Humble Pi: When Math Goes Wrong in the Real World

by Matt Parker

Amazingly, one specific Value-at-Risk calculation was being done in a series of Excel spreadsheets with values having to be manually copied between them.

The traders regularly gave their portfolio positions “marks” to indicate how well or badly they were doing. As they would be biased to underplay anything that was going wrong, the Valuation Control Group (VCG) was there to keep an eye on the marks and compare them to the rest of the market. Except they did this with spreadsheets featuring some serious mathematical and methodological errors.

Billions of dollars were lost in part because someone added two numbers together instead of averaging them. A spreadsheet has all the outward appearances of making it look as if serious and rigorous calculations have taken place. But they’re only as trustworthy as the formulas below the surface.

I’d made it very clear that I did not want to change the old signs (even I can appreciate that that might be a misuse of taxpayer funds). I just wanted them to update Statutory Instrument 2016 No. 362, Schedule 12, part 15, symbol 38 (I did my homework! Told you I was serious.) so all future signs would be correct. But that was not enough to please many people.

This was not just a picture of a different type of soccer ball: it could not even be a ball. That feels like a grand statement: that you could never make a ball out of hexagons. But I can state with complete mathematical confidence that it is impossible to make a ball shape out of only hexagons, even if they are distorted hexagons.

A friend of mine (hilariously) bought me a pair of soccer socks because they had the classic soccer ball–sign pattern with all hexagons, but because a sock (ignoring the toe) is a cylinder, that is fine. His gesture was both genius and cruel; the socks were simultaneously right and wrong.

So the signs remain incorrect. But at least now I have a framed letter from the UK government saying that they don’t think accurate math is important and they don’t believe street signs should have to follow the laws of geometry.

In this case, it was important because the Diamond Crystal Salt Company was already mining through the ground below the lake and Texaco had to avoid drilling into the pre-existing salt mines. Spoiler: they messed up the calculations. But the results were more dramatic than what you’re probably imagining.

The drill made it down 370 meters before the drilling platform in Lake Peigneur started to tilt to one side. The oil drillers decided it must be unstable, so they evacuated. Arguably, the salt miners had an even bigger surprise when they saw water coming toward them.

Thanks to good safety training, the mining crew of about fifty people was able to evacuate safely. But how much water could the mine take? The lake had a volume of around 10 million cubic meters of water to give. But the salt below had been mined since 1920, and the mines now had a volume greater than the volume of the lake above.

As the water gushed down, earth was eroded and salt dissolved. Soon, the 36-centimeter hole had become a raging whirlpool 400 meters in diameter.

Not only did the entire lake empty into the salt mine, but the canal joining the lake to the Gulf of Mexico reversed direction and started to flow backward i...

Because of the miscalculation of a triangle, a freshwater lake that was only about three meters deep was completely drained and refilled from the ocean. It’s now a four-hundred-meter-deep saltwater lake, and this has brought a complete change in plants and wildlife.

I’m more interested in geometry mistakes where someone has not properly thought through the shapes involved—situations where the geometry itself is wrong, not just the working out. Which brings me to one of my favorite hobbies: finding pictures of the moon that have stars shining through them.

a circle is just the line around the circumference, and a disk is completely filled in. A frisbee is a disk; a hula hoop is a circle. But I’m going to use them interchangeably, as they are in normal language.)

For while we sometimes cannot see the moon, it is still there as a silhouette. Which is why I get upset when a crescent moon is shown with stars visible through the middle of it!

Worse, there are Texas vehicle license plates that celebrate NASA’s presence in the Lone Star State. The space shuttle taking off on the left is surprisingly accurate, ascending to the side instead of directly up. This may look incorrect, but the space shuttle needed a huge amount of sideways speed to be able to get into orbit. Space is not that far away: as I type, the International Space Station is at an altitude of only 262 miles. But for something to stay in that orbit, it needs to be moving around the Earth at about 17,000 miles per hour, that is, 4.75 miles every second.

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.

Many lives have been saved or lost as a consequence of the simple geometry of which way a door should open.

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.

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 of conjuring, marionettes, and illusions billed as “the greatest treat for children ever given.” Around two thousand largely unsupervised children between the ages of seven and eleven were crammed into the theater. Nothing caught on fire but, equally frenzy-inducing to this age group, there was the sudden promise of free toys.

Not only did the doors at the bottom of the stairs open inward, 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 line, the children all rushed down the stairs to be the first one out. One hundred eighty-three kids died in the crush against the doors.

Woooof

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.

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.

During the Apollo program, NASA had to decide if its spacecraft cabin hatches should open inward or outward. A door that opened outward 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.

Exiting the spacecraft involved releasing the pressure then pulling the hatch inward. 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 liftoff), a fire broke out. An oxygen-rich environment, plus 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.

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 outward, for safety reasons.

The first Saturn V rocket launch thus became known as Apollo 4, giving us the niche bit of trivia that Apollo 2 and Apollo 3 never existed.

For takeoff, the space shuttle had two of these boosters, each of which weighed 650 tons and, amazingly, used metal as fuel: they burned aluminum. Once the fuel was spent, the boosters were jettisoned by the shuttle at an altitude of over 25 miles and eventually deployed parachutes so they would splash down into the Atlantic Ocean. Reuse 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 refueled. As they slammed into the ocean, the boosters were basically empty tubes.

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.

In front of the media, Feynman put some of the O-ring rubber in a glass of ice water and showed that it no longer sprang back. And the January 28 launch had taken place on a very cold day. Case closed.

Feynman also uncovered a second problem with the seals between the booster sections, a subtle mathematical effect that 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 ...

I immediately recognized them as “shapes of constant width.” I love these shapes and have written about them extensively before.* Despite not being circles, they always have the same-sized diameter from any direction you wish to measure it.

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.

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-and-flat distortions....

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.”

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 labeled “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.

The problem is that, if a cog is going clockwise, any other cog it is meshed with will have to spin counterclockwise.

For a three-cog mechanism like this to work, two of the cogs would need to be unmeshed from each other.

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 that 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.

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.

A chain of cogs will spin clockwise, counterclockwise, clockwise, counterclockwise . . . 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 a counterclockwise one.

What I found shocked me. On Bruce Rushin’s website is the original design that won the competition back in the late 1990s: it has twenty-two cogs. It would have worked!

He made his design mechanically correct not because he thought that was better but, rather, to avoid angry e-mails. When the Royal Mint turned Bruce’s plate-sized design into an actual coin only 28.4 millimeters across, they had to lose some of the finer details, and three cogs were the victim of this simplification.

Troll or genuine: either way, TheJosh has taken a perfect stance, which is wrong yet supported by enough plausible misconceptions that it is possible to argue about it at length. Which he does, utilizing two classic math mistakes: counting from zero and off-by-one errors.

Counting from zero is a classic behavior of programmers. Computer systems are often being used to their absolute limit, so programmers are sure not to waste a single bit.