Tawaf Crowd Flow Around the Kaaba

Crowd

Pilgrims 400

Free-flow walking speed 1.10 m/s

Jam density (crowd-control spacing) 5.5 p/m²

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Avg speed

0.00 m/s

Avg density

0.00 p/m²

Peak density

0.00 p/m²

Avg circuit progress (of 7)

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Tawafs completed (total)

How it works

Tawaf is the Islamic ritual of walking counterclockwise around the Kaaba seven times (one full Tawaf = 7 ashwat/circuits). This sim places pilgrims on seven concentric rings around a central cube and moves each one along its ring, but the walking speed is not constant — it follows the same pedestrian fundamental diagram used to model real crowd flow at the Mataf:

v(ρ) = v_free · max(1 − ρ/ρ_jam, 0.05)

ρ  = local crowd density (people / m²), measured per
     ring-sector cell each frame: count / (arc-length × ring-width)
v_free = free-flow walking speed you set
ρ_jam  = density at which the crowd effectively grinds to
         a crawl (a Greenshields-style linear speed-density law)

Each pilgrim's angular speed is ω = v(ρ)/r, so pilgrims on outer rings (larger r) sweep a slower angular rate for the same linear speed. When a pilgrim's accumulated angle reaches 7 full turns, that Tawaf is recorded as complete and they are re-seeded onto a fresh ring to keep the crowd flowing continuously.

Pilgrims — total agent count; more agents packed onto the same seven rings raises local density.
Free-flow speed — the speed each pilgrim would walk in an empty plaza.
Jam density — how much personal space real crowd-control barriers and marshalling leave per person before flow chokes; lower it to see gridlock form near the Kaaba's corners.
Density heatmap — colors each pilgrim from blue (free-flowing) to red (jammed) by the local density they are currently experiencing.

Real-world relevance: this is the same density-speed relationship crowd-safety engineers use to plan the Hajj Mataf's ring capacity and barrier spacing, where the 2015 Mina crush made this exact congestion physics a matter of public-safety engineering.

Free-flowing Slowing Jammed

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