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Our sidewalk interactions are "fundamentally anticipatory in nature," according to scientists.

Big city sidewalks can feel like an inexplicable dance of elbows and shopping bags and baby strollers and pigeons and texting. But a group of crowd scientists has whittled the chaos to its core and found that, far from unpredictable, foot traffic follows a mathematical formula elegant for its simplicity. From Shibuya Crossing to Times Square we're all performing invisible calculus: computing other people's speeds and trajectories and adjusting our own accordingly.

Or something like that. In a recently published paper, to be presented this week at a conference, a research team led by computer scientist Ioannis Karamouzas of the University of Minnesota propose "a simple and universal law governing pedestrian behavior." The law suggests our sidewalk interactions are "fundamentally anticipatory in nature"—meaning all those fellow walkers that seem oblivious are actually projecting, and thereby avoiding, future collisions in real-time:

Remarkably, this simple law is able to describe human interactions across a wide variety of situations, speeds and densities.

From a scientific perspective, pedestrian motion is highly difficult to model, because a walker's decisions could emerge from any number of factors on a given sidewalk, from fast-walkers to phone-talkers to poorly scooped dog poop. Previous theories figured that adjacent pedestrians respond a lot like charged particles. When they get too close to one another, they push away—a force known as "interaction energy."

Karamouzas and company found that comparison wasn't quite right. Case in point: pedestrians moving in parallel opposite directions can maintain their speeds and courses even as they scrape shoulders. Instead, by studying 1,500 pedestrian trajectories from real-world and laboratory settings, the researchers determined that "projected time to a potential future collision" influenced interaction energy.

So it's mental anticipation, more than physical reaction, that guides our movements. That explains why we plow ahead when others are following a straight course but shuffle our steps when someone's walking out of whack. And it's not until a collision feels imminent that we bother to do anything: interaction energy disappeared outside of three seconds to contact.

Kevin Hartnett explained the technical details well in the Boston Globe last fall, as the early research emerged:

The closer two people get to colliding, the more energy they expend getting out of each other’s way. To be technical, they found that the interaction between individuals in a crowd could be described as 1 over the square of the time to collision: As a collision becomes more imminent, the energy you apply to avoiding it goes up drastically.

When the researchers tested their new walking formula on pedestrian simulations, they found it a close match to real human behavior. In the videos below, the red and blue sims "reproduce a wide variety of important pedestrian behaviors": they spontaneously form walking lanes, arch through narrow passages, form zipper patterns at bottlenecks, slow for congestion, and anticipate collisions. Here's a hallway simulation:

And here's a red-and-blue crossing flow, as if simulating an intersection outside Capitol Hill:

The new work builds on a growing and fascinating science of how we walk. Pedestrians tend to pass on the proper side via unspoken maneuvers, even if that means heading right in a country that drives on the left. In especially crowded cultures, pedestrians seem to keep a faster stride even approaching a pack. Walkers who travel in groups slow down sidewalk streams because rather than break into single file when they hit a crowd, they flex into a V to keep talking.

This newest pedestrian law a isn't perfect representation of walking behavior. It didn't capture the shock waves that occur in super dense crowds, for instance. Still, it's a clear improvement on the old ways of envisioning pedestrian motion; collaborator Stephen Guy told Science News the researchers had found "the most humanlike model that exists right now." We'll be on the lookout for the next step forward.

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