Candidates for a suite of educational STEM games in the Repulsion template — each an instrument, not an illustration, with a thesis that is surprising, a phenomenon that is beautiful before it is explained, a dial, and a game whose losing case is the lesson.
Four generators each proposed eight candidates from a different vantage — a physicist who builds exhibits, a mathematician who writes for Quanta, a complex-systems researcher, and a game designer who chose for mechanics first. Three judges (a curator, a physics teacher, a sceptical engineer) scored all 32 on five criteria: educational, playful, visually compelling, honest, template fit. Duplicates were merged (percolation, traffic, the arcsine law each came in twice). The tiers below are my curation of their ranking; the scores are the judges' mean out of 25.
What the template demands, restated as a checklist: a one-paragraph thesis; look first (phenomenon beside a control); a statistic accumulating live against a theory curve; a deeper diagnostic that nails the identification; a dial across a family with labelled landmarks; a game with a score; a provenance section that says what is exact, what is surmise, what is fitted. Single file, no network at runtime, every number measured.
Each has a thesis I had not seen made visual before, an exact theory to measure against, and a game that is genuinely tense. Together they cover the three domains.
Tile a diamond-shaped board with dominoes, choosing one of the 2n(n+1)/2 tilings uniformly at random. Outside a perfect inscribed circle the tiling is frozen — every domino aligned, no choice left; inside, it churns. Nobody put a circle in the rules. The page measures π by counting frozen dominoes.
Randomness at the smallest scale can produce exact determinism at the large scale, with a boundary curve that appears nowhere in the rules.
A snapshot of a magnet: spins up or down at some temperature you must name. Hot is noise; cold is one colour; between, islands. It is an easy guessing game everywhere on the temperature axis — until within a couple of percent of 2.269, where it becomes impossible, and impossible for a reason: islands come in every size at once, so the picture has no scale to compare against. The page's own simulation finds Onsager's 2/ln(1+√2) to three digits.
At the critical temperature a material looks the same at every magnification, which is exactly why you cannot read its temperature from a picture — the diverging correlation length has eaten the only ruler you had.
Aim rays — electrons in a clean semiconductor, sound in the ocean, starlight through plasma — at a landscape of very weak, very smooth bumps that deflect nothing by more than a thousandth of a radian. The beam should spread into fog. Instead it collects into branches: bright filaments that split, persist, and carry many times the average intensity. The first branch forms at a distance scaling as the −2/3 power of the bump strength, for all four systems, because branching is focusing, and focusing points are caustics. This is why stars twinkle and where rogue waves come from.
Smooth randomness focuses before it blurs — weak, gentle disorder concentrates a beam into branches and rare bright spots long before it spreads it out.
Mark α, 2α, 3α… around a circle, keeping fractional parts. After n marks the circle is cut into n arcs — and those arcs have at most three distinct lengths, for any α and any n, forever, with the largest always the sum of the other two. Which three is dictated by the continued fraction of α, and that is why sunflowers place seeds at 137.5°: the golden ratio has the slowest-converging continued fraction of any number.
How well a number can be approximated by fractions is directly visible as the pattern of gaps its multiples leave on a circle — and the golden ratio wins because it is the hardest number to approximate.
Each would be a fine page; they rank behind Tier 1 on either build cost, cliché risk, or overlap with work already done.
Two chemicals, one slow activator and one fast inhibitor, and a uniform mixture breaks into spots or stripes with a spacing set by the chemistry alone. So a bigger dish makes more stripes, not wider ones. The page measures the selected wavelength from the live field and puts it beside the prediction from three lines of linear algebra on the Jacobian.
Diffusion-limited aggregation, coloured not by age but by the live harmonic measure, so tips burn while fjords go cold. Two exact theorems pull in opposite directions — Beurling's bound and Makarov's theorem that the set receiving the arrivals has dimension exactly one — on an object nobody can solve (dimension ≈ 1.71, measured for forty years, derived by nobody).
Pulse-coupled fireflies (Mirollo–Strogatz, 1990): from almost any start, any number of them reach perfect unison in finite time and can never come apart, because once two fire together they have merged. The whole theorem rests on the charging curve bending downward; straighten it and sync becomes impossible.
The Talbot effect: a grating re-images itself in empty space at a fixed distance; at any rational fraction p/q of that distance the field is exactly q shifted copies weighted by Gauss sums; at an irrational fraction the intensity is a fractal of dimension exactly 3/2. Number theory drawn in light, exact to machine precision.
A ring road with no cause for a jam gets one anyway, drifting backwards at exactly 1 − 2ρ. Strip the cars to one rule and the model is exactly solvable; spill a packed block of cars and the front roughens as the cube root of time, with fluctuations following the Tracy–Widom law — the same distribution as the top eigenvalue of a random matrix.
| Name | Domain | Thesis in a line | Condition | Score |
|---|---|---|---|---|
| Oligarchy | complex | Fair, zero-expectation multiplicative bets condense all wealth onto one person; the ensemble mean rises while every individual falls. | Must read as a statement about multiplicative processes, with its assumptions and the standard criticisms of econophysics stated bluntly, or it is dishonest by framing. | 22.3 |
| Crossing | cross | At the percolation threshold the crossing probability depends only on the rectangle's shape — Cardy's exact formula — for square, hex, bond, or random discs alike. | My own top seed; the panel ranked it mid-table because percolation is familiar. Cardy's curve with four lattices collapsing onto it is the fresh part — lead with that. | 20.5 |
| Horizon | physics | Each extra digit of measurement precision buys only a fixed small amount of prediction time (the Lyapunov horizon, measured). | Strong lesson, weaker visual; the double pendulum is a cliché. Lorenz with a "how many days ahead" forecasting game may be the better vehicle. | 22.0 |
| Collusion | physics | No pre-arranged plan lets two separated players win the CHSH game more than 3 times in 4 — but a shared entangled pair wins 85%. | The only quantum candidate; the game is the experiment itself. Needs a careful, non-mystical thesis. | 20.3 |
| Gasket | mathematics | Apollonian circle packings: passing every congruence test is not the same as being possible (the local–global conjecture and its 2023 disproof). | Beautiful and current, but the game is hard to make tense. | 22.7 |
| Noble | physics | In the standard map the last orbit to survive the onset of chaos is the golden-mean torus (KAM). | Pairs naturally with Sunflower — the same number for a different reason. Consider as Sunflower's sequel. | 22.0 |
| Lead | mathematics | In a fair coin game, one side leading almost the whole way is the typical outcome (the arcsine law). | Cheap, counter-intuitive, but visually thin; good as a short piece. | 20.3 |
| Return | physics | FPUT recurrence: a nonlinear chain hands its energy back because it sits beside an integrable system (Toda) it cannot see. | Deep and honest, but slow to be impressive; the Toda-invariant panel is both the point and the easiest thing to get quietly wrong. | 20.8 |
Quench (Kibble–Zurek; excellent physics, but the barcode visual is thin and the exact-solvability argument is graduate-level); Wake and Roughen (long-time tails, KPZ — important, not playful); Bias, Threshold, Patience (Chebyshev's bias, the secretary problem, longest increasing subsequence — each a lovely fact, none with a game that teaches by losing); Contagion (epidemic thresholds on networks — strong lesson about variance, but the visual is a network diagram); Murmuration (Vicsek flocking — beautiful, but the measurable thesis, giant number fluctuations, is subtle); Catastrophe (the quasispecies error threshold — niche); Revival (Riemann's non-differentiable function — stunning, no game); Ratchet (Parrondo's paradox — one day, but a curiosity rather than a principle); Sojourn (duplicate of Lead).
Two of my own pre-panel seeds were not generated and deserve a second look as wildcards: Benford (leading digits of Fibonacci numbers, powers of two, and any public-domain table — game: spot the fabricated ledger; lesson: scale invariance has a fingerprint) and Sandpile (Bak–Tang–Wiesenfeld — game: bet on the size of the next avalanche; lesson: a power law has no typical event). Both are cheap, computable, and game-native.
Tier 1 plus Tier 2 gives nine pages: three physics (Fever, Branch, Carpet), two mathematics (Arctic, Sunflower), four complex systems (Morphogen, Lightning, Flash, Phantom), with Repulsion as the tenth and the template. Several rhyme on purpose: Repulsion and Phantom both end at random-matrix universality from opposite directions; Fever's "name the temperature" and Repulsion's "which one is the primes" are the same game — the un-guessable case is the lesson; Sunflower and Noble are the golden ratio twice for different reasons; Arctic and Crossing are both about exact shapes that emerge from coin flips.
Recommended order: Arctic first (the most startling image, an exact theorem, a π measurement, and a self-test that makes verification clean), then Sunflower (two days, cheapest), then Fever (the physics anchor and the closest sibling to Repulsion), then Branch. Each build follows the Repulsion process: data/kernel built and adversarially verified first, page second, five-lens review third.
Method note: 32 candidates from four Opus generators, scored by three Sonnet judges; full proposals with all five instruments, data plans, hues and risks for every candidate are preserved in the session's brainstorm file.