At 3:14 a.m. on Tuesday, January 19, 2038, many of our modern microprocessors and computers are going to stop working. And all because of how they store the current date and time. Individual computers already have enough problems keeping track of how many seconds have passed while they are turned on; things get worse when they also need to keep completely up-to-date with the date. Computer timekeeping has all the ancient problems of keeping a calendar in sync with the planet plus the modern limitations of binary encoding.

When the first precursors to the modern internet started to come online in the early 1970s, a consistent timekeeping standard was required. The Institute of Electrical and Electronics Engineers (IEEE) threw a committee of people at the problem, and in 1971 they suggested that all computer systems could count sixtieths of a second from the start of 1971. The electrical power driving the computers was already
coming in at a rate of 60 Hertz (vibrations per second), so it simplified things to use this frequency within the system. Very clever. Except that a 60- Hertz system would exceed the space in a 32-digit binary number in a little over two years and three months. Not so clever.

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. 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.

Yet here we are. More than halfway there and we’re still on the same system. The clock is literally ticking. 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 num- ber of seconds that have passed since January 1, 1970. If you’re reading this before Wednesday, May 18, 2033, it is still coming up on 2 billion seconds. What a party that will be. Sadly, in my time zone, it will be around 4:30 a.m. I remember a boozy night out on February 13, 2009, with some friends to celebrate 1,234,567,890 seconds having passed, at just after 11:31 p.m. My programmer friend Jon had written a program to give us the exact countdown; everyone else in the bar was very confused as to why we were celebrating Valentine’s Day half an hour early.

Celebrations aside, we are now well over halfway through the count-up to destruction. After 2,147,483,647 seconds, everything stops. Microsoft Windows has its own timekeeping system, but MacOS is built directly on Unix. More importantly, many significant computer processors in everything from internet servers to your washing machine will be running some descendant of Unix. They are all vulnerable to the Y2K38 bug.

I don’t blame the people who originally set up Unix time. They were working with what they had available back then. 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 first edition of this book was published in 2019, and occasionally I think about ways to future-proof it. Maybe I’ll include “at the time of writing” or carefully structure the language to allow for things to change and progress in the future so that it doesn’t go completely out of date. You might be reading this after the 2 billion second mark in 2033; I’ve allowed for that. But at no point do I think about people reading it in 2087. That’s sixty-eight years away!

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.

Once we live in an entirely 64-bit world, we are safe. The question is: will we upgrade all the multitude of microprocessors in our lives before 2038? We need either new processors or a patch that will force the old
ones to use an unusually big number to store the time.

Here is a list of all the things I’ve had to update the software on recently: my lightbulbs, a TV, my home thermostat, and the media player that plugs into my TV. I am pretty certain they are all 32-bit systems. Will they be updated in time? Knowing my obsession with up-to-date firmware, probably. But there are going to be a lot of systems that will not get upgraded. There are also processors in my washing machine,
dishwasher, and car, and I have no idea how to update those. It’s easy to write this off as a second coming of the Y2K “millennium bug” that wasn’t. That was a case of higher level software storing the year as a two-digit number, which would run out after 99. With a massive effort, almost everything was updated. But a disaster averted does not mean it was never a threat in the first place. It’s risky to be complacent because Y2K was handled so well. Y2K38 will require updating far more fundamental computer code and, in some cases, the computers themselves.

From HUMBLE PI: When Math Goes Wrong in the Real World by Matt Parker, publishing on January 21, 2020 by Riverhead, an imprint of Penguin Publishing Group, a division of Penguin Random House LLC. Copyright © 2019 Matt Parker. First published in Great Britain as HUMBLE PI: A Comedy of Maths Errors by Allen Lane, an imprint of Penguin Random House UK, 2019.

My numerical idiocy aside, Facebook has trained an AI to solve the toughest of math problems. Real superstring stuff. In effect, FB has taught their neural network to view complex mathematical equations “as a kind of language and then [treat] solutions as a translation problem for sequence-to-sequence neural networks.”

This is actually quite a feat since most neural networks operate on an approximation system: they can figure out if an image is of a dog or a marmoset or a steam radiator with a reasonable amount of certainty but precisely calculating figures in a symbolic problem like b – 4ac = 7 is a whole different kettle of fish. Facebook managed this by not treating the equation like a math problem but rather like a language problem. Specifically the research team approached the issue using neural machine translation (NMT). In short, they taught an AI to speak math. The result was a system capable of solving equations in a fraction of the time that algebra-based systems like Maple, Mathematica, and Matlab would take.

“By training a model to detect patterns in symbolic equations, we believed that a neural network could piece together the clues that led to their solutions, roughly similar to a human’s intuition-based approach to complex problems,” the research team wrote in a blog post released today. “So we began exploring symbolic reasoning as an NMT problem, in which a model could predict possible solutions based on examples of problems and their matching solutions.”

Essentially the research team taught the AI to unpack mathematical equations much in the same way that we do for complex phrases, like the example below. Instead of breaking out the verbs, nouns and adjectives, the system silos the various individual variables.

The researchers focused primarily on solving differential and integration equations, but, because those two flavors of math don’t always have solutions for a given equation, the team had to get tricky in generating training data for the machine learning system.

“For our symbolic integration equations, for example, we flipped the translation approach around: Instead of generating problems and finding their solutions, we generated solutions and found their problem (their derivative), which is a much easier task,” the team wrote and which I vaguely understand. “This approach of generating problems from their solutions — what engineers sometimes refer to as trapdoor problems — made it feasible to create millions of integration examples.”

Still, it apparently worked. The team achieved a success rate of 99.7 percent on integration problems and 94 percent and 81.2 percent, respectively, for first- and second-order differential equations, compared to 84 percent on the same integration problems and 77.2 percent and 61.6 percent for differential equations using Mathematica. It also took FB’s program just over half a second to arrive at its conclusion rather than the several minutes it required for existing systems to do the same.

The team started by establishing the constraints on lengths and angles that need to be met for producing any given cut pattern, and then use a numerical optimization approach to determine the generic patterns themselves (such as their orientation, number and size). From there, it’s a matter of using mechanical analysis to manage the deployment path and its stability. The researchers crafted 2D and 3D models to verify that the method worked.

The result is pretty, as you might imagine, but it could also be highly practical once put into use. You could produce clothes, vehicle surfaces or other objects using just a sheet. This wouldn’t always work due to the nature of kirigami (you don’t really want a poncho full of holes), but it’s also just the beginning. The Harvard group hopes to bring origami into the mix, allowing even more complicated objects that would only need the right algorithm to come to life.

Calculus has traditionally been applied in the “hard” sciences like physics, astronomy, and chemistry. But in recent decades, it has made inroads into biology and medicine, in fields like epidemiology, population biology, neuroscience, and medical imaging. We’ve seen examples of mathematical biology throughout our story, ranging from the use of calculus in predicting the outcome of facial surgery to the modeling of HIV as it battles the immune system. But all those examples were concerned with some aspect of the mystery of change, the most modern obsession of calculus. In contrast, the following example is drawn from the ancient mystery of curves, which was given new life by a puzzle about the three-dimensional path of DNA.

The puzzle had to do with how DNA, an enormously long molecule that contains all the genetic information needed to make a person, is packaged in cells. Every one of your ten trillion or so cells contains about two meters of DNA. If laid end to end, that DNA would reach to the sun and back dozens of times. Still, a skeptic might argue that this comparison is not as impressive as it sounds; it merely reflects how many cells each of us has. A more informative comparison is with the size of the cell’s nucleus, the container that holds the DNA. The diameter of a typical nucleus is about five-millionths of a meter, and it is therefore four hundred thousand times smaller than the DNA that has to fit inside it. That compression factor is equivalent to stuffing twenty miles of string into a tennis ball.

On top of that, the DNA can’t be stuffed into the nucleus haphazardly. It mustn’t get tangled. The packaging has to be done in an orderly fashion so the DNA can be read by enzymes and translated into the proteins needed for the maintenance of the cell. Orderly packaging is also important so that the DNA can be copied neatly when the cell is about to divide.

Evolution solved the packaging problem with spools, the same solution we use when we need to store a long piece of thread. The DNA in cells is wound around molecular spools made of specialized proteins called histones. To achieve further compaction, the spools are linked end to end, like beads on a necklace, and then the necklace is coiled into ropelike fibers that are themselves coiled into chromosomes. These coils of coils of coils compact the DNA enough to fit it into the cramped quarters of the nucleus.

But spools were not nature’s original solution to the packaging problem. The earliest creatures on Earth were single-celled organisms that lacked nuclei and chromosomes. They had no spools, just as today’s bacteria and viruses don’t. In such cases, the genetic material is compacted by a mechanism based on geometry and elasticity. Imagine pulling a rubber band tight and then twisting it from one end while holding it between your fingers. At first, each successive turn of the rubber band introduces a twist. The twists accumulate, and the rubber band remains straight until the accumulated torsion crosses a threshold. Then the rubber band suddenly buckles into the third dimension. It begins to coil on itself, as if writhing in pain. These contortions cause the rubber band to bunch up and compact itself. DNA does the same thing.

This phenomenon is known as supercoiling. It is prevalent in circular loops of DNA. Although we tend to picture DNA as a straight helix with free ends, in many circumstances it closes on itself to form a circle. When this happens, it’s like taking off your belt, putting a few twists in it, and then buckling it closed again. After that the number of twists in the belt cannot change. It is locked in. If you try to twist the belt somewhere along its length without taking it off, counter-twists will form elsewhere to compensate. There is a conservation law at work here. The same thing happens when you store a garden hose by piling it on the floor with many coils stacked on top of each other. When you try to pull the hose out straight, it twists in your hands. Coils convert to twists. The conversion can also go in the other direction, from twists to coils, as when a rubber band writhes when twisted. The DNA of primitive organisms makes use of this writhing. Certain enzymes can cut DNA, twist it, and then close it back up. When the DNA relaxes its twists to lower its energy, the conservation law forces it to become more supercoiled and therefore more compact. The resulting path of the DNA molecule no longer lies in a plane. It writhes about in three dimensions.

In the early 1970s an American mathematician named Brock Fuller gave the first mathematical description of this three-dimensional contortion of DNA. He invented a quantity that he dubbed the writhing number of DNA. He derived formulas for it using integrals and derivatives and proved certain theorems about the writhing number that formalized the conservation law for twists and coils. The study of the geometry and topology of DNA has been a thriving industry ever since. Mathematicians have used knot theory and tangle calculus to elucidate the mechanisms of certain enzymes that can twist DNA or cut it or introduce knots and links into it. These enzymes alter the topology of DNA and hence are known as topoisomerases. They can break strands of DNA and reseal them, and they are essential for cells to divide and grow. They have proved to be effective targets for cancer-chemotherapy drugs. The mechanism of action is not completely clear, but it is thought that by blocking the action of topoisomerases, the drugs (known as topoisomerase inhibitors) can selectively damage the DNA of cancer cells, which causes them to commit cellular suicide. Good news for the patient, bad news for the tumor.

In the application of calculus to supercoiled DNA, the double helix is modeled as a continuous curve. As usual, calculus likes to work with continuous objects. In reality, DNA is a discrete collection of atoms. There’s nothing truly continuous about it. But to a good approximation, it can be treated as if it were a continuous curve, like an ideal rubber band. The advantage of doing that is that the apparatus of elasticity theory and differential geometry, two spinoffs of calculus, can then be applied to calculate how DNA deforms when subjected to forces from proteins, from the environment, and from interactions with itself.

The larger point is that calculus is taking its usual creative license, treating discrete objects as if they were continuous to shed light on how they behave. The modeling is approximate but useful. Anyway, it’s the only game in town. Without the assumption of continuity, the Infinity Principle cannot be deployed. And without the Infinity Principle, we have no calculus, no differential geometry, and no elasticity theory.

I expect in the future we will see many more examples of calculus and continuous mathematics being brought to bear on the inherently discrete players of biology: genes, cells, proteins, and the other actors in the biological drama. There is simply too much insight to be gained from the continuum approximation not to use it. Until we develop a new form of calculus that works as well for discrete systems as traditional calculus does for continuum ones, the Infinity Principle will continue to guide us in the mathematical modeling of living things.

Excerpted from INFINITE POWERS by Steven Strogatz. Copyright © 2019 by Steven Strogatz. Reprinted by permission of Houghton Mifflin Harcourt Publishing Company. All rights reserved.