Moving visuals on this page — measured at 0.26% brightness change per frame and zero flashes per second; your reduce motion setting stops them. What moves, and how it was measured
Mathematical plates that draw themselves — prime counts, orbits and lattices rendered live rather than shipped as pictures. It runs continuously while on screen, and it pauses on its own whenever it scrolls out of view.
We measured it rather than guessing: the largest frame-to-frame change in average brightness is 0.26%, and it produces zero flashes per second against the threshold of three set by WCAG 2.3.1. That is well under the limit, and it is not a promise about any individual: if moving images are a problem for you, you should not have to assess ours.
Everything animated on this site honours your system’s reduce motion setting — turn it on in your operating system or browser and these pages stop moving, with nothing else lost. All the mathematics is in the text and the still figures.
Entry 3 of 3 in Curiosities
Axelrod’s tournament, run again in your tab
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Math: the iterated prisoner’s dilemma with the classic payoff matrix R=3, S=0, T=5, P=1. Fifteen strategies, round-robin including self-play, 200 turns per match, 25 repetitions — 3,000 matches, 600,000 turns. Method: written from scratch, no game-theory library; the site’s rule is that no third party does the work for us. Dev: the same tournament exists twice, once in Python (research/axelrod/tournament.py) and once here, sharing a seeded mulberry32 so the two can be compared digit for digit. They agree to 0.0 points across all fifteen positions. That cross-check is the only real evidence the port did not drift, and it caught two genuine bugs before this shipped.
Replicating Axelrod & Hamilton (1981), doi:10.1126/science.7466396 — the paper that showed cooperation can win without anyone intending it. The claim under test is not “Tit for Tat wins”. It is the structural one: strategies that never defect first take the whole top of the table, and every strategy willing to strike first sits below them. Press the button and watch it happen.
Nothing runs until you press the button. It takes well under a second.
The result is not robust, and that is the interesting part
The tournament above is noiseless: every strategy plays exactly the move it intends. Real cooperation is not like that — signals get lost, hands slip, intentions are misread. Add a probability that each move comes out inverted and re-run the identical field, and the finding reverses. Two cooperators running Tit for Tat, after a single accidental defection, fall into an echo of mutual retaliation that neither one chose and neither can leave.
We swept it. Same fifteen strategies, same payoffs, same 200 turns and 25 repetitions, one parameter added:
Two things happen as the noise rises. The clean split breaks almost immediately — by bisection it survives to 0.594% and not beyond, and repeating the bisection under five unrelated seeds gives 0.59%, 0.64%, 0.60%, 0.56%, 0.58%, so the figure is a property of the field and not of one lucky run. Then the character of what wins changes: forgiving strategies that recover from an accident (Generous Tit for Tat, Pavlov’s win-stay lose-shift) climb past rigid ones, and by 20% the top of the table belongs to the unforgiving. Cooperation does not merely become harder; the recipe for it changes.
Whose result this is. The qualitative finding is not ours — noise in the iterated prisoner’s dilemma has been studied since Molander (1985), and Nowak & Sigmund’s work on Generous Tit for Tat and Pavlov in the early 1990s is where the strategies that survive it were identified. What is ours is the arithmetic on this field: nobody else has run these particular fifteen strategies at these settings, so the 0.594% is a fact about our replication rather than a fact about the world. We state it because it is checkable, not because it is new.
Two caveats. Axelrod received fourteen entries plus RANDOM; five of them (Nydegger, Tideman & Chieruzzi, Downing, Graaskamp, Stein & Rapoport) rely on chi-squared tests, Bayesian opponent models and internal state that the published descriptions do not specify tightly enough to rebuild without inventing, so they are left out and the scores here are not comparable entry-by-entry with the published 504.5 / 500.4 / 485.5. And Tit for Tat does not come first in this field — Tit for Two Tats does, which is precisely what Axelrod himself said would have happened had Rapoport submitted it.