Module 1, lesson 1 · 35 minutes

What is a crowd? Space, bodies and the number that matters

One number describes a crowd better than any other: how many people share each square metre. This lesson makes 2, 4 and 6 per square metre physically real.

You’ll be able to

estimate a crowd’s density by eye to within about 1 p/m², and say why your first guess is probably wrong;
explain why 2, 4 and 6 p/m² are qualitatively different states, not just bigger numbers;
state Keith Still’s definition of crowd dynamics.

Suppose the floor of a small lift measures 1.1 m by 1.1 m, and the plate by the buttons says it takes eight people. Fill it. Eight people in 1.21 m² is 6.6 people per square metre. Later in this lesson you will meet that exact number: it is the density at which clothed bodies run out of floor.

You have probably stood in a lift like that and come to no harm. Nobody was pushing, the doors opened after a few seconds, and you could step out. Hold on to that. What makes a crowd dangerous is not how many people it contains, nor even how tightly they stand for a moment. It is what packing does to the forces between bodies, and to everyone’s ability to get away.

To reason about any of that you need a way of describing how packed a space is, and a feel for what the numbers mean. That is this lesson.

The number that matters

Crowd density, written ρ (rho), is the number of people per square metre of floor, in p/m². Keith Still, a crowd scientist whose work runs through this course, recommends always quoting it this way round, people per square metre, rather than as square metres per person. The reason is practical: with people per square metre, a bigger number always means a more dangerous crowd, so nobody misreads a table in a hurry.

The two forms are reciprocals of each other, and both appear in the literature, so learn to flip between them:

Density and space per personSpace per person = 1 ÷ density. Our arithmetic.
Density (p/m²)	Space each (m²)	People in a 1 m × 1 m square
1	1.00	1
2	0.50	2
4	0.25	4
5	0.20	5
6	0.17	6

Density, not head count, is what the field organises itself around. Still defines his subject in terms of it: “Crowd Dynamics can be defined as the study of the how and where crowds form and move above the critical density of more than one person per square metre”. So for Still, crowd dynamics begins where density passes one person per square metre.

Speaking to CNN in November 2022, Still put the same idea in a sentence: “It’s not to do with the numbers of people, it’s due to the exposure of high density”. Exposure is the right word. Density is a condition people are in for a length of time, at a particular place.

What a body takes up

Before asking what densities feel like, ask what is physically possible. Seen from above, a clothed adult takes up about 0.15 m² of floor, according to Oberhagemann’s 2012 vfdb technical report on crowd densities at major public events. Divide one square metre by 0.15 m² and you get about 6.6 bodies: the density at which people are standing shoulder to shoulder and chest to back with no floor left between them.

Crowds can go beyond that, but not voluntarily. Oberhagemann reports that densities of 8 p/m² or more occur only under external pressure, with bodies compressed against each other, and puts the theoretical maximum at about 8.2 p/m². Using Weidmann’s smaller minimum body area of 0.085 m² instead, the purely geometric limit would be 11.8 p/m².

One square metre at 2, 4 and 6 p/m². Bodies are drawn 0.5 m across the shoulders and 0.3 m front to back, a 0.15 m² box, matching Oberhagemann’s figure for a clothed adult. The body shape is our drawing convention.

About 6.6 p/m² is where clothed bodies fill the floor. Beyond 8 p/m², people are being compressed by force from outside. Every safety threshold you will meet sits below these, for reasons the next section explains.

2, 4 and 6: three different states

It would be natural to think of density as a dial, with danger rising smoothly as it turns. Oberhagemann describes something closer to changes of state:

At 2 p/m² people are “completely decoupled”. If someone falls, nobody else is affected. Germany uses 2 p/m² as its regulatory basis for planning and evacuation.
At 5 p/m² there is still limited room to absorb forces.
At 6 p/m² force can no longer be absorbed. Oberhagemann puts the critical density “around 6.0 p/m²” and advises keeping sections of a crowd below 5–6 p/m².

The physics behind this is the subject of Module 2, but the intuition is simple. When there is floor between people, a shove is taken up by someone stepping back. When there is none, the person you push has nowhere to go, so the push passes through them to the next person, and the next. A crowd at 6 p/m² behaves less like a collection of individuals and more like a solid that transmits force.

What do real events reach? Oberhagemann reports voluntary peaks of 5–6 p/m² at major events, though usually no more than 4, and event-wide averages below 2. The averages are reassuring; the peaks are where the risk is.

The practitioner’s ladder

Guidance and practice have settled on a handful of anchor values. Learn them with their sources, because they mean different things.

Density anchors and what they meanEach value is quoted for a specific purpose. Standing figures do not apply to moving crowds.
p/m²	What it marks	Source
2	Typical UK event safety planning limit; Germany’s planning basis; “completely decoupled”	Still; Oberhagemann
4	Guidance figure for moving queueing systems	Still
4.7	Maximum for standing viewing areas at sports grounds (47 people per 10 m²)	Green Guide
5	“this should be the UPPER limit for standing/viewing spaces”; labelled “Maximum” on Still’s risk scale	Still
6	“High Risk” on Still’s scale; force can no longer be absorbed	Still; Oberhagemann

Still adds a warning about the Green Guide’s 4.7: it “fails to highlight that the demographics (individual sizes) need to be factored into the density analysis”. A terrace of adults in winter coats and a terrace of teenagers in T-shirts do not fill the same floor at the same count.

Set the density to 2, then 4, then 6 p/m². At each, look at the densest square metre (outlined) and count the bodies in it.
Starting at 3 p/m², raise the density a step at a time and watch the dashed escape path from you, ringed in the middle. Find the density at which it disappears.
Press “New arrangement” a few times at that density. Does the path always vanish at the same value?
Switch to the regular grid at 6.6 p/m², then go back to irregular. Which looks more like a real crowd?
Push past 6.6 p/m² and watch the bodies start to overlap.

Somewhere between about 4.5 and 5.5 p/m² the last channel of open floor closes and the escape path disappears. Below that, people can shuffle aside to let you out; above it, the only way out is to push, and the crowd starts to behave like a solid.

The “escape path” is our modelling choice: a connected channel of open floor, at least 2 cm between body outlines, from you to the edge. It is not a published threshold, which is why it moves a little with each arrangement. Each body is coloured by the density of its neighbourhood, graded on Fruin’s Levels of Service for queuing and standing areas, with bands E and F hatched; lesson 4 explains the bands. Bodies outlined in red overlap their neighbours. Descriptors are from Oberhagemann, Still and the Green Guide.

A 20 m × 30 m square holds 3,000 people. What is the density? Is anyone at risk if nobody pushes?

The area is 600 m², so the density is 3,000 ÷ 600 = 5 p/m². That is Still’s upper limit for standing and viewing, with only limited room left to absorb forces. Whether anyone is at risk depends on what happens next: a static, calm crowd at 5 p/m² can stand safely, but anything that adds pressure (a surge towards a stage, a blocked exit, people arriving from behind) has very little slack to take it up. And 5 p/m² is the average: some patches will be denser.

Why your eyes get it wrong

Density is easy to define and hard to see. Still gives two reasons. The first is that “Crowds rarely pack in regular formation”. People cluster around friends, barriers and sight lines, so a crowd that averages 3 p/m² may hold patches at 5 and gaps at 1. Helbing and colleagues, measuring the Jamarat Bridge approach at Mina in 2006, found that “the maximum local densities are twice as high as the average densities”.

The second is the camera. “Density can appear higher with lower CCTV angles”. A camera mounted low looks across the tops of heads, so each row hides the floor behind the row in front. From directly above, the same crowd shows its gaps.

The same evenly spaced crowd seen from a low camera and from overhead. At a shallow angle each head hides the floor behind it, so the crowd looks solid. Schematic, not to scale.

Two ways to estimate density by eye

Count a square metre. Pick a few representative patches about 1 m across, count heads in each, and average. Still describes two ways of sizing a crowd, “a count or a density check”; this is the density check in miniature, and it is why anything of known size in a CCTV image, such as paving or barriers, is so useful. Module 7 returns to it in counting crowds from images.

Use spacing. In a roughly even crowd, if neighbours stand about d metres apart, centre to centre, the density is about 1 ÷ d². People about 1 m apart are at roughly 1 p/m²; about 0.7 m apart, roughly 2; about 0.5 m apart, roughly 4. This is geometry, not a published rule, and it overestimates spacing in clustered crowds, but it gets you within about 1 p/m² of a uniform crowd quickly.

An average over a whole arena can look comfortable while the area in front of a stage or a gate is far denser. When someone quotes a density, ask over what area, and at what moment.

On a CCTV still you can see that people in front of a stage are standing about 0.45 m apart, centre to centre. Roughly what density is that, and what does the ladder above say about it?

1 ÷ 0.45² ≈ 4.9 p/m². That is at the Green Guide’s standing maximum and close to Still’s upper limit of 5 p/m². It is also an estimate from a single camera angle: if the camera is low, the true spacing may be larger than it looks, and if this is the densest patch, the crowd’s average will be lower. Check from a second view and count a square metre if you can.

“Danger is about how many people there are.” It is about density and force. Forty thousand people spread out are safe; a few hundred packed into a narrow alley may not be.
“6 p/m² is physically impossible.” It is observed at events, and crowds under external pressure reach 8 p/m² or more.
“A crowd has one density.” Local densities can be twice the average. Always ask where, and over what area.
“The 4.7 p/m² in the Green Guide means 4.7 is safe anywhere.” It is a maximum for standing spectators on terraces, and Still notes it ignores body size. Lesson 2 explains why moving crowds reach trouble at lower densities.