Keith Still has taken that work and expanded upon it through computer simulation and experiments with volunteers. He uses a measure of people per square meter—nearly the same as a square yard—and differentiates the requirements for a crowd that is moving and one that is not. At two people per square meter, even a moving crowd is fine. Add two more and movement becomes awkward. Add another, resulting in five people per square meter, and you begin to flirt with disaster. At six people per square meter, no space is left between individuals, and people are hemmed in and unable to control their movements, whether to stop or go. No one would willingly enter into such a mob, but crowds of the unwilling are compacted by the progression of the masses behind them and by physical constraints such as walls, fences, gates, doorways, stairways, up ramps, and slight turns or changes in direction. As the crowd in a given space exceeds 80 percent of the space’s capacity, the compression accelerates. In the real world, densities of seven, eight, or nine people per square meter are not uncommon.

Even at that extreme, people are not yet dying, but beyond five people per square meter the crowd has effectively formed into a single mass through which energy can be transmitted. It is more like a liquid than an assembly of solids, and the laws of fluid dynamics start to apply. Someone shoves, someone stumbles, and the effect is amplified by others. The impulses move through the crowd and rebound with increasing intensity. They are a prelude to death. From within the crowd they appear as sudden mass movements, impossible to resist, 10 feet in some direction, 10 feet in another. People caught up in them are in serious trouble. They need to leave, but cannot. They need to raise their hands into a boxing position to protect their chests, and turn 90 degrees to the flows, because from side to side the rib cage is less compressible than it is from front to back. If they are strong and lucky, they may succeed in this, though not in the highest-density crowds. Above all, they need to stay on their feet, although if a progressive crowd collapse occurs, this will be impossible to do. Then it’s a question of luck—whether they end up at the top of a pile or the bottom.

Shock waves are implicated in most crowd crushes, but not all. For instance, large crowds moving down stairways have repeatedly suffered mass casualties because someone tripped: 354 dead in 1942 on the stairs leading to an air-raid shelter in Genoa, Italy; 173 dead in 1943 on the stairs leading to another air-raid shelter, in the London Underground station at Bethnal Green; 21 dead and more than 50 injured in 2003, during an urgent exit from a second-floor nightclub in Chicago. Shock waves are a more insidious matter. They capture people long after the possibility of avoidance has vanished. Shock waves certainly accounted for the soccer deaths in Sheffield. They also accounted for the deadliest day of the war in Iraq—August 31, 2005—when a million Shiite pilgrims gathered at a Baghdad shrine and rumor spread of an impending suicide attack. The crowd did not respond to the rumor by panicking, as was widely reported, but quite reasonably began to leave the area. Thousands tried a bridge over the Tigris River, only to find that on the far side the exit from the bridge was heavily gated. In the crush that developed as people continued to cross, the shock waves grew so powerful that the guardrails gave way, dropping hundreds into the river. The fall to the river amounted to a lucky escape, but only for those who could swim. In all, 965 people died, most on the bridge, and by compression asphyxiation.

Admittedly, that was in the hell of Iraq during a chaotic time. But the problems exist even in the most orderly societies. In Duisburg, Germany, for instance, 21 people died and more than 500 were injured in 2010 at the entrance to a music festival called the Love Parade. A huge crowd was trapped in a sheer-walled concrete channel that the event’s organizers—who were worried about gate-crashers—had stupidly designated as the way in. The police were almost as incompetent. Their attempt to control the crowd added to the pressures. Fruin was the first to make the point that police are often poorly prepared to handle such masses of people, because their emphasis is on maintaining public order, and it is crowd management, not officious control, that is needed. In this case proper management would have entailed metering the pedestrian flow far upstream of the potential choke points; instead the police waded into the thick of things and tried to set up blockades. Inevitably they were overwhelmed. Videos exist on YouTube that show the shock waves developing and capture the screams of the victims. The point is that these were neither zealots following the dictates of an ancient prophet, nor even die-hard soccer fans. They were fresh-faced Germans who just wanted to celebrate life. But the density of the crowd condemned them.

III. The Saudi Dilemma

The obvious solution is to avoid big crowds. When it comes to the hajj, however, Muslims do not have a choice. This places the rulers of Saudi Arabia in a typically Saudi-style bind—one that is largely of their own making, and impossible to undo. The Saudis are conservative Wahhabis, true believers, and they take their hajj responsibilities seriously, for both religious and geopolitical reasons. Their problem goes back to the Prophet Muhammad, who was not only a big-picture man but also a micro-manager who issued edicts on all manner of subjects: how to go about one’s day; how to dress; how and what to eat; how to have sex; how to wash; when to pray. His words on any subject became law, subject to relatively little interpretation over the centuries because he was the final prophet.