Humble Pi: When Math Goes Wrong in the Real World

by Matt Parker

Pepsi took active steps to protect itself from future problems and re-released the ad with the Harrier increased in value to 700 million Pepsi Points. I find it amazing that they did not choose this big number in the first place. It’s not like 7 million was funnier; the company just didn’t bother to do the math when choosing an arbitrary large number.

We know a million, a billion, and a trillion are different sizes, but we often don’t appreciate the staggering increases between them.

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.

Three is a different kind of middle. It’s the logarithmic middle, which means it’s a middle with respect to multiplication rather than addition: 1 × 3 = 3; 3 × 3 = 9. You can go from one to nine either by adding equal steps of four or multiplying by equal steps of three. So the “multiplication middle” is three, and that is what humans do by default, until we are taught otherwise.

A logarithmic scale is a valid way to arrange and compare numbers, but mathematics also requires the linear number line.

All humans are stupid when it comes to learning formal mathematics. This is the process of taking what evolution has given us and extending our skills beyond what is reasonable.

As a species, we have learned to explore and exploit mathematics to do things beyond what our brains can process naturally.

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.

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.

Luigi’s breakthrough was to keep the standard every-fourth-year leap year of the Julian calendar but to take out three leap days every four hundred years. Leap years were all the years divisible by four, and all Luigi suggested was to remove the leap days from years which were also a multiple of 100 (except those that were also a multiple of 400). This now averages out to 365.2425 days per year; impressively close to the desired tropical year of around 365.2422 days.

England (and, by extension at the time, North America) clung to the old Julian calendar for another century and a half, during which time their calendar not only drifted another day away from the seasons but was also different from the one used in most of Europe.

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.

England (which still included parts of North America) swapped over in 1752, realigning its dates by removing eleven days from September. Thus, September 2, 1752, was followed by September 14, 1752.

Russia did not swap calendars until 1918, when it started February on the 14th rather than on the 1st to bring themselves back into alignment with everyone else on the Gregorian calendar.

Despite all these improvements, our current Gregorian calendar is still not quite perfect. An average of 365.2425 days per year is good, but it’s not exactly 365.2421875. We’re still out by twenty-seven seconds a year. This means that our current Gregorian calendar will drift a whole day once every 3,213 years. The seasons will still reverse once every half a million years. And you will be alarmed to know that there are currently no plans to fix this!

At 3:14 a.m. on Tuesday, January 19, 2038, many of our modern microprocessors and computers are going to stop working.

Computer timekeeping has all the ancient problems of keeping a calendar in sync with the planet plus the modern limitations of binary encoding.

So the system was recalibrated to count the number of whole seconds since the start of 1970. This number was stored as a signed 32-digit binary number, which allowed for a maximum of 2,147,483,647 seconds: a total of over sixty-eight years from 1970.

They were sure that, by the year 2038, computers would have changed beyond all recognition and no longer use Unix time. Yet here we are. More than halfway there and we’re still on the same system. The clock is literally ticking.

on February 13, 2009, with some friends to celebrate 1,234,567,890 seconds having passed,

at just after 11:31 p.m.

After 2,147,483,647 seconds, everything stops. Microsoft Windows has its own timekeeping system, but MacOS is built directly on Unix.

They are all vulnerable to the Y2K38 bug.

Some steps have already been taken toward a solution. All the processors that use 32-digit binary numbers by default are known as 32-bit systems. When buying a new laptop, you may not have paused to check what the default binary encoding was, but Macs have been 64-bit for nearly a decade now, and most commonly used computer servers have gone up to 64 bits as well. Annoyingly, some 64-bit systems still track time as a signed 32-bit number so they can play nicely with their older computer friends, but for the most part, if you buy a 64-bit system, it will be able to keep track of time for quite a while to come.

The largest value you can store in a signed 64-bit number is 9,223,372,036,854,775,807, and that number of seconds is equivalent to 292.3 billion years. It’s times like this when the age of the universe becomes a useful unit of measurement: 64-bit Unix time will last until twenty-one times the current age of the universe from now—until (assuming we don’t manage another upgrade in the meantime) December 4 in the year 292,277,026,596 CE, when all the computers will go down. On a Sunday.

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.

So the day you are enjoying now is exactly the same as the day it was four hundred years ago.

Now you can safely reply and say that nothing in the Gregorian calendar can happen less frequently than once every four hundred years. JUST FOR FUN.

And, given that there are only four possible month lengths and seven different starting days, there are actually only twenty-eight possible arrangements for the days of a month. So stuff like this actually happens every few years. (Not based on Chinese feng shui.)

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.

that time suddenly jumped by a day and the plane freaked out and decided that shutting everything down was the best course of action.

it’s a successful building. Except, during the summer of 2013, it started setting things on fire.

all the reflective glass windows accidentally became a massive concave mirror—a kind of giant lens in the sky able to focus sunlight on a tiny area.

the same thing happened at the Vdara Hotel in Las Vegas in 2010. The curved glass front of the hotel focused sunlight and burned the skin of hotel guests lounging by the pool.

Well, the Vdara Hotel was also designed by Rafael Viñoly,

when London’s Millennium Bridge was unveiled in 2000, it had to be closed after only two days.

people walking on it would set the bridge swinging.

The Millennium Bridge had been accidentally tuned to around 1 Hertz. But not in the normal up-and-down direction; it wobbled from side to side.

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.

This sideways resonance would only be a problem if enough people walked perfectly in step. This is the “synchronous” in “synchronous lateral excitation” from pedestrians.

the movement of the bridge affected the rhythm at which they were walking. This formed a feedback loop:

Broughton Bridge the people crossing it had to do all the work themselves. The bridge was built in 1826, and people crossed it with no problem at all until 1831. It took a troop of soldiers all marching perfectly in sync to hit the resonant frequency. The 60th Rifle Corps of seventy-four soldiers were heading back to their barracks at about midday on April 12, 1831. They started to cross in rows of four and pretty quickly noticed that the bridge was bouncing in rhythm with their steps. This was apparently quite a fun experience and they started to whistle a tune to go with the bouncing. Until about sixty soldiers were bouncing on the bridge at once and it collapsed.

Humans and carriages have some level of built-in suspension; they can deal with a road surface that is moving around a bit. A train has no such tolerance. The track needs to remain absolutely stationary, which makes for some very stiff railway bridges.

In late 1846, a railway bridge designed by engineer Robert Stephenson was opened over the Dee River in Chester.

In May 1847, the bridge was modified slightly: extra rock and gravel were added to keep the tracks from vibrating and to protect the bridge’s wooden beams from burning embers produced by the steam engines.

However, the first train to cross after the work did not make it to the other side.

It turns out that, as well as vibrating up and down and side to side, bridges can also twist in the middle.

The bridge twisted to the side and the carriages were dumped into the river below. Eighteen people were injured, and five died.

This is a common theme in human progress. We make things beyond what we understand, as we always have done.