus (and this is really the crux of the thing) some sort of hardcoded behavior that makes the risk of conflicting independent choices vanishingly small.

A concrete example of that last part is TCAS. [0] That's what birds seem to have built in that other animals lack.

> it's much easier to form a "wave" in 2d, and this wave covers a much larger portion of their potential movement options

Not true. Waves in a 3D medium are themselves 3D. Go check out a fluid dynamics simulation.

I think what you might be trying to get at is dissipation within a volume. Cubed versus squared. But that's not the issue here. Dense pockets will occur from time to time in such a system regardless of dimensionality. The question is collision frequency for a given density. Birds are ~0 whereas humans (and most other animals) are really quite bad.

> birds don't explode upon most collisions

It's a fair point (and humorous) however irrelevant because my entire point there is that I've never seen birds collide to begin with. Even if they did explode it wouldn't matter.

Note that unlike human pilots autonomous aircraft are capable of forming dense swarms without colliding. (At least assuming the software was designed with that in mind.)

[0] https://en.wikipedia.org/wiki/Traffic_collision_avoidance_sy...

If we just go by what we've seen it's all irrelevant, since I've never seen a flock of birds with anywhere near a comparable 3d density to a dangerously dense 2d crowd. Even the densest flocks still have visible gaps between the birds, I'd guess the density would need to be at least 5x higher.

We were never making an exact comparison though. The point you've been arguing against (unless I misunderstand) is that creatures such as birds and fish have some special hardcoded behavior that the rest of us don't. I don't think spacing the humans out a bit more is going to solve the problem here; the aircraft example makes that point nicely.

In fact even your objection about visible gaps serves as a case in point. A flock of hundreds of thousands of starlings somehow avoids such dangerous densities even in the face of transient "pressure" waves traveling through it. Meanwhile human groups appear to willingly form such densities without thinking twice. Sounds like a hardcoded behavior to me.

If you really want an example with more density then consider schools of fish. Of course unlike birds it's not so easy for most people to observe fish. I don't actually know for certain that they never collide or get injured by being crushed. I seriously doubt it though.

The point I'm arguing against is that a 2d crowd of people and a 3d flock of birds are anywhere near comparable. They aren't, they have fundamentally different mechanics, problems, behaviors etc. You're comparing apples to elephants.