The Universe of Discourse

Mon, 21 Dec 2015

This is page 23 (the last) of the Cosmic Call message. An explanation follows.

This page is a series of questions for the recipients of the message. It is labeled with the glyph , which heretofore appeared only on page 4 in the context of solving of algebraic equations. So we might interpret it as meaning a solution or a desire to solve or understand. I have chosen to translate it as “wat”.

I find this page irritating in its vagueness and confusion. Its layout is disorganized. Glyphs are used inconsistent with their uses elsewhere on the page and elsewhere in the message. For example, the mysterious glyph , which has something to do with the recipients of the message, and which appeared only on page 21 is used here to ask about both the recipients themselves and also about their planet.

The questions are arranged in groups. For easy identification, I have color-coded the groups.

Starting from the upper-left corner, and proceeding counterclockwise, we have:

Kilograms, meters, and seconds, wat. I would have used the glyphs for abstract mass, distance, and time, and , since that seems to be closer to the intended meaning.

Alien mathematics, physics, and biology, wat. Note that this asks specifically about the recipients’ version of the sciences. None of these three glyphs has been subscripted before. Will the meaning be clear to the recipients? One also wonders why the message doesn't express a desire to understand human science, or science generally. One might argue that it does not make sense to ask the recipients about the human versions of mathematics and physics. But a later group expresses a desire to understand males and females, and the recipients don't know anything about that either.

Aliens wat. Alien [planet] mass, radius, acceleration wat. The meaning of shifts here from meaning the recipients themselves to the recipients’ planet. “Acceleration” is intended to refer to the planet's gravitational acceleration as on page 14. What if the recipients don't live on a planet? I suppose they will be familiar with planets generally and with the fact that we live on a planet, which explained back on pages 11–13, and will get the idea.

Fucking speed of light, how does it work?

Planck's constant, wat. Universal gravitation constant, wat?

Males and females, wat. Alien people, wat. Age of people, wat. This group seems to be about our desire to understand ourselves, except that the third item relates to the aliens. I'm not quite sure what is going on. Perhaps “males and females” is intended to refer to the recipients? But the glyphs are not subscripted, and there is no strong reason to believe that the aliens have the same sexuality.

The glyph , already used both to mean the age of the Earth and the typical human lifespan, is even less clear here. Does it mean we want to understand the reasons for human life expectancy? Or is it intended to continue the inquiry from the previous line and is asking about the recipients’ history or lifespan?

Land, water, and atmosphere of the recipients’ planet, wat.

Energy, force, pressure, power, wat. The usage here is inconsistent from the first group, which asked not about mass, distance, and time but about kilograms, meters, and seconds specifically.

Velocity and acceleration, wat. I wonder why these are in a separate group, instead of being clustered with the previous group or the first group. I also worry about the equivocation in acceleration, which is sometimes used to mean the Earth's gravitational acceleration and sometimes acceleration generally. We already said we want to understand mass , !!G!! , and the size of the Earth. The Earth's surface gravity can be straightforwardly calculated from these, so there's nothing else to understand about that.

Alien planet, wat. The glyph has heretofore been used only to refer to the planet Earth. It does not mean planets generally, because it was not used in connection with Jupiter . Here, however, it seems to refer to the recipients’ planet.

The universe, wat. HUH???

That was the last page. Thanks for your kind attention.

[ Many thanks to Anna Gundlach, without whose timely email I might not have found the motivation to finish this series. ]

Fri, 18 Dec 2015

I only posted three answers in August, but two of them were interesting.

• In why this !!\sigma\pi\sigma^{-1}!! keeps apearing in my group theory book? (cycle decomposition) the querent asked about the “conjugation” operation that keeps cropping up in group theory. Why is it important? I sympathize with this; it wasn't adequately explained when I took group theory, and I had to figure it out a long time later. Unfortunately I don't think I picked the right example to explain it, so I am going to try again now.

Consider the eight symmetries of the square. They are of five types:

1. Rotation clockwise or counterclockwise by 90°.
2. Rotation by 180°.
3. Horizontal or vertical reflection
4. Diagonal reflection
5. The trivial (identity) symmetry

What is meant when I say that a horizontal and a vertical reflection are of the same ‘type’? Informally, it is that the horizontal reflection looks just like the vertical reflection, if you turn your head ninety degrees. We can formalize this by observing that if we rotate the square 90°, then give it a horizontal flip, then rotate it back, the effect is exactly to give it a vertical flip. In notation, we might represent the horizontal flip by !!H!!, the vertical flip by !!V!!, the clockwise rotation by !!\rho!!, and the counterclockwise rotation by !!\rho^{-1}!!; then we have

$$\rho H \rho^{-1} = V$$

and similarly

$$\rho V \rho^{-1} = H.$$

Vertical flips do not look like diagonal flips—the diagonal flip leaves two of the corners in the same place, and the vertical flip does not—and indeed there is no analogous formula with !!H!! replaced with one of the diagonal flips. However, if !!D_1!! and !!D_2!! are the two diagonal flips, then we do have

$$\rho D_1 \rho^{-1} = D_2.$$

In general, When !!a!! and !!b!! are two symmetries, and there is some symmetry !!x!! for which

$$xax^{-1} = b$$

we say that !!a!! is conjugate to !!b!!. One can show that conjugacy is an equivalence relation, which means that the symmetries of any object can be divided into separate “conjugacy classes” such that two symmetries are conjugate if and only if they are in the same class. For the square, the conjugacy classes are the five I listed earlier.

This conjugacy thing is important for telling when two symmetries are group-theoretically “the same”, and have the same group-theoretic properties. For example, the fact that the horizontal and vertical flips move all four vertices, while the diagonal flips do not. Another example is that a horizontal flip is self-inverse (if you do it again, it cancels itself out), but a 90° rotation is not (you have to do it four times before it cancels out.) But the horizontal flip shares all its properties with the vertical flip, because it is the same if you just turn your head.

Identifying this sameness makes certain kinds of arguments much simpler. For example, in counting squares, I wanted to count the number of ways of coloring the faces of a cube, and instead of dealing with the 24 symmetries of the cube, I only needed to deal with their 5 conjugacy classes.

The example I gave in my math.se answer was maybe less perspicuous. I considered the symmetries of a sphere, and talked about how two rotations of the sphere by 17° are conjugate, regardless of what axis one rotates around. I thought of the square at the end, and threw it in, but I wish I had started with it.

• How to convert a decimal to a fraction easily? was the month's big winner. OP wanted to know how to take a decimal like !!0.3760683761!! and discover that it can be written as !!\frac{44}{117}!!. The right answer to this is of course to use continued fraction theory, but I did not want to write a long treatise on continued fractions, so I stripped down the theory to obtain an algorithm that is slower, but much easier to understand.

The algorithm is just binary search, but with a twist. If you are looking for a fraction for !!x!!, and you know !!\frac ab < x < \frac cd!!, then you construct the mediant !!\frac{a+c}{b+d}!! and compare it with !!x!!. This gives you a smaller interval in which to search for !!x!!, and the reason you use the mediant instead of using !!\frac12\left(\frac ab + \frac cd\right)!! as usual is that if you use the mediant you are guaranteed to exactly nail all the best rational approximations of !!x!!. This is the algorithm I described a few years ago in your age as a fraction, again; there the binary search proceeds down the branches of the Stern-Brocot tree to find a fraction close to !!0.368!!.

I did ask a question this month: I was looking for a simpler version of the dogbone space construction. The dogbone space is a very peculiar counterexample of general topology, originally constructed by R.H. Bing. I mentioned it here in 2007, and said, at the time:

[The paper] is on my desk, but I have not read this yet, and I may never.

I did try to read it, but I did not try very hard, and I did not understand it. So my question this month was if there was a simpler example of the same type. I did not receive an answer, just a followup comment that no, there is no such example.

Sat, 12 Dec 2015

This is page 22 of the Cosmic Call message. An explanation follows.

The 10 digits are:

 0 1 2 3 4 5 6 7 8 9

This page discusses properties of the entire universe. It is labeled with a new glyph, , which denotes the universe or the cosmos. On this page I am on uncertain ground, because I know very little about cosmology. My explanation here could be completely wrong without my realizing it.

The page contains only five lines of text. In order, they state:

1. The Friedmann equation which is the current model for the expansion of the universe. This expansion is believed to be uniform everywhere, but even if it isn't, the recipients are so close by that they will see exactly the same expansion we do. If they have noticed the expansion, they may well have come to the same theoretical conclusions about it. The equation is:

$$H^2 = \frac{8\pi G}3\rho + \frac{\Lambda c^2 }3$$

where !!H!! is the Hubble parameter (which describes how quickly the universe is expanding), !!G!! is the universal gravitation constant (introduced on page 9), !!\rho!! is the density of the universe (given on the next line), and !!\Lambda c^2!! () is one of the forms of the cosmological constant (given on the following line).

2. The average density of the universe , given as !!2.76\times 10^{-27} \mathrm{kg} ~\mathrm{m}^{-3}!!. The “density” glyph would have been more at home with the other physics definitions of page 9, but it wasn't needed until now, and that page was full.

3. The cosmological constant !!\Lambda!! is about !!10^{-52} \mathrm{m}^{-2}!!. The related value given here, !!\Lambda c^2!!, is !!1.08\cdot 10^{-35} \mathrm{s}^{-2}!!.

4. The calculated value of the Hubble parameter !!H!! is given here in the rather strange form !!\frac1{14000000000}\mathrm{year}^{-1}!!. The reason it is phrased this way is that (assuming that !!H!! were constant) !!\frac1H!! would be the age of the universe, approximately 14,000,000,000 years. So this line not only communicates our estimate for the current value of the Hubble parameter, it expresses it in units that may make clear our beliefs about the age of the universe. It is regrettable that this wasn't stated more explicitly, using the glyph that was already used for the age of the Earth on page 13. There was plenty of extra space, so perhaps the senders didn't think of it.

5. The average temperature of the universe, about 2.736 kelvins. This is based on measurements of the cosmic microwave background radiation, which is the same in every direction, so if the recipients have noticed it at all, they have seen the same CMB that we have.

The next article will discuss the final page, shown at right. (Click to enlarge.) Try to figure it out before then.

Sun, 06 Dec 2015

This is page 21 of the Cosmic Call message. An explanation follows.

The 10 digits are:

 0 1 2 3 4 5 6 7 8 9