Humble Pi: When Math Goes Wrong in the Real World

by Matt Parker

The twisting action of the bridge has since become known to engineers as “torsional instability,” which means that a structure has the capability to twist freely in the middle.

Torsional instability came back with a vengeance in the Tacoma Narrows Bridge in Washington. Designed in the 1930s,

Opened in July 1940, the bridge quickly proved that being cheap to build had come at a cost. The thin road surface would move up and down in the wind. This was not yet torsional instability but the classic up-and-down bounce that had troubled many a bridge. But it seems, in this case, there was not enough bounce for it to be dangerous. People were told that it was perfectly safe to drive across Galloping Gertie, as it had been nicknamed by the locals.

was the other villain of the Millennium Bridge: a feedback loop. A feedback loop that had teamed up not with resonance but with torsional instability.

the bridge that was built had flat metal sides, perfect for catching the wind. The actual feedback loop was “flutter.”

each time the twist went up or down, it would be helped along by the wind, and the size of the oscillations would increase. If

The sixty-story John Hancock Tower was built in Boston in the 1970s, and it was discovered to have an unexpected torsional instability. The interplay of the wind between the surrounding buildings and the tower itself was causing it to twist.

Lumps of lead weighing 330 tons were put in vats of oil on opposite ends of the fifty-eighth floor. Attached to the building by springs, the lead weights damp any twisting motion and keep the movement below noticeable levels.

In 1980 a walkway was built in Kansas City for the Hyatt Regency Hotel. Intricate calculations had been performed so that the walkway would appear to float in the air, supported by a few slender metal rods at second-story level above the hotel’s lobby.

But thanks to the certainty of mathematics, the engineers knew the supports would work before even a single bolt had been put in place.

Then, on July 17, 1981, while a crowd of people were using the walkways as viewing platforms, the bolts tore through the supporting box beams—and over a hundred people died. This is a sobering reminder of how easy it can be to make a mathematical mistake and for it to have dramatic consequences. Here, the design had been changed—but the calculations had not been redone.

the original design, each nut had to support the walkway directly above it and any people on that walkway. The subtle change no one had noticed was that, with the modification, the bottom walkway was now directly suspended from the top walkway. So, as well as supporting its own weight and the people on it, this top walkway also had the bottom walkway hanging from it. The nuts which previously had to hold only the top walkway were now holding up the entire structure.

Whenever a computer cannot decipher a location, it still has to fill something in, and so 0,0 became the default location. The island where bad data goes to die.

Phone numbers are a perfect example: despite being made from digits, they are not actually numbers.

The rule of thumb should be: if you’re not going to do any math with it, don’t store it as a number.

At the time of writing, there is no way to turn off scientific notation by default in Excel. Some would argue it is a simple change that would solve some major database problems for a lot of people. Even more people would argue that, realistically, it’s not Excel’s fault, as it should not be used as a database in the first place. But, if we’re being honest, it will be. And it can cause problems with far more prosaic features than scientific notation. Spell check is bad enough.

On the whole, doing any kind of important work in a spreadsheet is not a good idea. They are the perfect environment for mistakes to spawn and grow unchecked. The European Spreadsheet Risks Interest Group (yes, that is a real organization, one dedicated to examining the moments when spreadsheets go wrong) estimates that over 90 percent of all spreadsheets contain errors. Around 24 percent of spreadsheets that use formulas contain a direct math error in their computations.

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. Collecting and crunching data can be more complicated, and more costly, than people expect.

My main point was that there is a general feeling in society that math is not that important, that it’s OK not to be good at it. But so much of our economy and technology requires people who are good at math. I thought the government acknowledging that there is a difference between hexagons and pentagons would raise awareness of the value we should place on math and education. Five is a different number than six!

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 1980 the Texaco oil company was doing some exploratory oil drilling in Lake Peigneur, Louisiana.

Getting to space is easy. It’s staying there that’s difficult.

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.

A door either loves letting people into the room or is keen to get everyone out of the room.

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.

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 AS-204 was declared to be the first official Apollo flight, despite the fact that it “failed on ground test.” This had a weird follow-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 program, even though they were never given Apollo names. 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.

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.* Despite not being circles, they always have the same-sized diameter from any direction you wish to measure it.

there is Finding #5: “significant out-of-round conditions existed between the two segments.” NASA undone by simple geometry.

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.)

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

That is counting from zero in disguise. TheJosh has taken the 8th of the month as day zero, which makes the 9th of the month the first day he is counting. In which case, yes: the 22nd of the month is the fourteenth day to be counted. But that does not mean it is a total of fourteen days. Counting from zero breaks the link between what you’ve counted to and what the total is. Counting from zero to fourteen is a total of fifteen.

Off-by-one errors also explain a struggle I always had with music theory. Moving along piano keys is measured in terms of the number of notes encompassed: hitting C on a piano, skipping D, and then hitting E is an interval called a third, because E is the third note on the scale. But what really matters is not how many notes are used but the difference between them. This is the reverse-fence-post problem: music intervals count the posts when they should count the fence!*

You are always older than your age. When you are thirty-nine, you are not in your thirty-ninth year of life but your fortieth.

If you count the day of your birth as a birthday (which is hard to argue against), then when you turn thirty-nine, it is actually your fortieth birthday. True as that may be, in my experience, people don’t like it written in their birthday card.

It’s as if you need to leave some beer to ferment for the first four days of the month and you stop it on the morning of the fourth day. It has only been going for three days! The pontifices did the same thing, but with years instead of beers. If you start counting from the beginning of year one instead of year zero, then the start of year four is only three years later. Coincidentally, if you drink my homebrew beer, you will also feel like a year of your life is missing (I call it “leap beer”).

this problem had happened before and that, legally, it is important to remove the day-zero ambiguity. In their home state of Massachusetts, the court system has to deal with this problem when it comes to court orders, and so it has defined:

In computing any period of time prescribed or allowed by these rules, by order of court, or by any applicable statute or rule, the day of the act, event, or default after which the designated period of time begins to run shall not be included.

James proposes a day-zero rule that would mean that all contracts are required to acknowledge day zero—an initiative I fully support.

Off-by-one errors have been a problem for thousands of years, and I suspect they will continue to be a problem for thousands more.

Because calculating combinations can give such large numbers, they are often used in advertising. But very rarely do companies bother trying to get the answer correct. When combinatorist (a mathematician who specializes in combinatorics) Peter Cameron went to a pancake restaurant in Canada, he noticed that they advertised a choice of “1,001 toppings.” Being a combinatorist, he recognized that 1,001 is the total number of ways to pick 4 things out of a total of 14 options, so he figured they had 14 toppings and customers could choose 4. Actually, the restaurant had 26 toppings (he asked) and had picked “1,001” just because it sounded big. Had the owners done the math correctly, they would have known that those 26 toppings in fact allowed for 67,108,864 options. A rare case of marketing understatement.

8 different items. Posters around London promised that this gave the customer a choice of 40,312 options,

when their errors were pointed out, McDonald’s did not admit its mistake but doubled down on trying to justify its bad math.

There are going to be eight yes-or-no choices, so you have 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 28 = 256 total options.

If eating nothing does not technically count as a meal, then that leaves 255 meal options (although some people would argue that the “null meal” is their favorite thing to order from McDonald’s).

The number McDonald’s used was the result of a very different calculation. It’s the answer to the question of how many different ways you could arrange 8 menu items.

(At three meals a day, it would take nearly thirty-seven years to try them all. That’s a long time to spend in the House of Ronald.)

In reality, the number of combinations of 2 or more items from a menu of 8 things is 247. Much smaller than 40,312. Like a kind of anti-supersizing.

In the United States, ZIP codes are five digits long and go from 00000 to 99999: a total of just a hundred thousand options.

Now I live in the UK, and if I look up my home postcode, there are only thirty-two addresses. That’s it.

The UK makes it work by having longer postcodes and allowing letters, digits, and strategic spaces. There are some limitations on where letters and digits can be positioned, but this system still allows for a staggering 1.7 billion possible postcodes.