Humble Pi

by Matt Parker

  But on the right of the plate is a crescent moon and, uncomfortably close to it, is a star. At a glance, it looks like that star is shining through what would be the disc of the moon. I had to find out for sure, so I bought some out-of-use Texan licence plates online for an up-close inspection. I ended up with plates 99W CD9 and set about scanning and digitally filling in the rest of the moon. Sure enough, that lone star should be hidden by the moon. In this case WCD stands for ‘Wrong Celestial Design’.

  They’ve got ninety-nine wrong celestial designs, but the pitch ain’t one.

  Doors of death

  I find the geometry of doors, locks and latches fascinating. You’d think securing property is something to take seriously, but it seems a lot of people don’t think through the dynamics of how doors and gates work. I love spotting people who have bought a big lock, only to leave the screws holding it in place exposed. Or there still being enough space to slide the padlock out of the way without having to open it. If you spot any such examples, please do send me a photo. And definitely take a closer look whenever you think something is ‘locked’. It might not be.

  Incorrect and correct mounting of a latch.

  Forgot your keys? No worries, just remove a few screws instead.

  Hypothetical story: my wife and her family were visiting their home town and took me to the local cemetery, where a beloved family member was buried. Except we’d (hypothetically) not checked the opening hours and the cemetery gate was locked. I looked at the gate and realized that, if you lifted a part of the latch, the gates were free to swing away from the padlock. I mean, if that had happened in actual fact (rather than hypothetically), I would have been the hero of the day (and respectfully ‘relocked’ the gate after paying our respects, naturally).

  These are amateur-level mistakes where someone has been put in charge of a door and not thought it through. Thankfully, nowadays an expert will have planned the entrances and exits to a building, but it wasn’t always that way. Many lives have been saved or lost as a consequence of the simple geometry of which way a door should open.

  As a general rule, doors should open in the direction they would need to in an emergency. Because of the location of the hinges, a door opens easily in only one direction; every doorway has a bias one way or the other. A door either loves letting people into the room or is keen to get everyone out of the room. Most household doors open into rooms (to avoid the door blocking a hallway), so it’s slightly easier to get into a room than to leave it. Most of the time, this is not a problem: you wait a few seconds to open the door towards you and then step through. We don’t even think about it. Until there are hundreds of other people trying to do the same thing.

  Now you’re expecting me to tell you a story about a fire and everyone trying to get out of a building quickly. But I’m not going to. The direction a door moves can be important even without a fire to drive the panic. In 1883 the Victoria Hall Theatre in Sunderland, near Newcastle, was hosting a show by The Fays which claimed to be ‘the greatest treat for children ever given’. Around two thousand largely unsupervised children between the ages of seven and eleven were crammed into the theatre. Nothing caught on fire but, equally frenzy-inducing to this age group, there was the sudden promise of free toys.

  Children on the ground floor were given their toys directly from the stage, but the 1,100 kids on the upper level had to descend the stairs and receive their toys as they left the building, showing their ticket number as they did so. Not only did the doors at the bottom of the stairs open inwards, they had also been bolted slightly ajar so that only a single child could exit at a time, to make checking the tickets easier. With not enough adults to monitor the queue, the children all rushed down the stairs to be the first one out. A hundred and eighty-three kids died in the crush against the doors.

  It took half an hour for all the children to be evacuated from the stairwell. Rescuers frantically tried to remove the kids one by one through the gap in the doors; they were unable to open the door back into the stairwell to allow the children to get out. The deaths were all the result of asphyxiation. As is common in human stampede situations, the kids pushing forward at the top of the stairs had no idea the people at the bottom had nowhere to go.

  It’s easy, perhaps, to distance ourselves from these children. They died over a century ago. To remind myself that they were real people, I looked up the list of their names. Looking through, I found Amy Watson, a thirteen-year-old who took her younger siblings Robert (twelve) and Annie (ten) to the show. Their house was a half-hour walk through town and over the river to the theatre. All three died in the tragedy.

  If the doors had had the capacity to swing open in an emergency, then the casualties could have been far fewer; maybe there wouldn’t have been any. This, of course, occurred to everyone at the time and, after a national outcry (two investigations failed to attribute any blame), the UK parliament passed laws requiring exit doors to open outwards. Directly inspired by the Victoria Hall incident, the ‘crash bar’ was invented so a door could be locked from the outside for security reasons but be opened from the inside with a simple push.

  The US followed with its own disasters, and these were fires, not the promise of free toys. A fire in Chicago’s Iroquois Theater in 1903 killed 602 people and is to this date the deadliest single-building fire in US history. The material and design of the building made for a rapidly advancing blaze, but the limited useable exits, which opened inwards, added to the death toll. Subsequent changes to the fire code required outward-opening doors in public buildings, but it took a while for it to be widely implemented. In the 1942 fire that swept through Boston’s Cocoanut Grove Night Club, 492 people died. Fire officials directly attributed three hundred of these deaths to the inward-opening doors.

  The question of which way doors should open in other situations is not always so clear cut. How about a spacecraft? During the Apollo programme, NASA had to decide if its spacecraft cabin hatches should open inwards or outwards. A door that opened outwards would be easier for the crew to operate and could be rigged with explosive bolts that could blow the hatch off in an emergency, so that was the initial choice. But after the ocean splashdown of NASA’s second human spaceflight Mercury-Redstone 4, the hatch unexpectedly opened and astronaut Gus Grissom had to get out, as seawater started flooding in.

  So the first Apollo spacecraft cabin had a hatch that opened inwards. The cabin was kept slightly above atmospheric pressure and this pressure difference helped to hold the hatch shut. Exiting the spacecraft involved releasing the pressure then pulling the hatch inwards. But during a ‘plugs-out’ launch dress rehearsal (where the spacecraft was unplugged from support systems and fully powered up to test everything except the actual lift-off), a fire broke out. An oxygen-rich environment, and combustible nylon and Velcro (used to hold equipment in place), caused the flames to spread rapidly.

  The heat from the fire increased the air pressure in the cabin to the point where it was impossible to open the hatch. All three astronauts inside – Gus Grissom, Edward White II and Roger Chaffee – were trapped and died of asphyxiation from the toxic smoke. It took five minutes for the rescue crew to open the cabin hatch.

  It later came to light that the Apollo astronauts had already requested outward-opening hatches, as they would make leaving the cabin for spacewalks far easier. After the inquiry into the fire, as well as changing the concentration of oxygen and the materials used in the cabin, in all future NASA human spaceflights the hatches were changed to open outwards, for safety reasons.

  This tragedy led to a numerical quirk of the Apollo missions. Even though the spacecraft never launched, the mission with Gus Grissom, Edward White II and Roger Chaffee was retrospectively named Apollo 1 out of respect for them, rather than keeping its codename, AS-204. Officially, the first actual launch should have been named Apollo 1 but, in the event, AS-204 was declared to be the first official Apollo flight, despite the fact that it ‘failed on ground test’. This had a weird
knock-on effect because, now, two previous crewless launches (AS-201 and AS-202; AS-203 was a payloadless rocket test and so not an official launch) were also retrospectively part of the Apollo programme, even though they were never given Apollo names. The first human launch thus became known as Apollo 4, giving us the niche bit of trivia that Apollo 2 and Apollo 3 never existed.

  More than just O-rings

  When the space shuttle Challenger exploded shortly after launch on 28 January 1986, killing all seven people onboard, a Presidential Commission was formed to investigate the disaster. As well as including Neil Armstrong and Sally Ride (the first American woman in space), the commission also featured Nobel prize-winning physicist Richard Feynman.

  The Challenger exploded because of a leak from one of the solid rocket boosters. For take-off, the space shuttle had two of these boosters, each of which weighed over 500 tonnes and, amazingly, used metal as fuel: they burned aluminium. Once the fuel was spent, the solid rocket boosters were jettisoned by the shuttle at an altitude of over 40 kilometres and eventually deployed a parachute so they would splash down into the Atlantic Ocean. Re-use was the name of the shuttle game, so NASA would send boats out to collect the boosters and take them off to be reconditioned and refuelled.

  As they slammed into the ocean, the boosters were basically empty tubes. They were built with a perfectly circular cross-section, but the impact could distort them slightly, as could transporting them on their sides. As part of the refurbishment, they were dismantled into four sections, checked to see how distorted they were, reshaped into perfect circles and put back together. Rubber gaskets called O-rings were placed between the sections to provide a tight seal.

  It was these O-rings that failed during the launch of Challenger, allowing hot gases to escape from the boosters and start the chain of events which led to its destruction. Famously, during the investigation, Richard Feynman demonstrated how the O-rings lost their elasticity at low temperatures. It was vital that, as the separate sections of the booster moved about, the O-rings sprang back to maintain the seal. In front of the media, Feynman put some of the O-ring rubber in a glass of iced water and showed that it no longer sprang back. And the 28 January launch had taken place on a very cold day. Case closed.

  But Feynman also uncovered a second problem with the seals between the booster sections, a subtle mathematical effect which could not be demonstrated with the captivating visual of distorted rubber coming out of a glass of cold water. Checking if a cross-section of a cylinder is still circular is not that easy. For the boosters, the procedure for doing this was to measure the diameter in three different places and make sure that all three were equal. But Feynman realized that this was not sufficient.

  Writing about his investigation, Feynman recalled as a child seeing in a museum ‘non-circular, funny-looking, crazy-shaped gears’ which remained at the same height as they rotated. He did not note their name, but I immediately recognized them as ‘shapes of constant width’. I love these shapes and have written about them extensively before.fn1 Despite not being circles, they always have the same-sized diameter from any direction you wish to measure it.

  In his report, Figure 17 is a shape Feynman has drawn which is obviously not a circle but does have three identical diameters. He could have gone one step further. You could make thousands of diametric measurements of a shape of constant width, such as a Reuleaux triangle, and they would all come out exactly the same, despite the shape being very much not circular.

  FIGURE 17. This figure has all its diameters the same length – yet it is obviously not round!

  Feynman’s shape with three identical diameters next to a shape with infinitely many identical diameters. Both are obviously not circles.

  If a booster had been distorted into a Reuleaux triangle cross-section, then the engineers would have been able to spot this easily, but this kind of distortion could happen on a much smaller scale; it might not be visible to the naked eye but still be enough of a distortion to change the shape of the seal. Shapes of constant width often have a bump on one side and a flat section on the other to compensate.

  Feynman managed to sneak some time alone with the engineers who worked on these sections of the boosters. He asked if, even after the diameter measurements had been completed (allegedly confirming the shape was perfectly circular), they still had these bump–flat distortions.

  ‘Yes, yes!’ they replied. ‘We get bumps like that. We call them nipples.’ This was in fact a problem that occurred regularly, but it didn’t seem like anything was being done about it. ‘We get nipples all the time. We’ve been tryin’ to tell the supervisor about it, but we never get anywhere!’

  The final report bears all this out. The performance of the rubber O-rings was definitely the primary cause of the accident and remains the headline finding that most people remember. But as well as the O-ring findings, and recommendations for how NASA should handle communication between the engineers and management, there is Finding #5: ‘significant out-of-round conditions existed between the two segments’. NASA undone by simple geometry.

  For the love of cog

  As an ex-high-school teacher, I have a framed poster in my office claiming that ‘Education works best when all the parts are working’. It shows three cogs labelled ‘teachers’, ‘students’ and ‘parents’, all linked together. This poster has become an internet meme with the description ‘mechanically impossible yet accurate’ because three cogs meshed together cannot move. At all. They’re locked in place. If you want some movement, one of the three needs to be removed. (In my experience: parents.)

  Inspirational posters work best when all the parts are geometrically plausible.

  The problem is that, if a cog is going clockwise, any other cog it is meshed with will have to spin anticlockwise. The teeth lock together, so, if the ‘teachers’ cog is going clockwise, the teeth on the right will push the left side of the ‘students’ cog down, turning it anticlockwise. The problem is that the teeth of the ‘parents’ cog links through both the other cogs, grinding the whole thing, as well as parent–teacher interview night, to a halt.

  For a three-cog mechanism like this to work, two of the cogs would need to be unmeshed from each other. When the Manchester Metro released a poster to represent the parts of the city working together, people redesigned the cogs in 3D such that they could all spin in unison. In this example, the teeth of cogs 2 and 3 no longer touch each other so everything is now free to move.

  But sometimes it is unfixable. The newspaper USA Today ran a story in May 2017 reporting President Trump’s decision to renegotiate the North American Free Trade Agreement between the US, Canada and Mexico. In this case, the cogs are already in 3D and so are unambiguously in deadlock. The article discussed both how beneficial a trade agreement could be to all the member countries and how hard it is to get three countries to simultaneously work together. So I’m still undecided whether the three locked cogs were deliberate or not.

  If only public transport in Manchester was that easy to fix.

  Making cogs grate again.

  More cogs only makes things worse. Never put ‘teamwork cogs’ as a search term into a stock image website. For a start, if you’re not used to the cheese-tastic world of inspirational work posters, what you see will come as a shock. The next shock is that a lot of the diagrams supposed to be showing a team working like a well-oiled machine use a mechanism which would be permanently seized in place.

  Cogs and clockwork-like mechanisms are a stock example of things working together in unison; that’s why they are used in so many inspirational workplace posters. But here’s the thing: clockwork mechanisms are hard. They are difficult to build: one part in the wrong place and the whole thing stops working completely. The longer I think about it, the more I’m convinced that this does actually make a great analogy for workplace teamwork.

  I was prepared to pay for one stock image to use in this book. This is my favourite. The description is ‘Model of 3D figures on
connected cogs as a metaphor for a team’.

  But, to be honest, a four-way high-five as a symbol of teamwork has even more geometric problems.

  In 1998, in the lead-up to the millennium, a new £2 coin was released in the UK. There was a competition to design the back of the new coin (the Queen, by default, gets to design the front of the coin, with her face) and it was won by Bruce Rushin, an art teacher in Norfolk. Bruce designed a series of concentric rings, each representing a different technological age of humankind. The one for the Industrial Revolution was made from a ring of nineteen cogs. You can see where this is going – or rather not going anywhere.

  A chain of cogs will spin clockwise, anticlockwise, clockwise, anticlockwise … and so on. So if they loop back on themselves, there needs to be an even number of cogs so that a clockwise cog meets an anticlockwise one. Any odd number of cogs in a loop will come to a standstill. The nineteen cogs on the £2 would be completely locked up and unable to move at all.

  Of course, the internet spotted pretty quickly that the new £2 coin suffered from the same problem. The people complaining about it online ran the usual gamut of the curious to the insufferably smug. Someone even managed to get an official response out of the Royal Mint about the implausibility of the design.