Humble Pi: When Math Goes Wrong in the Real World

by Matt Parker

A million seconds from now is just shy of eleven days and fourteen hours. Not so bad. I could wait that long. It’s within two weeks. A billion seconds is over thirty-one years. A trillion seconds from now is after the year 33,700 CE.

Humans instinctively perceive numbers logarithmically, not linearly. A young child or someone who has not been indoctrinated by education will place three halfway between one and nine.

The universe has given us only two units of time: the year and the day. Everything else is the creation of humankind to try to make life easier.

The yearlong orbit of the Earth around the sun now takes 365 days, 6 hours, 9 minutes, and 10 seconds. For simplicity, we can call that 365¼ days.

The Northern Hemisphere summer occurs around the same point in the Earth’s orbit every year because this is where the Earth’s tilt aligns the north toward the position of the sun. After every 365-day year, the calendar year moves a quarter of a day away from the seasons. After four years, summer would start a day later. In less than four hundred years, within the lifespan of a civilization, the seasons would drift by three months. After eight hundred years, summer and winter would swap places completely.

If you ever have access to a friend’s phone, go into the settings and change their calendar to the Buddhist one. Suddenly, they’re living in the 2560s. Maybe try to convince them they have just woken up from a coma.

A political committee is rarely a good solution to a mathematical problem.

The Julian calendar is too short compared to the orbit. But it is too long compared to the seasons. Bizarrely, the seasons don’t even exactly match the orbital year.

As the Earth orbits, the direction it is leaning also changes, going from pointing directly at the sun to pointing away every 13,000 years. A calendar perfectly matching the Earth’s orbit will still swap the seasons every 13,000 years. If we factor the Earth’s axial precession (the change in how it leans) into its orbit, the time between summer solstices is 365 days, 5 hours, 48 minutes and 45.11 seconds. The movement of the Earth’s tilt buys us an extra 20 minutes and 24.43 seconds per orbit. So the true sidereal (literally, “of the stars”) year based on the orbit is longer than the Julian calendar, but the tropical year based on the seasons (which we actually care about) is shorter. It’s because the seasons depend on the tilt of the Earth relative to the sun, not on the actual position of the Earth.

SIDEREAL YEAR 31,558,150 seconds = 365.2563657 days 365 days, 6 hours, 9 minutes, 10 seconds TROPICAL YEAR 31,556,925 seconds = 365.2421875 days 365 days, 5 hours, 48 minutes, 45 seconds

Thankfully, if there’s one thing a pope can do, it’s convince a lot of people to change their behavior for seemingly arbitrary reasons.

the Gregorian calendar was backdated, recalibrating the year as if it, rather than the Julian option, had always been used. Through the use of pope power, it was decreed that ten dates would be taken from October 1582 and so, in Catholic countries, October 4, 1582, was directly followed by October 15. All this does of course make historical dates a bit confusing. When the English forces landed on Île de Ré on July 12, 1627, as part of the Anglo-French War, the French forces were ready to fight back on July 22. That is, on exactly the same day. At least, for both armies, it was a Thursday.

But astronomy does give Julius Caesar the last laugh. The unit of a light-year, that is, the distance traveled by light in a year (in a vacuum) is specified using the Julian year of 365.25 days. So we measure our current cosmos using a unit in part defined by an ancient Roman.

The engineers of the 1970s figured that someone else, further into the future, would fix the problems they were causing (classic baby-boomers). And to be fair, sixty-eight years is a very long time.

If you want to see the Y2K38 bug in action for yourself, find an iPhone. This may work for other phones, or the iPhone may one day be updated to fix this. But for now, the built-in stopwatch on the iPhone piggybacks on the internal clock and stores its value as a signed 32-bit number. The reliance on the clock means that, if you start the stopwatch and then change the time forward, the time elapsed on the stopwatch will suddenly jump forward. By repeatedly moving the time and date on your phone forward and backward, you can ratchet up the stopwatch at an alarming rate, until it hits the 32-bit limit and crashes.

In the short term, it is super easy and loops every few years. Allowing for the two types of year (leap and normal), and the seven possible days a year can start on, there are only fourteen calendars to choose from.

If you care about the sequence of years, the Gregorian calendar loops perfectly every four hundred years after a complete cycle of meta–leap years (the cycle of leaping leap years). So the day you are enjoying now is exactly the same as the day it was four hundred years ago.

In February 2007, six F-22s were flying from Hawaii to Japan when all their systems crashed at once. All navigation systems went offline, the fuel systems went, and even some of the communication systems were out. This was not triggered by an enemy attack or clever sabotage. The aircraft had merely flown over the International Date Line.

The official description for what went wrong was “synchronous lateral excitation” from pedestrians. It was the people walking on the bridge who caused it to wobble. Getting something as massive as the Millennium Bridge to start to wobble using brute force is a near-impossible challenge for a bunch of pedestrians, but this bridge was accidentally tuned to make it easy. Most people walk at about two steps per second, which means their body swings side to side once per second. A human walking is, for all bridge intents and purposes, a mass vibrating at 1 Hertz—which was the perfect rate to get the bridge wobbling. It matched one of the bridge’s resonant frequencies.

When the timing of your effort matches the frequency the swing is moving at, each push adds a little more energy into the system. This will build up with each push until the child is moving too fast to easily inhale and their screaming will finally cease.

The British Empire was all about trains, and British engineers prided themselves on their stiff upper bridges.

This is a common theme in human progress. We make things beyond what we understand, as we always have done. Steam engines worked before we had a theory of thermodynamics; vaccines were developed before we knew how the immune system works; aircraft continue to fly to this day, despite the many gaps in our understanding of aerodynamics. When theory lags behind application, there will always be mathematical surprises lying in wait. The important thing is that we learn from these inevitable mistakes and don’t repeat them.

I get upset when a crescent moon is shown with stars visible through the middle of it! Sesame Street is a repeat offender. In Ernie’s book I Don’t Want to Live on the Moon, the cover shows stars shining right through a crescent moon. And in a “C in space” segment, the moon looks surprisingly happy, despite the fact that these stars are shining through it. OK, yes, the moon having a face and emotions is not astronomically accurate either, but that is still no excuse for teaching children inaccurate geometry. I expect more from a supposedly “educational” program. The only explanation I can think of is that, in the extended Sesame Street universe, there are Muppet bases on the moon, and those are the dots of light we are seeing.

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 theater. All three died in the tragedy.

“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. 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. 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 counterclockwise. The problem is that the teeth of the “parents” cog links to both of the other cogs, grinding the whole thing, as well as parent-teacher interview night, to a halt.

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 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. 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.

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

he had actually been worried about the number of cogs, even though he did not think it was that important. 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.

Age is systematically rounded down: in many countries, a human is age zero for the first year of their life and increments to being one year old only after they have finished that whole period of their life. 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.

Computers just blindly follow the rules they are given and do the logical thing, with no regard for what may be the reasonable thing. This means that writing computer code involves trying to account for every possible outcome and making sure the computer has been told what to do. Yes, programming requires being numerate, but in my opinion it is the ability to think logically through scenarios that most unites programmers with mathematicians.

The rule changes have resulted in more jackpot rollovers, which make for much bigger prizes—the sorts of prizes that get media attention. And people are not buying tickets for the expected value; they are buying the permission to dream. Having a non-zero chance of winning a life-changing amount of money allows someone to dream about that version of their life. The more publicity a lottery draw gets and the more life-changing the prizes are, the bigger those dreams can be. Which is, arguably, better value.

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 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.

I love this view of accident management, because it acknowledges that people will inevitably make mistakes a certain percentage of the time. The pragmatic approach is to acknowledge this and build a system robust enough to filter mistakes out before they become disasters. When a disaster occurs, it is a system-wide failure, and it may not be fair to find a single human to take the blame.

As an armchair expert, it seems to me that the disciplines of engineering and aviation are pretty good at this. When researching this book, I read a lot of accident reports, and they were generally good at looking at the whole system. It is my uninformed impression that in some industries, such as medicine and finance, which do tend to blame the individual, ignoring the whole system can lead to a culture of not admitting mistakes when they happen. Which, ironically, makes the system less able to deal with them.

Not long after the June 2017 UK election, a Google search for “how long has theresa may been pm” gave her height in picometers. When it comes to measuring a leader’s body parts, trillionths of a meter is never the most convenient unit. Except maybe for Trump.

Many people I speak to say that, when they were at school, they were put off mathematics because they simply didn’t get it. But half the challenge of learning math is accepting that you may not be naturally good at it, but if you put the effort in, you can learn it. As far as I’m aware, the only quote from me that has been made into a poster by teachers and put up in their classrooms is: “Mathematicians aren’t people who find math easy; they’re people who enjoy how hard it is.”