A curling stone leaves your hand at the hack, glides down pebbled ice, and can hook markedly in its last few seconds. Late. Surprising. Almost alive.
A century of physicists have tried to explain that hook. They still disagree.
Inside the pinned Shegelski model, the late hook is a cube, not a line: Δy = (k·ω·μ·g / 6m)·τ³. A proof is a receipt.
Δy = (k·ω·μ·g / 6m)·τ³·shegelski_late_curl_cubic · Audrey/Friction/LateCurl.lean:162Spin a wine glass clockwise on a wet table and shove it forward. Under familiar tabletop conditions it often veers left. A common friction model for sliding spinners predicts a hook away from the direction of rotation because the two sides experience different relative motion.
Throw a clockwise in-turn down a sheet of curling ice. The rock hooks right. Toward the spin. Against the rule.
“A curling stone is a striking counterexample to familiar spinner intuition: it curls toward its rotation, and the mechanism remains contested.”
Mark Shegelski’s idea: the front half of the running band, plowing into fresh pebble tops, melts a slightly thicker water film than the trailing half. Less friction in front, more behind, and the rock pivots into its rotation.
Falsifiable: vary the ice temperature and measure whether the asymmetry and curl weaken together. Reported experiments provide evidence to compare with that prediction; this page does not claim universal agreement.
Harald Nyberg and Sture Hogmark at Uppsala photographed the ice after a stone passed and found tiny cross-scratches, etched by microscopic roughness on the running band, all leaning the same way. The next pass catches on those scratches like a needle in a record groove and is nudged sideways.
Falsifiable: scuff the sheet with sandpaper before each shot. The scratch-pivot model says curl drops to zero. Asymmetric melting says nothing should change.
Twenty years after the first paper, Shegelski returned with Edward Lozowski and a peace treaty. Maybe both are real. The pivot-slide model has the rock alternating between long stretches of asymmetric-melt sliding and brief scratch-driven pivots at the back of the band. The weight between the two depends on speed.
Falsifiable: high-speed video the last six metres. The blended model predicts a kink in the curl rate at a critical speed. Single-mechanism models predict a smooth curve.
“Three theories. One rock. We are not done arguing.”
A schematic cross-section through the running band. Contact is concentrated near a thin ring at the edge of the cupped underside.
A regulation stone weighs between 17.24 and 19.96 kilograms (WCF Rules of Curling, 2025). Competition stones are commonly associated with two granite sources: Ailsa Craig, an uninhabited granite plug in the Firth of Clyde off the Scottish coast, and Trefor, on the Llŷn Peninsula in north Wales. Ailsa Craig has supplied stones for over 150 years; the island is now a bird sanctuary and quarrying is rare.
Pick up a stone and turn it over. The bottom is concave. A thin ring touches the ice — the running band, about 6.4 centimetres in radius and a few millimetres wide. That narrow surface is where the stone and pebble interact. The material is selected for durability under repeated impacts and temperature cycling; those engineering demands help explain curling’s preference for particular granites.
And here is the scale that sharpens the puzzle: even within the running band, estimates put the actual contact area at any instant at a few square millimetres. Not centimetres. Millimetres. The ice is sprayed with frozen droplets, and the rock rides on a small set of droplet-tops at a time — a marble on a handful of grains of rice.
“Most of a curling stone is decoration. The math is in the ring.”
Turn the stone over and the contact question stops belonging to curling. In 1882 Heinrich Hertz worked out how two elastic bodies deform where they touch; Johnson’s Contact Mechanicsis the modern reference on every tribologist’s shelf. The running band pressing a pebble top is one instance of the same model.
The pinned source record proves the model’s scaling identity exactly: scale the radius by λ and the load by λ³, and the contact radius scales by λ4/3. No approximation, no regime restriction inside the model’s own definitions. On log-log axes the identity is a straight line of slope 4/3, and the same line carries a granite band, a skate rocker, and the tip of an atomic force microscope.
766.13 µmcontact radius a = λ4/3 · a₀, with a₀ = 766.13 µm for the granite band under its own weight
a(λR*, λ³F) = λ^(4/3)·a(R*, F)·hertz_scale_invariance · Audrey/Universality/ContactMechanics.lean:82The record proves the scaling identity of the defined Hertz model. It does not claim the model fits every physical contact, and the landmark radii above are characteristic sizes, not measurements of this sheet.
That is why the millimetre-scale contact patch matters. The same closed form that sizes the band’s footprint on a pebble top is the one a scanning probe group uses to read forces off a silicon wafer. The exponent does not ask what the bodies are made of.
“The 4/3 does not care whether the sphere is granite or a probe tip. That is the whole receipt.”
To prepare a standard sheet, an ice technician sprays water through a fine rose. The droplets freeze into pebble — a field of tiny frozen domes. Pebble concentrates contact at droplet tops and materially changes how the stone slides and curls; this page does not model an unpebbled-sheet counterfactual.
Pebble reduces the contact surface to a small set of droplet-tops that deform as the rock travels. Across forty-seven trials at Calgary Olympic Park, Penner reported μ = 0.0168 ± 0.0008 under the tested conditions. (Penner, Am. J. Phys. 2001.)
The technician chooses a temperature, a stroke pattern, a droplet size. Two technicians on the same arena ice can produce sheets that play differently. The ice tech is the instrument maker — the closest thing curling has to a Stradivarius. Too few pebbles and the rock skates without grip. Too many and the meltwater films merge and the curl saturates. There is a sweet spot.
“A man with a watering can decides how your rock will curl.”
Watch a draw in slow motion and its path often appears nearly straight before a late hook. The exact split depends on release and sheet conditions. Curlers recognize the pattern; competing physical models explain it differently.
In the simplified numerical account above, forward speed decays faster than spin, so the modeled spin-to-slide ratio rises late in the shot. Its chosen lateral-force law turns that ratio into a sharper bend near rest. Those assumptions generate the browser curve; measurements are needed to decide how well they describe a particular rock and sheet.
In the next game, look for a relatively straight early path and a larger late deflection. The proportions vary by release and sheet.
Illustrative browser integration. This simplified numerical trajectory is a visualization, not a calibrated physical prediction or theorem receipt.
Modeled hook: 0.20 m · Sheet covered: 23.2 m
Δy ∝ s³·shegelski_late_curl_cubicThe linked record is narrower: under its stated parameters and positive-mass hypothesis, it proves a cubic identity for a defined displacement expression. It does not certify this integrator, its slider values, or real-sheet accuracy.
“In this simplified model, the late hook follows from forward speed decaying faster than spin. The browser curve illustrates that account; it does not settle the competing physics.”
In a simplified account, sweeping warms and mechanically conditions the pebble, changing friction so the rock can travel farther and curl differently. The relative roles of heat, brushing, debris, and ice conditions are empirical questions, not a single theorem on this page.
In the cited experimental conditions, Penner reported unswept friction at μ = 0.0168 and a swept value near 0.0128. Day and Reid reported broom output on the order of a hundred watts for competition sweepers. Extrapolating either result to another sheet requires matching its conditions. (Penner 2001; Day & Reid 1994.)
Ten feet can be the difference between freezing to a guard and clipping it. Curling is unusual in how directly teammates can influence a shot after release.
“Sweeping isn’t magic. It’s a degree or two of warmth, twenty seconds of contact, and a hundred watts of will.”
Hammer is the last-rock advantage. In the displayed tied-game Markov model, its estimated value changes by end and generally rises late in the game. That result is conditional on the model’s transition estimates and tied ten-end setup.
Kostuk, Willoughby and Saedt modelled a curling game as a Markov chain in 2001 and estimated the value of hammer across a tied ten-end game under that model. Their work is one influential quantitative treatment of end-by-end strategy. (Kostuk, Willoughby & Saedt, Eur. J. Oper. Res. 2001.)
In the displayed model, blanking with hammer in end one compares 0.45 expected points now with 0.45 next end; in end nine it compares 0.60 with 0.70. Those are model outputs under its assumptions, not universal shot prescriptions.
“In this Markov model, the estimated value of hammer rises late. The table is a model output, not a universal strategy theorem.”
Illustrative score-state heuristic. This advisor sees only end, score difference, and hammer. It ignores the stones, ruleset, skill, ice, and opponent, so its prompts are neither optimal-strategy theorems nor outcome guarantees.
Two loaded simulation datasets appear below: a thousand modeled draws and a thousand modeled hit-and-stick takeouts, selected under a release-noise objective. The plots sample those files and fit descriptive least-squares lines. They are not measurements, Pareto certificates, or proofs of an optimal release.
In this simulated draw slice, greater launch speed is associated with greater distance from the button: within the selected range, the modeled shots overshoot farther. That fitted association describes this file, not every draw.
In the simulated hit-and-stick slice, launch speed and distance from the button have the opposite fitted association. The sign reversal belongs to these generated samples, objective, and parameter range; it is not a universal takeout prescription.
“One modeled objective, two descriptive trends. Neither plot certifies an optimum.”
Three pinned source records back this waddle. Each names its theorem, its file and line, its exact statement, and the scope it does not step outside. The interactive figures above are illustrations; these cards are the artifacts.
(p : ShegelskiParams) (ω μ g m τ : ℝ) (hm : 0 < m) : lateCurlDisplacement p ω μ g m τ = (p.k * ω * μ * g / (6 * m)) * τ^3
A conditional model identity inside the source's defined Shegelski late-curl model. It is not an empirical validation of curling physics.
open the pinned record(h : HertzPair) (F lam : ℝ) (hF : 0 ≤ F) (hlam : 0 < lam) : hertzContactRadius (h.scale lam hlam) (lam^3 * F) = lam ^ ((4 : ℝ) / 3) * hertzContactRadius h F
An algebraic property of the source's defined Hertz model. It does not prove the model applies to every physical contact.
open the pinned record{Catalogue : Type*} [Fintype Catalogue] (n k : ℕ) (trace : Catalogue → (Fin k → BoundedMat n)) (htrace : Function.Injective trace) : Fintype.card Catalogue ≤ Fintype.card (BoundedMat n) ^ kA necessary counting bound for injective encodings of a finite catalogue. It does not derive a logarithmic round bound or an identification strategy.
open the pinned recordThe full source graph for every waddle lives in the proof observatory.
This page mixes cited measurements, reported physical hypotheses, a simplified browser integrator, a score-state heuristic, and generated simulation data. The interactive outputs are illustrations; linked formal records apply only to their displayed definitions and hypotheses.
If you want to compare a question about your rock or ice with the assumptions behind these models, write to us.
Two finite-model essays next door: Onitama — board and card envelopes separated from legal-state and search claims — and Signal — a response-alphabet inequality separated from strategy and sufficiency.
The reading room is free. Three chat turns a day with the model are free once you sign in. Patrons are named on this waddle’s wall, and $150 of patronage crystallizes into an acrylic disc that ships by mail.
“Curling deserves better than folklore.”
Good curling.