This is the whole course in one sitting. Every idea you’ve met — the neutron that either escapes or splits another atom, the amplification factor k that decides everything by which side of 1 it sits on, the geometric fizzle versus the exponential runaway, R0 and herd immunity, network effects and S-curves and moats, Granovetter’s crowd thresholds, hysteresis that refuses to tip back, and the honest warning that most things never tip at all — shows up here as questions. Hold the through-line in your head: a self-amplifying process lives or dies by whether each event causes, on average, more than one more. Below 1 it dies quietly; above 1 it runs away. Take a breath. There is no going back once you commit — rather fitting for a course on irreversibility.
How this exam works
This is a real exam, not a practice quiz. Questions appear one at a time. Once you submit an answer it is locked for good — there is no going back, no retry, and no restart. Your score stays hidden until the very end. A few questions ask you to select all that apply (read those carefully — partial credit is not a thing here). You need 70% to pass. Ready when you are.
The single idea at the heart of the whole course is the amplification factor k — the average number of new events each event causes. What defines "critical mass"?
Select an answer to continue.
Big picture
Critical mass, in one picture
- Critical mass
- The nuclear origin
- Neutrons escape through the surface while fission fills the volume, so volume as r cubed beats surface as r squared and a big enough lump keeps k above 1 - sub-critical dies, critical holds steady in a reactor, super-critical explodes
- The general threshold
- Any reinforcing loop tips when its per-step gain k crosses 1 - below 1 it decays to a finite seed over 1 minus k, above 1 it grows without bound, and seed size is not k
- Tipping points everywhere
- Epidemics with R0 and herd immunity at 1 minus 1 over R0, network effects and adoption S-curves and moats, Granovetter crowd thresholds and bank runs, autocatalysis and forest-fire percolation
- Why change feels sudden
- The lily pad full on day 30 is half full on day 29, so exponentials look like nothing then everything - diagnose by the ratio between successive waves, not the current level
- Lock-in and hysteresis
- Some tips do not tip back - reversing means overshooting far past the flip point, as in ice-albedo, collapsed fisheries and QWERTY, so prevention beats cure and moats and habits defend themselves
- Where the model lies
- Do not extrapolate a line toward a cliff, remember most things never tip, k is not constant, you know only which side of 1 and which way it moves, and not every fast change is a threshold crossing
- The nuclear origin
Key takeaways
You now hold the whole model. Critical mass is decided by one number — the amplification factor k, the average number of new events each event causes — and by which side of 1 it sits on: below 1 a process decays to a finite total (about seed / (1 - k)), above 1 it runs away exponentially. The nuclear origin shows why size matters: neutrons escape through the surface while fission fills the volume, so a big enough lump keeps enough neutrons to cross the threshold; reactors pin k at 1, bombs push it past. The same shape reappears as R0 and herd immunity (1 - 1/R0), network effects, S-curves and moats, Granovetter crowd thresholds and bank runs, autocatalysis and forest-fire percolation. Because exponentials look like “nothing, then everything” (the pond is half-covered on day 29), you diagnose by the ratio between successive waves, not the current level. Some tips are sticky — hysteresis and lock-in mean reversing costs far more than preventing, so near an irreversible tip prevention beats cure. And stay honest about where the model lies: don’t extrapolate a straight line toward a cliff, remember most things never tip, k isn’t constant, you usually know only which side of 1 and which way it’s moving, and not every fast change is a threshold crossing.