INTERACTIVE COMPANION · EXPLAINER · GENERAL SCIENCE

The Dating Bench

A live model accompanying “How Anybody Knows How Old Anything Is”

Two benches. The first one lets you watch atoms decay, one at a time, and see the exponential law assemble itself out of nothing but coin flips. The second takes a radiocarbon age and calibrates it against the real IntCal20 curve, which is where you find out that a single clean measurement can map onto three separate centuries.

Both run entirely in your browser with no libraries, and every number they print can be checked by hand against the equations printed beside them.

Bench 1. Decay, atom by atom

Each dot is one atom. On every tick of the clock, each surviving atom independently decides whether to decay, with probability \(p = 1 - e^{-\lambda \Delta t}\) where \(\lambda = \ln 2 / T_{1/2}\). There is no exponential curve in the simulation. The curve you see growing on the right is drawn from the surviving count, and the pale line behind it is the analytic law, added afterwards so you can see how well the coin flips reproduce it.

The prediction for the spread, which is the part worth watching, is the binomial standard deviation \(\sigma_N = \sqrt{N_0 f (1-f)}\) with \(f = e^{-\lambda t}\). At exactly one half-life with 1,000 atoms that is \(\sqrt{1000 \times 0.25} = 15.81\) atoms. Run it a few times and see how often you land more than two of those away from 500.

Population 1,000 atoms Start at 1,000, which is where the article's numbers come from. Then drop to 100 and run it three or four times: the trace stops being a curve and becomes a staircase, and the half-life you would read off it moves by tens of percent between runs. Push it to 20,000 and the trace welds itself to the analytic line. Half-life 3.0 seconds This only changes how fast the clock runs, never the shape. That is the point: a nuclide with a half-life of five thousand years and one with a half-life of five billion do the identical thing at different speeds. Show the analytic curve Turn it off, run a 100-atom population to four half-lives, and try to guess where the smooth law would have been. Then turn it back on. Nobody in the club got it right.

elapsed: 0.00 half-lives surviving: 1000 analytic expectation: 1000.0 binomial s.d.: 0.0 deviation: 0.00 s.d.

Press pause at any moment to read the counts off.

Check by hand. With 1,000 atoms at exactly one half-life, the analytic expectation is 1000 × 0.5 = 500.0 and the binomial spread is sqrt(1000 × 0.5 × 0.5) = 15.81. Our article ran 4,000 of these worlds and got a mean of 499.91 with a spread of 15.797 against the predicted 15.811.

Bench 2. Calibrating a radiocarbon age

A conventional radiocarbon age is a laboratory measurement, not a date. Turning it into a date means asking which calendar years the IntCal20 curve predicts that radiocarbon age for, weighting every candidate year by how well it matches, and keeping the densest 95.4% of the resulting probability. That is what the panel below does, using the published curve resampled to ten-year knots and interpolated onto a one-year grid.

The naive answer, subtracting the radiocarbon age from 1950, is shown alongside so you can watch how far wrong it goes.

Measured radiocarbon age 2450 BP 2450 is the default because it lands on the Hallstatt plateau and produces the article's three-block answer. Now try 10,360, the worst value on the whole curve: four blocks and 419 calendar years. Then try 10,780, only 420 radiocarbon years away, which collapses to about thirty calendar years. Laboratory error ± 30 years Counter-intuitive one. Sitting on a plateau at 2450, tightening the error from 30 to 15 does not narrow the answer, it splits it into more blocks. Off a plateau, at 4280 say, halving the error really does halve the range. Money spent on precision only buys you something where the curve is steep. Window half-width 400 calendar years How much of the curve to draw either side. Widen it to 1,200 at a radiocarbon age of 9,600 to see how long the flat stretches really are.

naive 1950 minus age: 500 BC calibrated median: – 95.4% blocks: – total span: –

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Check by hand. At the default 2450 ± 30 the blocks should read 2362–2543, 2559–2618 and 2632–2701 cal BP, totalling 312 calendar years, with the calibrated median at 2517 cal BP. That median is 568 BC, while the naive subtraction says 500 BC: sixty-eight years out, and that is one of the smaller errors on the curve. Every one of those figures is printed in analysis/how-we-date-things-output.txt, section C3.

What you are looking at in the lower panel is not measurement noise. The laboratory did its job. The lumps are the atmosphere: centuries in which carbon-14 production and ocean uptake happened to cancel, leaving the radiocarbon clock parked on the same reading while the calendar moved on. The trees recorded it, which is the only reason anybody can correct for it.