The sorta-all-in-one guide

 kw: book reviews, nonfiction, compendiums

Who wants to know everything? Pick me! Not many folks have four college majors (yes, it cost more than usual). Just peruse the contents of this blog's roughly 2,000 book reviews on all subjects. So I couldn't pass up the chance to read The Complete Guide to Absolutely Everything*: Adventures in Math and Science [*Abridged] by Adam Rutherford and Hannah Fry.

Since the book is admittedly extracts of a potentially universal encyclopedia (perhaps one for which Wikipedia is a rehearsal), the authors felt free to choose items of interest to them. The nine chapters are riffs on nine subjects. I'll touch on three of them:

Chapter 3, "The Perfect Circle" starts with an insult attributed to Fritz Zwicky, "spherical bastard." Since a sphere has perfect symmetry and looks the same from every direction, this hypothetical jerk is always and everywhere the same. But can there be a perfect sphere, or even a perfect circle? I suppose there could be if matter were continuous rather than quantized into atoms and other "elementary" particles. Just to keep things interesting, the authors get into what a 4-D sphere would look like. To us 3-D creatures, it would look like a sphere, because its intersection with our 3-D spacetime would be a spherical "cut" from its hyperspherical reality. As a hypersphere "moved through" 3-D spacetime, it would first look like a tiny sphere that grew, stabilized briefly, then shrank again to a tiny sphere that then winked out.

Fun enough. Can anything material be truly spherical or circular? Soap bubbles look like spheres, but are subtly distorted by gravity, and by even the tiniest, shifting breeze. The orb of the Earth isn't a sphere (ignoring mountains for a moment), but an oblate spheroid, the stable compromise between self-gravity and the centripetal force of its rotation. Even if it had no rotation, there are mountains and trenches, of course, but shrunk to the size of a cue ball, it would be smoother than the cue ball. The four metal spheres in a super-gyroscope in one of the satellites are considered the smoothest, most perfect spheres ever made, but a strong microscope would enable us to see ultra-tiny defects in their surfaces. If we could get rid of every defect, however, the atoms or molecules of the ball force a limit below which the smoothness cannot be reduced. Consider this printed circle:

It looks pretty good, even though I deliberately made it rather small. I instructed Blogger to display it "original size", so it matches the pixels on your computer screen. But that screen does have pixels. The "pixels" of a piece of paper are smaller, of course. Here is a 16x blowup of part of the red circle:

Somewhere in this jaggedy band of red and pink pixels would run a line that represents the ideal circle I had PowerPoint draw for me. Is it possible to make a circle that is actually perfect? Clearly not. Is it possible to place some atoms such that they are on the exact locations a circular arc would pass through? To my figuration, at most 12 atoms, plus a 13th to mark the circle's center, could be so placed, using equipment such as an atomic force microscope (AFM) to push around atoms on the surface of a perfect atomic lattice such as a surface of pure silicon, oriented in a direction such that the Si atoms are in a square array:

• Place atom #1 where you want the center to be, nestled in a pocket between four Si atoms.
• Place #2 in such a pocket, located 5 spaces to the right.
• Place #3 in a pocket 4 to the right and 3 upwards of the center; the 3-4-5 triangle has a hypotenuse 5 units long.
• Place #4 in a pocket 3 to the right and 4 upwards.
• Place #5 in a pocket 5 spaces upwards.
• Continue around the circle.

At the end, the twelve peripheral atoms are all exactly 5 units from the center atom. The first person to do this will be the first person to create a "dotted line" that traces a perfect circle (within the limits of quantum vibration of the Si atoms!). To avoid insanity, I won't think about what is entailed in creating something with atoms at some exact distance from a known center, to form such a tracery on a perfect sphere.

Chapter 5, "A Brief History of Time" centers initially on high-speed investment algorithms that take advantage of the time lags in communication between different stock exchanges. Such algorithms have caused half a dozen "flash crashes", which came and went in milliseconds, and briefly (and fortunately, reversibly) destroyed around a trillion dollars of equity in world markets. A couple pages in, the authors ask "what is a second?", and lose their way. Here is a pair of sentences to which I take strong exception:

"If you want to measure how long a second it, it should simply be a matter of pointing a telescope straight up at a star in the sky and waiting until the same star comes back around to the same spot the next night—that is, an exact day later. If you divide the time elapsed by 86,400 (the number of seconds in the day), then you should end up with precisely the length of one second."

Nope!! This will only work if the star you focus upon is the Sun; especially, some unchanging feature of the Sun such as its east or west edge. This is a confusion between solar time and sidereal time. During the day in which the Earth rotates once, to point at the same feature on the Sun's surface (or edge), one solar day passes (which is unlikely to be exactly 86,400 seconds long, as the rest of the chapter describes). During that day, the Earth moves just under one degree along its orbit, so that it has to rotate that extra most-of-a-degree to point to the same feature again. If you begin with any other star, the time that passes will be one sidereal day, which has a length of 86,164.0905 seconds.

In this chapter we find a version of this diagram, which describes the Equation of Time. This shows the cumulative effect of variations in the length of an apparent solar day, and is the expected error of a sundial at various times during the year.

Two factors create this effect. Firstly, the Earth's orbit is not a perfect circle, but an ellipse. Using slightly rounded figures, our distance from the Sun varies from 91,407,000 miles in early January to 94,510,000 miles in early July. That means that in early April and early October, the Sun is offset from the center of the ellipse, as seen from Earth, by about 1.5 million miles. Thus, the solar day varies from 86,379 seconds to 86,429 seconds. Note that the difference from 86,400 is not symmetrical. This is because of the Earth's axial tilt. 

The contribution of axial tilt to the length of the day is more complex, so I won't try to explain. Instead, we can see from this diagram that it has two cycles per year (the purple dashed line), while the variation caused by the elliptical orbit has one cycle (the blue dot-dash line). These add to the total equation of time (the red solid line). This graph is from the German language Wikipedia.

Another expression of this mess is the Analemma, the infinity-shaped symbol printed on globes. Hardly anyone pays attention to it.

This is an example. The analemma represents the subsolar point at Noon, mean solar time, at a particular longitude for every day of the year. Some globes, as this one, have some explanation about it. Others just show the figure without much explanation. Probably only one person in 1,000 knows what the odd "8" on their globe means…of those who even have one.

This is just part of what makes the definition of "one second" far from obvious!

Chapter 6, "Live Free" asks "What is free will?", proceeds to tell why some scientists think there is no such things, then describes some conditions in which the thinking and attitude of an animal or person is affected by a chemical or a parasite. And then we find the possibility that we are still, somehow, capable of making decisions that seem to be free, and perhaps they are. 

From the other side, that of predicting the fate of the universe (or any part of it), the authors discuss chaos and quantum mechanics. I find this funny, both "haha" funny and "so odd" funny: Mathematical chaos isn't actually chaotic. It is repeatable if you always start from the same point.

We read of the Lorentz Butterfly, a seemingly unpredictable figure that represents near-cyclical patterns of weather. The origin of mathematical chaos came when Lorentz ran a simulation for a while, then stopped his computer program and wrote down the values of the parameters he was tracking. Then he let the program run a while more, seeing how it would to. Later he started the program with the values he had written down partway through, and was surprised that the ensuing trajectory soon went differently from what he had seen earlier! He realized that the program was calculating things to an accuracy of 15 decimals (48 bits), but he had written down the numbers with "only" seven decimals. The seemingly tiny difference from where the program started from during the second run made all the difference.

Mathematical "chaos" is better described as "sensitivity to initial conditions". This is seen in orbital mechanics. Predicting the position of a planet over many orbits is tricky. Every time the planet makes one orbit, the numbers that were added in the first half orbit all get subtracted out again. Tiny rounding errors pile up, and after a few orbits, they add up to substantial errors in the planet's position and velocity. Actually, in most systems that rely on numerical integration, the initial position error's size is doubled with every iteration. That's why it's best to use methods that permit one to take larger steps (usually called "higher order" methods). An error that is initially one-trillionth of the starting value will, in ten steps, grow to about 1,024 trillionths, or just over one-billionth. That doesn't seem so bad. However: ten more steps, and the error is more than one-millionth; ten more and it is one-thousandth; then a further ten, and the error is as big as the initial starting value, meaning that the planet is as much as half an orbit away from where you thought it should be. Going to a higher order method is part of the solution to such issues. Astrophysicists have numerous methods to stabilize and correct their calculations so they can predict the positions of planets and moons thousands or millions of orbits later.

The situation is worse in weather forecasting, which is what Edward Lorentz was working on. The weather models that are running on the world's largest supercomputers have millions of coupled differential equations, and they are getting better and better. Weather.com and Accuweather confidently predict the weather for up to 90 days. But in most parts of the world, going beyond a 3-day forecast is still pretty chancy. The truth is, long-range forecasts are adjusted "pattern" forecasts, based on similarity of history. The weather models all fall apart in 5-10 days, and sometimes less. The atmosphere is too big, and too much happens on too big a scale, and we have too few reliable weather stations taking the atmosphere's pulse. It's a wonder that even a 3-day forecast is any good at all.

Thus, in actuality, "chaos" just means "impossible to predict because there is way, way too little good data".

The quantum situation is different. Quantum uncertainty can be stated "impossible to predict because at the smallest scales, genuinely random influences occur." That means that atoms and electrons and protons and so forth can't be pinned down; they are subject to randomizing influences. The reason we can predict where a baseball is going is that, when the system of interest is composed of a trillion trillion atoms or more, those random influences mostly cancel out, to such a degree that we can neglect them. We only care if the ball is in the strike zone, while quantum effects on a baseball's path are measured in trillionths of a trillionth of a meter.

Interestingly, quantum effects on the path of ions and electrons in our neurons, which have axons with a diameter between 1 and 10 microns, can cause variations in the timing of a signal, and sometimes quench it altogether. Also, "shot noise" is the scattered arrival times of ions, which can change when or whether a certain synapse is triggered. This is one possible mechanism behind "free will."

But I like this description better, from an researcher who studies rats in mazes and such: "Given specific conditions of light, temperature, and location of food, the rat will do what the rat wants to do." A statement like that was once the motto of the Rat Runner's Digest. So if we feel like we have free will, it's like the proverbial duck: "Does it quack like a duck? Does it walk like a duck? It must be a duck."

If you want to know everything about everything, be prepared for a long process…like forever, to be precise. However, if you can settle for an 80-20 solution, this book provides a good start.