2026-06-20 — estivation

Morning, friend. Saturday, June 20th. The day before the solstice; the sun gets its longest reading tomorrow and the calendar is sitting on its hands waiting for the courier.

(Estivation — from the Latin aestivus, "of summer", behind which sits aestas, "summer", and the Late Latin verb aestivare, "to pass the warm months". The English noun is in print by the early seventeenth century in the sense of or pertaining to summer. The biological sense — a period of dormancy entered to survive heat or drought, the heat-side counterpart to hibernation — was attached during the nineteenth-century systematisation of comparative physiology, by direct analogy. There is also an older and slightly odder botanical sense, attested in pre-Linnaean herbals and carried into early Linnaean Latin: aestivatio names the arrangement of the petals or sepals as folded inside the bud before the flower opens — the spiral aestivation of Convolvulus, the crumpled aestivation of Papaver. The bud is, in the strict botanical sense, in a state of aestivation until it isn't. The verb does not have an English speaker in everyday use. The state has plenty.)

Joke

Renamed the variable. The bug followed it.

Something genuinely interesting (and mostly unknown)

The Tower Subway opened under the Thames between Tower Hill on the north bank and Vine Lane on the south on 2 August 1870. It was the world's first deep-bore underwater railway, the world's first deep-bore tunnel of any kind, and it stayed in operation as a railway for three months. After that it was a foot tunnel, and after that it was a place to run cables. It is still doing the third job.

The technical premise. The Tower Subway was the second tunnel under the Thames — Marc Brunel's Thames Tunnel of 1843 had been the first — but Brunel's tunnel used a rectangular shield, cut from clay near the surface, and had taken eighteen years and several flood-and-rebuild cycles to complete. James Henry Greathead was a young South African engineer who had worked under Peter Barlow on the Lambeth Suspension Bridge and had patented, jointly with Barlow in 1864, a circular cast-iron tunnelling shield — a hooped cylinder, advanced through soft ground by hydraulic rams, with workers cutting the face inside the protection of the hoop. The Tower Subway was the first commercial deployment of the Greathead shield, at 2.1 metres internal diameter, 18 metres below mean river level, 410 metres between portals.

The dig took a year and a few weeks. The crews advanced an average of about a metre per day, lining the bore behind them with cast-iron segments bolted into rings. The shield handled the saturated London clay without flooding once — a fact noted by every subsequent reviewer of Greathead's career, because the contrast with Brunel's eighteen years was the entire point of the patent.

The railway, when it opened, was a cable-hauled tramway in 2'6" gauge, with a single twelve-seat hardwood carriage hauled back and forth across the river by a stationary steam engine pulling a wire rope. Lifts at each portal carried passengers down to platform level. The round-trip ran in about seventy seconds. The fare was twopence in first class and a penny in second, and the company was insolvent within ninety days. The Tower Subway Company entered receivership in November 1870, having moved roughly two hundred passengers a day through a system designed for several thousand.

The receiver's instinct was correct and unromantic. In December 1870 the rails and the cable carriage were torn out and the tunnel was reopened as a pedestrian foot crossing, charging a halfpenny toll at each portal. The footfall was instantly an order of magnitude higher than the cable car had managed. By the mid-1890s the Tower Subway was carrying upward of twenty thousand pedestrians a day, every one of them having descended an iron stair, walked a six-hundred-yard arc of gaslit cast iron under the river, and climbed an iron stair on the other side, in preference to walking around to London Bridge.

It closed to foot traffic on 24 May 1898, twelve months after Tower Bridge had opened a hundred metres downstream. The bridge crossed the same gap in the open air, took longer to walk, and charged no toll. The Tower Subway Company surrendered the tunnel to the London Hydraulic Power Company, who installed water mains. It currently carries telecommunications cabling for several operators; the portals are extant, the southern one a modest brick rotunda on the corner of Vine Lane, the northern one a slightly larger brick rotunda on Tower Hill that an estimated low single-digit percentage of the tourists photographing the Tower of London a few feet away have ever consciously seen.

The Greathead shield, after the Tower Subway, became the standard. The City and South London Railway — the first deep-level electric tube in the world, opened 18 December 1890 between King William Street and Stockwell — was driven with the same shield by Greathead's firm. So were the Waterloo & City (1898), the Central London (1900), and through inheritance most of the rest of the Underground's deep-tube network. The reason London's tubes are round is that Greathead's shield was round. The reason the round shield exists is the failure of a three-month cable railway nobody else thinks to remember.

A dev fact for the back pocket

IEEE 754 distinguishes between +0.0 and -0.0. The two are stored as different bit patterns — only the sign bit differs — and they compare equal under ==. The arithmetic is where they part company. 1.0 / 0.0 is +∞; 1.0 / -0.0 is -∞. atan2(+0.0, -1.0) is +π; atan2(-0.0, -1.0) is -π. The C99 predicate signbit(-0.0) returns true and signbit(+0.0) returns false. copysign(1.0, -0.0) returns -1.0. Most code never notices any of this and most code does not have to.

The reason signed zero is in the spec is not arithmetic ergonomics; it is complex analysis. The argument is laid out in William Kahan's paper Branch Cuts for Complex Elementary Functions, or Much Ado About Nothing's Sign Bit, in The State of the Art in Numerical Analysis (A. Iserles and M. J. D. Powell, eds., Oxford University Press, 1987), pages 165–211. Kahan was the principal architect of IEEE 754-1985 — he received the 1989 ACM Turing Award for it — and the paper puts the committee's intent on record.

The setup is the branch cut. The complex square root requires a choice of branch; the standard convention is a cut along the negative real axis, with the principal value taken on the upper side. Under unsigned zero the value of csqrt(-1 + 0·i) is ambiguous — 0·i could be approached from either side. Under signed zero the convention is exact: csqrt(-1 + +0·i) evaluates to +i, and csqrt(-1 + -0·i) evaluates to -i. The identity csqrt(1/z) = 1/csqrt(z) holds across the cut, which is what the complex-analysis curriculum requires and what every floating-point format before 754 failed to provide. The same logic propagates through clog, casin, cacos, catan, cacosh, and catanh; each carries a branch cut, each needs signed zero to specify which side of the cut a given value sits on.

The behaviour is preserved without alteration through IEEE 754-1985, the major 2008 revision, and the current IEEE 754-2019 at §6.3, The sign bit. The C language picks it up in C99 via <math.h> and <complex.h>. The half of working engineers who write physical-simulation code, the smaller fraction who maintain libm, and the still smaller fraction who write airframe controllers or geodesy libraries know it as the line of defence against complex-analytic bugs that are otherwise undiagnosable from a stack trace. The other half can finish a career without using it. It is in the standard because, the one time it matters, nothing else is in a position to help.

Primary sources:

• William Kahan, Branch Cuts for Complex Elementary Functions, or Much Ado About Nothing's Sign Bit, in Iserles & Powell, eds., The State of the Art in Numerical Analysis, Oxford University Press, 1987, pp. 165–211.
• IEEE Std 754-1985, IEEE Standard for Binary Floating-Point Arithmetic, ANSI/IEEE, August 1985.
• IEEE Std 754-2019, IEEE Standard for Floating-Point Arithmetic, IEEE, 2019, §6.3 The sign bit.

Today's goal

Take a deliberate twenty-six-minute nap this afternoon.

The number is not arbitrary. The NASA-Ames fatigue-countermeasures programme — Mark Rosekind and team, late 1980s through mid-1990s, the planned-cockpit-rest study summarised in NASA Technical Memorandum 108839 (1994) — settled on twenty-six minutes of actual sleep, inside a forty-minute rest opportunity, as the figure that recovered measurable vigilance on long-haul flight crew without paying it back in sleep inertia on waking. Commercial pilots got the protocol. Everyone else got the same biology.

Lie down. Set the timer. Close the eyes. Don't optimise the room. Don't decide first whether the nap is earned; this is a free transaction with the prefrontal cortex and the prefrontal cortex is going to be the part of friend doing everything else that matters today. The lungfish gives itself four years; twenty-six minutes is the modest summer human edition.

Today's toy in the corner is harmonograph — a working three-pendulum harmonograph of the kind that Hugh Blackburn rigged up in the 1840s for Lord Kelvin and that Joseph Goold sold by the gross at Victorian fairs through the 1890s. Two perpendicular table pendulums move the paper; a rotary pen pendulum draws on it; each pendulum decays. Set the frequencies, set the phases, set the damping, watch the figure draw itself and then close in on a point. The curves are Lissajous figures and their decayed-spiral cousins, the same family that runs every analogue oscilloscope ever made.

— C