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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. 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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”. Back: the one-ATP step, the stall force, the work per step in k_BT and the 1993, 1999 and 2004 sources. 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The constant a sets a region's overall activity; the slope b is typically close to 1, so each step up in magnitude is about ten times rarer.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003elog₁₀ N = a − bM\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eN: number of quakes of magnitude M or more\u003c\/li\u003e\n\u003cli\u003eM: magnitude\u003c\/li\u003e\n\u003cli\u003ea: regional activity constant\u003c\/li\u003e\n\u003cli\u003eb: slope, usually near 1\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eGutenberg \u0026amp; Richter, “Frequency of earthquakes in California”, Bull. Seismol. Soc. Am. 34, 185 (1944).\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e b is not universal: about 0.8–1.2 is a common range for tectonic quakes, and volcanic or induced swarms can differ more. The power law is often linked to self-organised criticality, but that is debated.\u003c\/p\u003e\u003ch3\u003eOn the shirt\u003c\/h3\u003e\u003cp\u003eFront: Seisma, the pixel-art character, with “TEN TIMES RARER”. Back: the Gutenberg–Richter law, the ten-times-rarer rule, the regional range of b and the 1944 paper. 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In 1891 S. C. Chandler found a component with a period of about 14 months, about 433 days. Wilson and Vicente (1990), using 86 years of optical astrometry, estimated 433.0 ± 1.1 days with a quality factor Q ≈ 179; later estimates fall near 431 days. A rigid Earth of the same shape would wobble freely with a period of about 305 days; the elastic mantle, the fluid core and the oceans lengthen it.\u003c\/p\u003e\u003cp\u003eThe wobble damps, so something must keep exciting it. Richard Gross showed in 2000 that during 1985–1996 it was driven mainly by changes in ocean-bottom pressure, with the atmosphere second.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eT_Chandler ≈ 433 days   vs   T_rigid ≈ 305 days\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eT_Chandler: observed free-wobble period\u003c\/li\u003e\n\u003cli\u003eT_rigid: Euler period for a rigid Earth of the same shape\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eS. C. Chandler, Astronomical Journal 11, 59 (1891); Gross, Geophys. Res. Lett. 27, 2329 (2000).\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e Gross's result covers 1985–1996; the balance of excitation in other periods is still studied. The wobble is not precession, and its size in metres varies.\u003c\/p\u003e\u003ch3\u003eOn the shirt\u003c\/h3\u003e\u003cp\u003eFront: Wobbleon, the pixel-art character, with “THE CHANDLER WOBBLE”. Back: the wandering pole, the 433-day period, the rigid-Earth comparison, the ocean-pressure driver and Chandler's 1891 paper. 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W. Ekman, Arkiv för matematik, astronomi och fysik 2, no. 11 (1905).\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e The 45° and 90° hold for a steady, deep ocean with constant viscosity and no boundaries. Real oceans show a smaller turning angle because viscosity is not constant. In the Southern Hemisphere the turning is to the left.\u003c\/p\u003e\u003ch3\u003eOn the shirt\u003c\/h3\u003e\u003cp\u003eFront: Ekmara, the pixel-art character, with “THE EKMAN SPIRAL”. Back: the 45° and 90° rules, the spiral solution, the decay depth, the ideal-model note and Ekman's 1905 paper. 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Timing how long a sound stayed audible as he brought in seat cushions, he found that reverberation time is proportional to the room's volume divided by its total absorption. The 0.161 comes from 24 ln 10 \/ c with c ≈ 343 m\/s.\u003c\/p\u003e\u003cp\u003eHe went on to advise on Boston's Symphony Hall, which opened in 1900 and is often described as the first concert hall whose acoustics were designed with a quantitative formula. Its occupied reverberation time is about 1.9 s.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eT₆₀ = 0.161 V \/ A,   A = Σ S α\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eT₆₀: time for sound to fall by 60 dB (s)\u003c\/li\u003e\n\u003cli\u003eV: room volume (m³)\u003c\/li\u003e\n\u003cli\u003eA: total absorption (m²)\u003c\/li\u003e\n\u003cli\u003eS: area of each surface (m²)\u003c\/li\u003e\n\u003cli\u003eα: absorption coefficient of that surface\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eWallace Clement Sabine, Harvard, late 1890s; Boston Symphony Hall, 1900.\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e Sabine's formula is not exact: it fails for very absorbent rooms, where Eyring's version does better. Sabine's own calculation for Symphony Hall gave 2.31 s, as reported in E. Thompson's history The Soundscape of Modernity (2002); historians attribute the overestimate to how he counted audience absorption.\u003c\/p\u003e\u003ch3\u003eOn the shirt\u003c\/h3\u003e\u003cp\u003eFront: Sabina, the pixel-art character, with “THE ECHO HAS A CLOCK”. Back: Sabine's law, the absorption sum, the 60 dB definition and the Symphony Hall figure. 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L. Rijke put a hot metal gauze inside a vertical tube open at both ends and heard it sing loudly. The sound stops when the tube is laid horizontal and the upward air flow stops. Rayleigh gave the reason in 1878: an acoustic oscillation grows if heat is added where and when the pressure is high.\u003c\/p\u003e\u003cp\u003eIn an open–open tube the fundamental has a pressure node at each end. The drive is strongest with the gauze about a quarter of the way up from the bottom; a gauze in the upper half damps the sound instead. The same feedback causes dangerous combustion instability in rocket engines and gas turbines.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ef ≈ c \/ 2L   (loudest with the gauze at about L\/4 from the bottom)\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003ef: tone frequency (Hz)\u003c\/li\u003e\n\u003cli\u003ec: speed of sound in the warm air (m\/s)\u003c\/li\u003e\n\u003cli\u003eL: tube length (m)\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eRijke, Ann. Phys. 183, 339 (1859); Rayleigh, “The explanation of certain acoustical phenomena”, Nature 18, 319 (1878).\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e c is the warm-gas value, so the pitch depends on how hot the air is. Rijke used a hot gauze, not a flame; flame versions are variants.\u003c\/p\u003e\u003ch3\u003eOn the shirt\u003c\/h3\u003e\u003cp\u003eFront: Rijka, the pixel-art character, with “SINGING HEAT”. Back: the Rijke tube, Rayleigh's criterion, the L\/4 sweet spot, the tone formula and the 1859 and 1878 sources. 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