Humble Pi: When Math Goes Wrong in the Real World

by Matt Parker

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.

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.

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.

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. And this was put in place by members of the generation who in the sixty-eight years leading up to 1970 had seen humankind go from the Wright brothers inventing the first powered airplane to humans dancing on the moon. They were sure that, by the year 2038, computers would have changed beyond all recognition and no longer use Unix time.

Computers have indeed changed beyond recognition, but the Unix time beneath them is still there. If you’re running any flavor of Linux device or a Mac, it is there in the lower half of the operating system, right below the GUI. If you have a Mac within reach, open up the app Terminal, which is the gateway to how your computer actually works. Type in date +%s and hit Enter. Staring you in the face will be the number of seconds that have passed since January 1, 1970.

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.

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.

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.

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.

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 US Air Force has not confirmed what went wrong (only that it was fixed within forty-eight hours), but it seems that time suddenly jumped by a day and the plane freaked out and decided that shutting everything down was the best course of action.

The building at 20 Fenchurch Street in London was nearing completion in 2013 when a major design flaw became apparent. It had nothing to do with the structural integrity of the building; it was completed in 2014 and is a perfectly functioning building to this day, and was sold in 2017 for a record-breaking £1.3 billion. By all measures, it’s a successful building. Except, during the summer of 2013, it started setting things on fire.

The exterior of the building was designed by architect Rafael Viñoly to have a sweeping curve, but this meant that 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.

Even without a previous building having set something on fire, however, the mathematics of focusing light is very well understood. The shape of a parabola—that ubiquitous curve from whenever you had to graph any variation on y = x2 at school—will focus all directly incoming parallel light onto a single focal point.

Famously, when London’s Millennium Bridge was unveiled in 2000, it had to be closed after only two days. The engineers had failed to calculate that people walking on it would set the bridge swinging.

So, instead of being suspended from a rope hanging above like someone rappelling down a cliff, the ropes were pulled almost straight and held the bridge up, in effect functioning more like a tightrope. The steel ropes have to be very tight: the cables carried a tension force of about 2,200 tons.

Much like a guitar string, the more tension in a bridge, the more likely it is to vibrate at higher frequencies. If you gradually decrease the tension in a guitar string, the note it plays will get lower, until the string becomes too slack to play any note at all. 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.

Engineers have a lot of experience in stopping bridges from bouncing, and all the calculations were spot on for vertical movement. But the engineers who designed the Millennium Bridge underestimated the importance of lateral movement.

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.

From the Latin word resonare, which roughly means “echo” or “resound,” in the nineteenth century “resonance” became a scientific term to describe infectious vibrations.

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.

Playing the trumpet involves tightening your lips and using them to throw a cacophony of messy frequencies at it. But only those that match the resonant frequencies of the cavity inside the trumpet build up to audible levels.

The antenna is receiving a mess of different electromagnetic frequencies from TV signals, Wi-Fi networks, and even someone nearby microwaving leftovers. The antenna is then plugged into an electronic resonator, made of capacitors and coils of wire, that perfectly matches the specific frequency it wants to pay attention to.

A washing machine is incredibly annoying in that brief moment when the spin frequency matches the resonance of the rest of the machine: it takes on a life of its own and decides to go for a walk.

In July 2011, a thirty-nine-story shopping center in South Korea had to be evacuated because resonance was vibrating the building. People at the top of the building felt it start to shake, as if someone had banged the bass and turned up the treble. Which was exactly the problem.

On the Millennium Bridge, people did start to walk in step, because the movement of the bridge affected the rhythm at which they were walking. This formed a feedback loop: people stepping in sync caused the bridge to move more, and the bridge moving caused more people to step in sync.

This is how engineering progresses. Before the Millennium Bridge, the math of “synchronous lateral excitation” from pedestrians was not at all well understood.

In 1993 an investigation was carried out on a footbridge that wobbled sideways when two thousand people crossed it at the same time. Before that, there was a 1972 investigation into a bridge in Germany with similar problems when three hundred to four hundred people walked on it simultaneously. But none of this had seemingly made it into the building regulations for bridges. Everyone remained obsessed with vertical vibrations.

The vertical up-and-down impact from a human walking is around ten times greater than the side-to-side force, which is why the lateral movements had been ignored for so long. The vertical vibrations of bridges had been noticed much sooner.

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.

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.

It was not that the bridge could not support the extra weight but rather that the combination of length and mass opened up a whole new way for bridges to go wrong. It turns out that, as well as vibrating up and down and side to side, bridges can also twist in the middle.

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.

Most structures don’t have the right combination of size and length to twist noticeably, so torsional instability is forgotten about until a new construction dips just below the threshold where it manifests and then, suddenly, it’s back!

But the notoriety of this bridge’s collapse has come with a downside: the wrong explanation. To this day, the Tacoma Narrows Bridge disaster is held up as an example of the dangers of resonant frequencies.

A feedback loop that had teamed up not with resonance but with torsional instability. The sleekness of the design made it very aerodynamic. As in, the air made it dynamic. Whereas other proposed designs for the Tacoma Narrows Bridge had a metal mesh that wind could have blown through, the bridge that was built had flat metal sides, perfect for catching the wind.

Under normal circumstances, the bridge would twist in the middle a bit but quickly spring back to normal. However, with enough wind the flutter feedback loop could drive torsional instability to very noticeable levels.

If the upwind side of the bridge were to lift slightly via some classic torsional twisting, then it would act like an airplane wing and be pushed higher by the wind. When it rebounded and dipped down, the wing...

It turns out that torsional instability can also affect buildings. 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.

Despite being designed in line with current building codes, torsional instability found a way to twist the building, and people on the top floors started feeling seasick. Once again, it was tuned mass dampers to the rescue!

The only downside is that mathematics and experience now allow humans to build structures beyond what our intuition can comprehend.

We are approximation machines. Math, however, can get straight to the correct answer. It can tease out the exact point where things flip from being right to being wrong, from being correct to being incorrect, from being safe to being disastrous.

During construction, a seemingly innocuous change was made to the design, and the engineers did not properly redo the calculations. No one noticed that the change would fundamentally alter the underlying mathematics. And it pushed the walkway over the edge.

In 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.

because of the alteration in the design the upper-walkway bolts were under about twice that load, estimated to have been 9,690 kilograms per bolt. This was more than the box beams could handle, so one of the bolts in the middle was ripped out. This meant that the remaining bolts were each bearing even more load, so they all failed in quick succession, causing the walkway to collapse.

In the mid-1990s, a new employee of Sun Microsystems in California kept disappearing from their database. Every time his details were entered, the system seemed to eat him whole; he would disappear without a trace. No one in HR could work out why poor Steve Null was database kryptonite.

Null is still a legitimate surname and computer code still uses NULL to mean a lack of data. A modern variation on the problem is that a company database will accept an employee with the name Null, but then there is no way to search for them.