DOMINOES - credits, sources, and what this app made up ====================================================== This is an independent, from-scratch browser app. It is not a reimplementation of any commercial product, and no other party's code, art, branding or trademarks appear in it. Dominoes itself has no author and no owner. The European double-six set has been played since the eighteenth century and the game is in the public domain everywhere. What DOES have authors is the physics of a toppling line, and this app is checked against them. 1. THE SOURCES, AND WHAT EACH ONE ACTUALLY GAVE ----------------------------------------------- [W] "Dominoes", English Wikipedia. https://en.wikipedia.org/wiki/Dominoes Read through the MediaWiki API on 2026-09-11. - A European double-six set is 28 tiles. - Tiles are "normally twice as long as they are wide". - The total pip count of a double-six set is n(n+1)(n+2)/2 = 168. - The BLOCK game: two players, seven tiles each from a shuffled stock; play alternately onto either end matching pips; a player who cannot play draws until they can; the game ends when a player is out or when neither can move, and then "whoever caused the block receives all of the remaining player points not counting their own". - The DRAW game: a player may draw as many tiles as they like before playing and may not pass while the stock holds tiles; "the score of a game is the number of pips in the losing player's hand plus the number of pips in the stock. Most rules prescribe that two tiles need to remain in the stock." - "Players often play tiles at right angles when the line of play gets too close to the edge of the table", which is why the line here snakes. - Doubles are laid crosswise; the spinner rule (doubles branching the line) is described as a variant, so this app lays doubles crosswise but does NOT branch on them. [T] "Domino toppling", English Wikipedia. https://en.wikipedia.org/wiki/Domino_toppling Domino Day records: smallest tile 7 mm, largest 4.8 m, longest chain 4,491,863 tiles (Leeuwarden, 2009). [VL1] J. M. J. van Leeuwen, "The Domino Effect", arXiv:physics/0401018v1 (2004); published as Am. J. Phys. 78, 721-727 (2010), doi:10.1119/1.3406154. PDF downloaded and read for this app (188,901 bytes, sha256 6098d31f...). - The model: tiles rotate only and never slide; after a collision they stay in contact, so the collision is fully inelastic; the train of leaning tiles is a soliton. - I about the bottom edge = m(h^2+d^2)/3; time scale sqrt((h^2+d^2)/(3gh)); velocity scale sqrt(gh). - Centre-of-mass height of a tile tilted by theta: 2 z = h cos(theta) + d sin(theta). - Contact geometry: h sin(theta_i - theta_i+1) = (s+d) cos(theta_i+1) - d. - Contact angle arcsin(s/h); stacking angle arccos(d/(s+d)). - Released potential energy P = [s h - d sqrt(s^2+2sd)] / [h (s+d)]. - TABLE 1, the asymptotic soliton speed in units of sqrt(gh) for d/h = 0.179, at nine separations and four friction coefficients. Those 36 numbers are reproduced verbatim in tools-oracle.js and are the third oracle this app is checked against. - His central correction: conserving the train's total angular momentum through a collision (Shaw's law) is wrong, because successive tiles turn about different axes. - He estimates the mutual friction of wooden tiles at about 0.2 by tilting the table until they slip over each other. [VL2] J. M. J. van Leeuwen, "Domino Magnification", arXiv:1301.0615v2 (2013). PDF downloaded and read for this app (151,429 bytes, sha256 3131946a...). - d/h = 0.14583 is quoted as the ratio of a standard domino. That is 7/48 exactly. - Frictionless, topple-only, scale-invariant growth; q = 4 for solid tiles (mass follows volume), q = 3 for hollow ones (mass follows surface); the inertia ratio between neighbours is r^-(q+1). - The two-tile survival condition, his equations (9), (12), (15) and (17), implemented independently in tools-oracle.js as the fourth oracle. - The largest workable growth ratio rises with separation to about 2 near s/h = 0.5 and then falls along r = 1/s, which is simply the requirement that the smaller tile can still reach the larger one. - "Whitehead describes a demonstration with a magnification factor r = 1.5." [WH] L. A. Whitehead, "Domino 'chain reaction'", Am. J. Phys. 51, 182 (1983), doi:10.1119/1.13456. ONE PAGE, BEHIND THE AIP PAYWALL, AND NOT READ FOR THIS APP. The citation was verified through Crossref's metadata API: author Lorne A. Whitehead, University of British Columbia; volume 51; page 182; February 1983. Everything this app says about it is what [VL2] reports about it, and nothing more. [EJ] C. J. Efthimiou and M. D. Johnson, "Domino Waves", SIAM Review 49, 111-120 (2007), doi:10.1137/s0036144504414505. Crossref-verified; not read directly. [DW] D. Ding, C. Lau, J. Westerhof, L. van der Hoeven, L. Kampstra, P. van der Beek and I. Ostanin, "How Fast are Domino Waves?", arXiv:2204.07997v4 (2022). PDF downloaded and read for this app (1,619,288 bytes, sha256 a7bdd6c2...). Criticises the instantaneous-collision theory for predicting an infinite propagation speed as the gap goes to zero and for missing the collective nature of the contact. [SP] L. Dalla Pola, L. Darmendrail, E. Galantay and A. Muller, "Listen! A smartphone inquiry on the domino effect", arXiv:2303.17231v1 (2023). PDF downloaded and read for this app (1,011,508 bytes, sha256 58dabb0c...). Measured by recording the collision clicks: 118 cm/s at s/h = 0.34 and 91 cm/s at s/h = 0.58, on tiles 4.1 cm tall with d/h = 0.146. Their Figure 4 - total elapsed time against tile number is a straight line after about 5 to 10 tiles - is the behaviour the second oracle in tools-harness.js reproduces. [SH] D. E. Shaw, "Mechanics of a chain of dominoes", Am. J. Phys. 46, 640-642 (1978). Cited by [VL1] and [VL2]; not read directly. 2. HOW THE SOURCES WERE CHECKED ------------------------------- Some servers answer every path with the same page, so an HTTP 200 is not evidence that a document was retrieved. Everything above that came off the network was either downloaded as a PDF and hashed - the four arXiv papers have four different sha256 prefixes and four different byte counts, so they are four different documents - or looked up in Crossref's metadata API, which returns a different record per DOI and would not have invented one. CORRECTION TO A SOURCE. Reference [11] of van Leeuwen's "Domino Magnification" cites Whitehead as "Am. J. Phys, 2 (1983) 182". The volume is 51, not 2; Crossref's record for doi:10.1119/1.13456 gives volume 51, page 182, February 1983. It is a typographical slip in a bibliography and nothing turns on it, but it is recorded here because this app quotes that reference and cannot check the paper itself. A NUMBER THIS APP COULD NOT REPRODUCE. The Domino Toppling article [T] lists, among the Domino Day 2008 records, a "fastest topple of 30 metres of domino tiles (4.21 sec)". That is 7.1 metres a second. A wave on ordinary tiles runs at roughly 1.37 sqrt(gh), which for a 48 mm domino is about 0.94 m/s; to reach 7.1 m/s by that scaling the tiles would have to be about 2.7 m tall. Whatever that record measured, it was not a line of ordinary dominoes toppling one another, and this app does not reproduce it. 3. WHAT THIS APP CHOSE, BECAUSE NOTHING PUBLISHED FIXES IT ----------------------------------------------------------- RECONSTRUCTED - marked as choices on the page's limits section too: * The tile is 48 x 24 x 7 mm. The two RATIOS are sourced ([W] for 2:1, [VL2] for d/h = 0.14583 = 7/48); the millimetres are this app's rounding. No manufacturer is claimed. * Tile-on-table friction, 0.35 by default. Nobody publishes one for a plastic domino on a table. It is a slider, and the app reports what it does. * Tile-on-tile friction defaults to 0.20, which is [VL1]'s own estimate for wood, used here as a default rather than as a measurement. * Restitution 0. Every source treats the collision as fully inelastic; [DW]'s whole point is that a finite stiffness changes the answer, and this app does not model one. * The target score for a match, who opens a game, and how the computer decides what to play. None of those are in [W] for the two-hand game. * Every layout - the curve, the S-bend, the spiral, the branch, the fan and the growing chain - and all the colours, the camera work and the tile art. * The physics step and the number of contact-solver sweeps. Both were MEASURED against the published table rather than picked; the measurements are in tools-harness.js and the resulting error is quoted on the page. 4. WHAT THE ENGINE DOES THAT THE PUBLISHED MODEL DOES NOT ---------------------------------------------------------- Every published treatment gives a tile ONE degree of freedom: it may topple, and it may not slide. That makes tile-on-table friction irrelevant by construction. Here a tile has TWO - a tilt and a slide of its pivot along its own fall line, held by Coulomb friction against the table - so the two frictions are different numbers doing different jobs. The cost and the benefit were both measured: * With the pivot PINNED, which is exactly the published assumption, this engine reproduces all thirty-six cells of [VL1]'s Table 1 to within about one per cent, and most of them to a few hundredths of one. * With the pivot FREE, the same runs agree up to a gap of about half the tile height and then diverge, by up to about eight per cent at the widest spacing - because that is exactly where the striker lands low on the next tile and shoves it forward instead of over. [VL1] predicts that in words: "the foremost domino hits the next at a low point, such that the tendency to slip is larger than to topple, as the moment arm becomes very small." * The free pivot produces an UPPER edge to the spacing window that the published model cannot have, and that edge moves with the table friction. The numbers are on the page. What the engine still cannot represent is stated on the page's limits section: a tile that twists, tips sideways, leaves the table, or slides so far that it ends up inside the pile. 5. LICENCE ---------- The code, the art and the text of this app are released under the MIT licence; see LICENSE.txt. The game of dominoes is in the public domain. The papers cited above are the property of their authors and publishers and are not reproduced here - only their published numbers are quoted, with the citation attached.