Cycles and Predictions
A brief summary of a brief history
No, this isn’t another post about bikes; instead, this is a post about the talk I gave for a New Scientist event last week, the last currently scheduled “live” event of the promotional cycle for A Brief History of Timekeeping. (There are multiple reasons for the quotes around “live” in that, which I will explain at the end of this post…) This was the only time I’ve done this version of the book talk, and the video won’t be freely available, so I thought I’d do a sort of text summary of the basic ideas, just so I can point to it later should the need arise.
This was part of New Scientist’s “Big Thinkers” series, so I didn’t want to give the navigation-themed version of the talk again, but to try to tie into some grander intellectual history. The angle I took for this was to focus on a kind of tension that sits at the very heart of the activity of tracking and measuring time, between cyclical and linear approaches.
That is, the key central process of timekeeping— the Big Idea, if you will—is the counting of cycles. Time is what you measure with a clock, and a clock is a thing that “ticks:” that undergoes some regular cycle whose repetitions can be counted to quantify the passage of time. The ticks can be very fast, like the 9,192,631,700 oscillations per second of the microwaves in a cesium clock, or incredibly slow like the 1,400-year Sothic cycle measured by the ancient Egyptian calendar, but in the end, it’s all counting “ticks”
While timekeeping necessarily involves cycles, though, our experience of time is undeniably linear. Despite the best efforts of theoretical physicists to cook up schemes where people could murder their own grandparents, we’re all on a one-way trip from the past into the future, moving at a steady rate of one second per second.
So there’s a kind of tension there, in that we’re measuring a one-way trip using cyclical processes. This is reflected in a lot of different ways through the history of human civilization, but one of the most common is an interest in prediction. That is, an attempt to identify and understand natural cycles, and then not just use them to mark time, but also to run them forward to say how things will be at some future time.
The oldest forms of timekeeping, dating back thousands of years, before written language, use the motion of the Sun. Its daily motion from rising in the east(-ish) to setting in the west(-ish) is the most fundamental and important “tick” in history. There’s also a slower motion of the rising and setting points along the horizon— rising and setting more to the north in the Northern Hemisphere summer, and rising and setting more to the south in the Northern Hemisphere winter— that we’ve been tracking for millennia. The passage tomb at Newgrange, built around 3300BCE, is a giant solstice marker, picking out the day when the sun rises at its southernmost point. It’s a clock that “ticks” once a year.
(This also has an obvious predictive character— the day it marks is the shortest of the year, and from that point on, the days will get longer and warmer. While we don’t know exactly who they were or what they were doing, it’s pretty obvious that this would be important for an agrarian society to know.)
There’s a third screamingly obvious cycle in the sky, though, namely the phases of the Moon. Over the course of 29-and-a-bit days, the Moon goes from a thin crescent appearing just after sunset to a full disk to a thin crescent disappearing just before sunrise. The shape of the Moon changes rapidly enough to make it a really useful time scale for a lot of human activities, so lots of civilizations have understandably made use of it.
The problem, though, is that none of these cycles are “nice” multiples of one another. Taking the day as the most fundamental of ticks, the length of a tropical year is 365.24217 days (from one summer solstice to the next) and the length of a lunar month is 29.53059 days. Both of those have lots of decimal places because we’ve been measuring them for thousands of years, and neither of those is a “nice” multiple of the other. That means that reconciling them in a way that lets you make predictions is a difficult problem, requiring making some choices.
Maybe the simplest choice you can make is to follow the path of the Islamic calendar, which is strictly lunar— twelve months, each starting and ending with the appearance of a new crescent moon. This allows you to unambiguously predict the phase of the moon on a given date in a given year, but it doesn’t tell you anything at all about the seasons. A twelve-month lunar calendar comes out to around 355 days, well short of a tropical year, so the Islamic calendar shifts relative to the seasons. That’s why the month of Ramadan, when devout Muslims fast during the day, fell in the early (Northern Hemisphere) spring this year, but was in high summer several years ago, and will be in the winter several years hence.
The other relatively simple choice you can make is to ignore the Moon, as in the Gregorian calendar used in the West and as a civil calendar for international trade. This has a set of twelve months with fixed lengths, plus some leap year rules that add a day every now and them, and the end result is a year whose average length matches the tropical year to within a few minutes. The allows you to unambiguously predict the season of a given date in a given year, and to know when the solstices and equinoxes will be to within a day or so, but it tells you nothing at all about the phase of the moon.
If you want to track both solar and lunar cycles, you end up with something like the Jewish calendar, which has twelve fixed-length months that match the length of the lunar cycle, but also a set of rules for inserting a thirteenth month every few years, to keep the months in the correct seasons. There are also some rules relating to the days of the week that can add days, giving the Jewish calendar six different year lengths in total. This lets you say with some confidence what season a given date will fall in, and what the phase of the moon will be, but at the cost of a good deal of complexity.
Each of these calendar systems lets you make predictions about how the world will be on a future date. Exactly what you can predict is a matter of choice, reflecting the particular priorities of the culture that produced the calendar in question.
These are the three most obvious approaches to making a calendar, and most Eurasian civilizations follow some version of these— the traditional calendars in China and India used variants of the lunar-with-an-occasional-extra-month scheme. This is probably not too surprising, given that they were all in intermittent contact with one another. These aren’t the only possible choices though, but you’ve got to go to very isolated cultures to find something significantly different, the most spectacular version of which is the Mayan calendar.
The Maya in Central America around 1000CE used a set of two interlocking calendar cycles: one had eighteen months with twenty days each, plus a five-day “unlucky” period at the end of each year, for a total of 365 days identified by a name and a number. The other used a system of twenty names of deities plus a set of thirteen numbers (each day you move to the next name and the next number), for a total of 260 days identified by a name and a number. The four-character designation of a day— name and number from the 365-day agricultural calendar and name and number from the 260-day ritual calendar— repeats once every 18,980 days, or around 52 years.
Nobody knows quite what the origin of the 260-day cycle was— the Spanish conquistadores were distressingly thorough about stamping out as much of the Mayan culture as they could— but it’s possible that this reflects astronomy in the tropics. In addition to the usual solstices and equinoxes, people at the latitudes where the Maya flourished will see two “zenith crossing” days per year, where the sun rises north of east and sets north of west, but is directly overhead at noon. The larger interval between these is close to 260 days in the places where the earliest Mayan sites are found, and that period is the best season for warfare and agriculture. So it’s at least plausible that counting this off was the origin of the 260-day ritual calendar.
Whatever the reason, the Maya tracked this very assiduously, and were also keen obbservers of other phenomena in the sky. One of the handful of Mayan texts that escaped being consigned to the fire by the Spanish is the “Dresden Codex,” and contains a number of pages devoted to tables of numbers predicting the rising and setting of the planet Venus. This is the third brightest object in the sky, and much more complicated than either the Sun or the Moon, appearing just after sunset or just before sunrise, moving up the sky on subsequent nights at the same time, then back down to disappear in the glare of the sun, and then reappear some time later in the other phase. The motion is complicated, but there’s a regular pattern to it, and the Dresden Codex tables correctly predict the dates of the first rising and setting as either morning or evening star to within a few days over a period of several hundred years. They even synch up with the Mayan ritual calendar in such a way that the reappearances fall on “auspicious” dates associated with the god identified with Venus. Again, we see this meshing of cycles, and a strong interest in predicting the future. There are even indications that this was considered essential for warfare, such as a set of murals at Bonampak that some interpret as crediting a victory in battle to knowing the right point in the Venus cycle to attack.
We don’t know much about how the Mayan astronomer-priests tracked the position between the risings and setting predicted in the Dresden Codex— those damned Spanish priests again—but the basic idea is pretty straightforward. You measure the angle between the planet or other object of interest and a star whose position is known, using an instrument that can be as simple as a forked stick. Do that for at least three fixed stars, and you can uniquely specify a position on the sky. Astronomers all over Eurasia have been doing this for millennia, and those records have been carefully preserved.
The best astronomer ever to do this with the naked eye was probably Tycho Brahe, who parlayed the resources available to him as a wealthy Danish nobleman into a magnificent observatory on the island of Hven, where he compiled years of data on the positions of the planets. He hoped to use this to determine the structure of the solar system, but died before it could be completed, and his data passed to Johannes Kepler, who cracked the problem. Kepler used Brahe’s observations of Mars to try to construct a circular orbit that would predict its position, but found there was no way to make it work perfectly. The error was tiny— about a quarter of the width of the full moon— but Brahe’s observations were so good that Kepler felt this couldn’t be brushed aside, and it led him to a new understanding of the operation of the planets.
Kepler’s new system had the planets moving in elliptical orbits with the Sun at one focus, changing speeds in such a way that they covered equal areas in equal times, and it worked brilliantly. Astronomers using it could predict fleeting events such as a transit of Venus, where the planet passes between the Earth and Sun for a short period. Explaining why these orbits were elliptical, and what held them in orbit, was one of the principal problems that spurred Isaac Newton to write the Principia Mathematica, transforming the science of physics. Newton’s laws of motion and theory of gravitation gave a solid conceptual reason for Kepler’s system, and the two together became central to our understanding of the universe.
Using Newton’s laws and Kepler’s theories, you can start to predict the orbit of the Moon in detail, which turns out to be a fiendishly difficult problem (Newton himself got it badly wrong) because the gravitational pull of the Sun on the Moon can’t be ignored. This makes the orbit very complex, and it took some of the best mathematical minds of the 18th century— notably the great Leonhard Euler— years to crack it. Once they did, careful work by the German catrographer and astronomer Tobias Mayer— measuring the position and grinding through caculations— led to a set of tables that could be used to predict the Moon’s position against the background stars for years into the future, with enough precision to serve as a clock. These were put in a more sailor-friendly form by the Astronomer Royal, Nevil Maskelyne, and formed the backbone of the Nautical Almanacs that were key to navigation through the 1800’s.
In the same span of time, between Kepler and Mayer, the emerging new physics led to a great leap in human-scale timekeeping with the invention of the pendulum clock. First proposed in the 1630’s by Galileo (who was basically completely blind by this point), the first pendulum clock was built by Christiaan Huygens and Solomon Coster in 1658 (though Huygens got into the inevitable priority dispute with Robert Hooke), and within a decade this had led to mechanical clocks accurate to minutes per year (as opposed to minutes per day, for pre-pendulum mechanical clocks). This, combined with the invention of the telescope, drove a further revolution in our understanding of the universe.
In the early 1670’s, the Danish astronomer Ole Christensen Rømer was working for Jean-Domonique Cassini at the Paris Observatory, making observations of the moons of Jupiter, and he noticed a pattern in the timing of the eclipses of Io. Once per orbit, Io dips into Jupiter’s shadow and disappears for a short time, and these eclipses occur in a very regular and predictable way that can be used as a precise clock for measuring longitude (which is why Cassini and company were interested in this). When compared to a mechanical clock, though, Rømer noticed that at some times of the year, each eclipse showed up very slightly earlier than expected, while at other times, each showed up a little late.
He correctly attributed this to the light from the eclipses traveling at a finite speed— we don’t see the eclipse until the light from it reaches the Earth, and that takes some time. When the Earth’s orbit is bringing it closer to Jupiter, the distance the light needs to cover gets slightly shorter during the time between eclipses, and the eclipses show up early. When the Earth’s orbit is carrying it away from Jupiter, on the other hand, the distance gets slightly longer, and the eclipses show up late. The total difference is a matter of minutes, but it let Rømer make a decent estimate of the speed way back in the 1670’s; more importantly, it established the finiteness of that speed.
Fast-forward into the late 1800’s and early 1900’s, and that finite speed became the cornerstone of a new revolution in physics. Einstein’s theory of relativity (which was not solely his, as George FitzGerald, Hendrik Lorentz, and Henri Poincare all had important pieces before Einstein) can be arrived at by taking that finite speed of light as a fundamental feature of the universe, a constant that has to be agreed upon by all observers no matter how they’re moving. This forces a fundamental reconsideration of what it means to measure an interval in time and what it means to measure a distance in space, with the surprising result that moving clocks will run slow, and moving objects will shrink. Einstein carried this further with General Relativity, which brings gravity in: time passes more slowly in the presence of gravity, so a clock at sea level will tick slightly slower than an identical clock at the top of a mountain.
These results run very counter to our intuition, but they’re confirmed empirical facts, measured in any number of experiments. My favorite is a 2010 experiment from a group at NIST in Boulder, who used aluminum-ion clocks that “tick” 1.21 quadrillion times a second to demonstrate it. The built two identical clocks and held the ion in one steady while setting the other in motion at speeds ranging from a slow walk to a fast drive, and found that the moving ion’s “tick” slowed down by exactly the predicted amount. Then they held both ions at rest, but jacked one clock up by about a foot (in American units) and found that its “tick” sped up by exactly the amount predicted by General Relativity. Strange as they seem, the effects of relativity are unquestionably real, and in a very deep way, it all traces back to the drive to track and most importantly predict the passage of time, a path that runs from the builders of Newgrange through naked-eye astronomers like the Maya and Tycho Brahe, to folks like Einstein to modern-day scientists building atomic clocks of unparalled precision.
I teased at the start that I would explain the scare quotes around “live,” so as a reward for reading this far…
The first reason is the obvious one: this was an online streaming event, not an in-person talk. But more than that, the streaming presentation was not, in fact, done live, but was pre-recorded, for the very sensible reason that it would’ve sucked for the Internet to go out in the middle of a live stream of the talk. So, I went into my office about a week before the live event, and recorded it on a stream with just me, the host, and a tech from New Scientist.
Then, on the day of the event, I went back to my office, and watched the stream along with however many folks signed up, then came on actually live and did Q&A after the talk. They reminded me a whole bunch of times to make sure I was doing the Q&A from the same place and wearing the same clothes, which had the effect of making me incredibly paranoid— I spent the whole week terrified that I was either going to test positive for Covid and be barred from my campus office, or take a foul ball to the face at one of The Pip’s baseball games and show up with a black eye that would ruin the illusion.
Happily, neither of those things happened, so both in the recording and during the Q&A I looked pretty much like this test shot I took to make sure I looked okay, and all my various Easter eggs were visible in the background:
So, that’s my talk in text form, and a bit of the funny story behind it. If you’d like more of this kind of thing, here’s a button you can click:
If you’d like to challenge any of my descriptions, ask for a clarification, or just guess which items in that photo I wanted to be on camera and why, the comments will be open:
I thought the scare quotes around "live" was going to actually be explained by something related to timekeeping and general relativity, and maybe a deep question or two about "can something be 'live' in an absolute sense."
But I'll never give up a good chance at a Bookshelf Judging.