Humble Pi

by Matt Parker

  The first official height of Mount Everest was 29,002 feet. This is the kind of specific figure you would expect after decades of measurement and calculation. The Great Trigonometrical Survey (GTS) had been started by the British in 1802 as a comprehensive survey of the Indian subcontinent. In 1831 Radhanath Sikdar, a promising maths student from Kolkata who excelled at the spherical trigonometry required for geodetic surveying, joined the GTS.

  In 1852 Sikdar was working his way through the data from a mountain range near Darjeeling. Using six different measurements to calculate the height of ‘Peak XV’, the number dropped out to be around 29,000 feet. He burst into his boss’s office to tell him he had discovered the tallest mountain in the world. The GTS was by then run by Andrew Waugh, who, after a few years of double-checking the height, announced in 1856 that Peak XV was the tallest mountain on Earth and named it after his predecessor, George Everest.

  But the rumour is that Sikdar’s original number was 29,000 feet exactly. In this case, all those zeros were significant figures – but the public would not see them as such. They would assume the value was ‘about 29,000 feet’. And people might not accept a new claim for title of tallest mountain on the planet if the calculations looked like they were insufficiently precise. So an extra two fictitious feet were added. At least, that is how the story goes. The official recorded height in 1856 was definitely 29,002 feet, but I cannot find any evidence that the initial calculations gave 29,000 feet exactly. Or even where the original rumour about the rounding started.

  But even if this specific case is not true, I have no doubt that many seemingly precise values have been subtly changed away from an accidentally round number to make them look as precise as they truly are.

  Significant significance

  In February 2017 the BBC reported a recent Office for National Statistics (ONS) report that in the last three months of 2016 ‘UK unemployment fell by 7,000 to 1.6 million people.’ But this change of seven thousand is well below what the number 1.6 million had been rounded to. Mathematician Matthew Scroggs was quick to point out that the BBC was basically saying that unemployment had gone from 1.6 million to 1.6 million.

  A change below the precision of the original number is meaningless. Some people pointed out that a change of seven thousand jobs was within the scope of a single company shutting down and not a meaningful number for looking at changes in the economy as a whole. This is true, and it is why the ONS was rounding unemployment numbers to the nearest hundred thousand in the first place.

  The BBC story was later updated with more details about the statistics the ONS had actually released:

  The ONS is 95 per cent confident that its estimate of a fall in unemployment of 7,000 is correct to within 80,000, so the drop is described as not being statistically significant.

  So, in reality, the ONS was confident that unemployment had changed somewhere between an increase of 73,000 and a decrease of 87,000. In other words, the unemployment levels had not changed much, and it appeared they were maybe a little bit better rather than a little bit worse. That is a different message to the take-away stat of ‘unemployment fell by 7,000’, and I’m glad the BBC updated the article to add more details.

  Lumped together

  Changing the clocks at daylight saving time can cause people a lot of stress. Forget about it and you’ll either show up at work an hour early and embarrassed or an hour late and fired. I actually look forward to the clocks going back, because of that extra hour of sleep. Except I don’t squander it right away: I save it up for a few days, until I really need it. I’ve seriously considered taking an hour off every Friday night, when it will barely be noticed, and spending it on Mondays with an extra hour’s lie-in.

  The clocks going forward an hour does not have the same advantages: an hour of your life vanishes. But being a bit sleepy is not as bad as it gets; on the Monday after the clocks go forward there is a 24 per cent increase in heart attacks. Daylight saving time is literally killing people.

  Or rather it is literally killing people on that one specific day. The Monday after the clocks go forward and people lose an hour of sleep does show an increase in heart attacks above the average expected for a Monday (which is already peak heart-attack time). And on the Tuesday after the clocks go back, gifting us a bonus hour of sleep, heart attacks go down by 21 per cent. It’s a matter of timing. This is not a case where combining numbers and rounding has caused a problem, but rather, by lumping all the data together, it has revealed what is actually going on.

  There had been some prior research showing that heart attacks seemed to be linked to daylight savings, so the University of Michigan got their best cardiovascular people on to it. They crunched the Blue Cross Blue Shield of Michigan Cardiovascular Consortium database for all the time changes between March 2010 and September 2013. The study which found this result did a good job for controlling for all sorts of factors, including compensating for the fact that a day of twenty-five hours is going to have an extra 4.2 per cent of everything.

  But what makes the results misleading is how big the window is. That 24 per cent increase is lumping all heart attacks over the one day into the same category. The researchers looked at what the average number of heart attacks on a Monday would be for different times across the year, and the Monday after daylight saving is 24 per cent above what is to be expected. But if you go from looking at one day to a whole week, the effect disappears completely. The weeks after changes in daylight saving time had the expected number of heart attacks. They were just distributed differently within the week.

  It seems the clocks going forward and depriving people of sleep did cause extra heart attacks, but only in people who would have had a heart attack at some point anyway. The heart attack merely happened sooner. And, likewise, the clocks going back gave people a rest and bought them a few more days until their heart turned on them. This could be relevant information for a hospital planning its staffing around when the clocks go forward, but it does not mean daylight saving time is net dangerous.

  So we now know that the clocks going forward and back does not increase the number of heart attacks (but rather, a lack of sleep can bring on a heart attack that would have happened anyway). It angers me that whenever daylight saving time is discussed in the media, this statistic about heart attacks is brought up with no mention that it is misleading and that the total count for the week should be used. It’s happened once (on a BBC radio programme) even as I’ve been writing this book, and it causes me a lot of stress. Ironically, the misuse of this statistic in the media each time we have daylight saving probably does increase my personal chance of a heart attack!

  9.49

  Too Small To Notice

  Sometimes the seemingly insignificant bits which get rounded off or averaged out are actually very important. As the precision in modern engineering gets ever finer, humans find themselves working with machines that require tolerances beyond what our eyesight can manage and our sense of touch can handle.

  When the Hubble Space Telescope was put into orbit in 1990 at a cost of about $1.5 billion the first images which came back were disappointing. They were out of focus. At the heart of the telescope was a 2.4-metre-wide mirror which was supposed to be able to focus at least 70 per cent of the incoming starlight to a focal point, giving a sharp image. But it appeared to be bringing only 10 per cent or 15 per cent of the light into focus, leaving a blurry mess.

  NASA frantically set about trying to work out what was going wrong. After much head-scratching from the engineers and optics experts it was deduced that the mirror must be the wrong shape. When it was being made, the mirror was ground into a paraboloid shape and it was slightly off. Much like a reflective building in the hot sun, a paraboloid is the perfect shape to direct all the incoming light on to one small spot. But creating a sharp image required more accuracy than merely hitting a lemon with enough light to burn it. The mirror needed to be an exact paraboloid of a very specific type.

  What the Hub
ble initially saw and what the image should have looked like.

  The team investigating the problem considered all sorts of other errors, including the fact that the mirror was made under 1G of gravity and was now operating in 0G. It turns out the mirror was made and assembled perfectly. It had just been perfectly made to the wrong paraboloid. After much analysis it was determined that the primary mirror in Hubble had a conic constant (a measure of parabolaness) of −1.0139 when it needed to be −1.0023.

  Not that you could tell by looking at it. The edges of the 2.4-metre mirror were 2.2 micrometres lower than they should have been. That’s 2.2 thousandths of a millimetre. To construct the mirror to such ridiculous accuracy in the first place, beams of light had been bounced off the surface, forming complex interference patterns which changed with the slightest variation in distance. This was such a delicate operation that the wavelength of light had to be used to measure the shape.

  The main Hubble mirror under construction. ‘I can really see myself polishing that mirror.’

  The error was in the optics which shone the light on the mirror to analyse its shape. They had been set up in a way which would give the wrong conic constant; the official report said it was a 1.3 millimetre misplacement. News coverage said the error was a spare washer in the wrong place, but that isn’t in the official report. A repair mission was flown to the space telescope to add in corrective optics. A space-telescope contact lens, of sorts.

  Mecca for mistakes

  Many systems are accurate enough most of the time but break in ‘edge cases’ where errors can be amplified. An app which points towards Mecca has to know where both the phone and Mecca are only to a low degree of accuracy to point in the right direction from most places on the planet. Until the phone is held right next to the Kaaba (a building at the centre of Islam’s most important mosque).

  I would lose all faith in that app.

  If the bolt fits

  I have ordered some strange things off the internet over the years, but nothing was quite as difficult to track down from obscure specialist websites as the two piles of bolts on my desk in front of me. On the left I have some A211-7D bolts and on the right some A211-8C bolts. They are on my desk as a result of me contacting several suppliers of aerospace parts and equipment. Opposite is just one of each.

  I’ve had to be careful to keep close track of them, as it is hard to distinguish between them. The packages they came in are labelled but, once you take them out, there are no markings on either of the bolts to say if it is a 7D or an 8C. In theory, the 7C is 0.026 inches wider (about 0.66 millimetres) than the 8C but, rolling the two between my fingers, it is fairly hard to tell which is which. The thread on the 7D is also finer than on the 8C, but that is hard to spot. Thankfully, the 8C are 0.1 inches (around 2.5 millimetres) longer, which a careful alignment will reveal.

  Two very different bolts. Whatever you do: don’t mix them up.

  So I certainly feel sorry for the shift maintenance manager working the night shift on 8 June 1990 for British Airways in Birmingham Airport. He removed ninety bolts from the windscreen of a BAC 1-11 jet airliner and noticed they would need replacing, but they were unmarked. Taking one bolt with him, he climbed back down the safety raiser (an elevated platform) used to reach the front of the plane and headed off to the storeroom. After painstakingly comparing the bolt he was holding to all the other various bolts in the parts carousel, he correctly identified it as an A211-7D bolt. I now appreciate what a feat that was. He reached in to get more and discovered there were only four or five left.

  I can really empathize with the guy. It wasn’t even his job to replace the windscreen but, because they were short-staffed that night, and he was the manager, he stepped in to avoid further delays. It had been a few years, but he had done these windscreen changes before while working for BA and a quick flick through the aircraft maintenance manual had assured him it was as straightforward as he remembered. In the aircraft accident report which was published just over a year and a half later, our friend the shift maintenance manager is never named (and rightly so). I like to think of him as Sam (Shift mAintenance Manager). I imagine Sam standing there at 3 a.m., working on a job that wasn’t really his to do, holding about four of the bolts he needed ninety of.

  So Sam gets into a car and drives out of the hangar and over to a second parts store under the International Pier across the airport. It’s raining. He’s still clutching one of the bolts he removed from the windscreen. Unlike the main storeroom, which has a stores supervisor, this second store is unstaffed. Sam pulls up and finds the carousel, but the whole area is dimly lit. He would normally wear glasses for close-up reading, but he didn’t bother at work because his eyesight was good enough, but now, to access the bolt drawers, he blocks the only light source. The drawers are not even properly labelled. Sam resorts to comparing bolts manually. Eventually he manages to find some matching bolts. They must be A211-7Ds. Spoiler: they were not.

  Wait, Sam thinks, part of the windscreen has an extra ‘fairing strip’ of metal to improve aerodynamics, making it slightly thicker. Six of the bolts have to be longer. Damn it, why did he bring only one random bolt! Sam makes a call and grabs enough of what he thinks are A211-7Ds, along with six A211-9Ds, which are a bit longer. Back in the car and back out in the rain.

  He gets to the main hangar and goes to grab the torque wrench he needs to put the bolts in with. Torque wrenches are designed to disengage when a bolt has reached the correct tightness, to avoid overtightening. But it’s not on the tool board. It has gone missing. Sam, if you ever read this, I feel for you, man.

  The store manager does have a torque-limiting screwdriver, though, except it has not been properly calibrated and so they are not supposed to use it. Sam and the stores supervisor set it to release at 20 foot-pounds of turning force and give it a few test goes. It seems fine. Sam can finally get to work.

  Except the screwdriver has a socket which does not match the screwdriver bit Sam needs to use. So he has to hold a No. 2 Phillips screwdriver bit into the screwdriver’s socket while he works. And it does not clip into place; if he lets go, it will fall out. Several times the screwdriver bit fell to the ground and Sam had to clamber down to retrieve it. Leaning out from the safety raiser, he can just reach the windscreen to screw in the bolts, which is now a two-hand job. Using both hands means that Sam can no longer tell if the screwdriver is releasing because the correct torque has been achieved or slipping because the bolt is the wrong size.

  It’s nearly 5 a.m. and Sam is almost done. But the longer A211-9D bolts he grabbed for the thicker section don’t fit. I like to imagine Sam banging the torque screwdriver against the side of the aircraft as he weeps quietly. Maybe he invented some new swear words. In the end, he decided that the bolts he originally took out were not that bad after all. He grabbed six of them and put them back in. At last, he had finished.

  Twenty-seven hours after Sam had been (probably) swearing at the BAC 1-11 jet airliner, it was sitting on the runway as flight BA5390, ready to take eighty-one passengers and six members of crew to Malaga in Spain. I don’t know if you have ever been to either Birmingham, England, or Malaga, Spain, but I have and I can confirm that Malaga is a significant upgrade. Everyone on board was in high spirits.

  Thirteen minutes after take-off, the airliner was at around 17,300 feet altitude and the stewards were about to start the food-and-drink service. There is a loud bang as the windscreen fails and explodes outwards, causing the cabin to decompress in under two seconds. The air became foggy from the rapid change in pressure.

  Steward Nigel Ogden rushed back on to the flight deck, to find the co-pilot trying to regain control of the aircraft because the pilot had been sucked out of the window, colliding with the control column on the way out and disengaging the autopilot. Well, he’s almost out of the window. He’s caught on the windscreen frame, so his legs are still inside the aircraft. Ogden managed to grab the pilot’s legs to stop him from flying out of the window compl
etely.

  The co-pilot, Alistair Atcheson, was able to regain control of the aircraft and land it, with Captain Tim Lancaster dangling half out of the window. The crew had taken it in turns to hold on to his legs. Everyone survived, including Captain Lancaster, who spent twenty-two minutes outside the aircraft, made a full recovery and went back to being a pilot.

  It’s an incredible story. An amazing tale of a crew responding to a sudden and catastrophic disaster and managing to land the aircraft with no lives lost. But I’m equally amazed at how the windscreen could fail in the first place. There are so many checks in place that something like that should not be able to happen.

  The short and unfair answer is that Sam used the wrong bolts. When he was fumbling around in the unstaffed parts carousel under the International Pier at Birmingham Airport he did not pull out A211-7D bolts, as he thought, but rather A211-8Cs. The 8C bolts had a slightly smaller diameter, which meant they could be ripped out of the thread designed to hold 7D bolts in place. As I look at both bolts in the clear light of day in my office, I could easily make that mistake now, without all the extra pressure Sam was under.

  It is our nature to want to blame a human when things go wrong. But individual human errors are unavoidable. Simply telling people not to make any mistakes is a naive way to try to avoid accidents and disasters. James Reason is an Emeritus Professor of Psychology at the University of Manchester whose research is on human error. He put forward the Swiss Cheese model of disasters, which looks at the whole system, instead of focusing on individual people.