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The gravity, twist and period relations below are the standard torsion-balance equations.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eF = G M m \/ r²,   κθ = F L,   T = 2π √(I \/ κ)\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eF: gravitational force (N)\u003c\/li\u003e\n\u003cli\u003eG: gravitational constant (m³ kg⁻¹ s⁻²)\u003c\/li\u003e\n\u003cli\u003eM, m: large and small masses (kg)\u003c\/li\u003e\n\u003cli\u003er: centre-to-centre distance (m)\u003c\/li\u003e\n\u003cli\u003eκ: torsion constant (N·m\/rad)\u003c\/li\u003e\n\u003cli\u003eθ: twist angle (rad)\u003c\/li\u003e\n\u003cli\u003eL: lever arm (m)\u003c\/li\u003e\n\u003cli\u003eT: swing period (s)\u003c\/li\u003e\n\u003cli\u003eI: moment of inertia (kg·m²)\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eHenry Cavendish's paper was read to the Royal Society on 21 June 1798, using a torsion balance devised by John Michell. 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At low energy her path is neat; push her harder and her trail frays into dust.\u003c\/p\u003e\u003ch3\u003eThe Physics\u003c\/h3\u003e\u003cp\u003eIn 1964 Michel Hénon and Carl Heiles asked whether a star moving in an axisymmetric galaxy has a third conserved quantity besides energy and angular momentum. They reduced the question to a two-degree-of-freedom model and integrated orbits numerically.\u003c\/p\u003e\u003cp\u003eAt low energy the Poincaré section shows smooth closed curves, as if a third integral exists. As the energy rises, the curves break up and a chaotic sea spreads between surviving islands. The potential has three-fold symmetry, and its three saddle points sit at energy 1\/6; above that, orbits can escape.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eH = ½(p_x² + p_y²) + ½(x² + y²) + x²y − y³\/3\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eH: total energy (dimensionless units)\u003c\/li\u003e\n\u003cli\u003ex, y: positions\u003c\/li\u003e\n\u003cli\u003ep_x, p_y: momenta\u003c\/li\u003e\n\u003cli\u003eSaddle energy: E = 1\/6\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eHénon \u0026amp; Heiles, Astronomical Journal 69, 73 (1964). It became a textbook early example of chaos in a Hamiltonian system.\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e It was not the first chaos ever found: Poincaré came first. Whether a real galaxy behaves like this toy model is a modelling question.\u003c\/p\u003e\u003ch3\u003eOn the shirt\u003c\/h3\u003e\u003cp\u003eFront: Helia, the pixel-art character, with “CHAOS IN THE GALAXY”. 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When the system is nudged, the rational rings around her shatter, and hers holds.\u003c\/p\u003e\u003ch3\u003eThe Physics\u003c\/h3\u003e\u003cp\u003eTake an integrable Hamiltonian system, whose motion winds around nested invariant tori, and perturb it slightly. Kolmogorov announced, and Arnold and Moser proved, that most tori survive. Those whose frequency ratios are “sufficiently irrational” (a Diophantine condition) persist, slightly deformed. Tori with rational, resonant frequency ratios break into island chains and chaotic layers. For small perturbations the surviving set has large measure.\u003c\/p\u003e\u003cp\u003eFor one model, the Chirikov standard map, Greene found numerically in 1979 that the last surviving rotational torus has the golden mean as its rotation number, and breaks at K ≈ 0.971635.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003e|k · ω| ≥ γ \/ |k|^τ\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003ek: any nonzero integer vector\u003c\/li\u003e\n\u003cli\u003eω: frequency vector of the torus\u003c\/li\u003e\n\u003cli\u003eγ, τ: positive constants of the Diophantine condition\u003c\/li\u003e\n\u003cli\u003eGolden mean: (1 + √5)\/2 = 1.618…\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eKolmogorov 1954; Arnold, Russ. Math. Surveys 18(5), 9 (1963); Moser 1962 (area-preserving twist maps); Greene, J. Math. 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Summarising those measurements, Schnitzer, Visscher and Block (2000) put the motor's maximum load at about 5.5–7.5 pN.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eW = F × d = 6 pN × 8 nm = 48 pN·nm ≈ 12 k_BT\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eW: work per step (pN·nm)\u003c\/li\u003e\n\u003cli\u003eF: load force, near stall (pN)\u003c\/li\u003e\n\u003cli\u003ed: step size (nm)\u003c\/li\u003e\n\u003cli\u003ek_BT: thermal energy, 4.1 pN·nm at 298 K\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eSvoboda et al., Nature 365, 721 (1993); Visscher et al., Nature 400, 184 (1999); Schnitzer et al., Nat. Cell Biol. 2, 718 (2000); Yildiz et al., Science 303, 676 (2004).\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e The stall force is not always 6 pN; it depends on conditions. 48\/4.1 is 11.7, so “about 12 k_BT”.\u003c\/p\u003e\u003ch3\u003eOn the shirt\u003c\/h3\u003e\u003cp\u003eFront: Kinessa, the pixel-art character, with “8 NM PER STEP”. 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