The Universe of Discourse


Sun, 17 Jun 2018

Bounded does not imply totally bounded

I somehow managed to miss the notion of totally bounded when I was learning topology, and it popped up on stack exchange recently. It is a stronger version of boundedness for metric spaces: a space !!M!! is totally bounded if, for any chosen !!\epsilon!!, !!M!! can be covered by a finite family of balls of radius !!\epsilon!!.

This is a strictly stronger property than ordinary boundedness, so the question immediately comes up: what is an example of a space that is bounded but not totally bounded. Many examples are well-known. For example, the infinite-dimensional unit ball is bounded but not totally bounded. But I didn't think of this right away.

Instead I thought of the following rather odd example: Let !!S!! be the closed unit disc and assign each point a polar coordinate !!\langle r,\theta\rangle!! as usual. Now consider the following metric:

$$ d(\langle r_1, \theta_1\rangle, \langle r_2, \theta_2\rangle) = \begin{cases} r_1, & \qquad \text{ if $r_2 = 0$} \\ \lvert r_1 - r_2 \rvert, & \qquad\text{ if $\theta_1 = \theta_2$} \\ r_1 + r_2 & \qquad\text{ otherwise} \\ \end{cases} $$

The idea is this: you can travel between points only along the radii of the disc. To get from !!p_1!! to !!p_2!! that are on different radii, you must go through the origin:

A circle, with the
center marked.  A shortest-distance path is drawn in blue between two
blue points on the same radius, and in red between two red points on different
radii.  The blue path goes straight from one blue point to the other.
The red path goes from one point straight to the origin,
then straight to the other point.

Now clearly when !!\epsilon < \frac12!!, the !!\epsilon!!-ball that covers each point point !!\left\langle 1, \theta\right\rangle!! lies entirely within one of the radii, and so an uncountable number of such balls are required to cover the disc.

It seems like this example could be useful in other circumstances too. Does it have a name?


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Mon, 21 May 2018

More about disabling standard I/O buffering

In yesterday's article I described a simple and useful feature that could have been added to the standard I/O library, to allow an environment variable to override the default buffering behavior. This would allow the invoker of a program to request that the program change its buffering behavior even if the program itself didn't provide an option specifically for doing that.

Simon Tatham directed me to the GNU Coreutils stdbuf command which does something of this sort. It is rather like the pseudo-tty-pipe program I described, but instead of using the pseudo-tty hack I suggested, it works by forcing the child program to dynamically load a custom replacement for stdio. There appears to be a very similar command in FreeBSD.

Roderick Schertler pointed out that Dan Bernstein wrote a utility program, pty, in 1990, atop which my pseudo-tty-pipe program could easily be built; or maybe its ptybandage utility is exactly what I wanted. Jonathan de Boyne Pollard has a page explaining it in detail, and related packages.

A later version of pty is still available. Here's M. Bernstein's blurb about it:

ptyget is a universal pseudo-terminal interface. It is designed to be used by any program that needs a pty.

ptyget can also serve as a wrapper to improve the behavior of existing programs. For example, ptybandage telnet is like telnet but can be put into a pipeline. nobuf grep is like grep but won't block-buffer if it's redirected.

Previous pty-allocating programs — rlogind, telnetd, sshd, xterm, screen, emacs, expect, etc. — have caused dozens of security problems. There are two fundamental reasons for this. First, these programs are installed setuid root so that they can allocate ptys; this turns every little bug in hundreds of thousands of lines of code into a potential security hole. Second, these programs are not careful enough to protect the pty from access by other users.

ptyget solves both of these problems. All the privileged code is in one tiny program. This program guarantees that one user can't touch another user's pty.

ptyget is a complete rewrite of pty 4.0, my previous pty-allocating package. pty 4.0's session management features have been split off into a separate package, sess.

Leonardo Taccari informed me that NetBSD's stdio actually has the environment variable feature I was asking for! Christos Zoulas suggested adding stdbuf similar to the GNU and FreeBSD implementations, but the NetBSD people observed, as I did, that it would be simpler to just control stdio directly with an environment variable, and did it. Here's the relevant part of the NetBSD setbuf(3) man page:

The default buffer settings can be overwritten per descriptor (STDBUFn) where n is the numeric value of the file descriptor represented by the stream, or for all descriptors (STDBUF). The environment variable value is a letter followed by an optional numeric value indicating the size of the buffer. Valid sizes range from 0B to 1MB. Valid letters are:

U unbuffered

L line buffered

F fully buffered

Here's the discussion from the NetBSD tech-userlevel mailing list. The actual patch looks almost exactly the way I imagined it would.

Finally, Mariusz Ceier pointed out that there is an ancient bug report in glibc suggesting essentially the same environment variable mechanism that I suggested and that was adopted in NetBSD. The suggestion was firmly and summarily rejected. (“Hell, no … this is a terrible idea.”) Interesting wrinkle: the bug report was submitted by Pádraig Brady, who subsequently wrote the stdbuf command I described above.

Thank you, Gentle Readers!


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Sun, 20 May 2018

Proposal for turning off standard I/O buffering

Some Unix commands, such as grep, will have a command-line flag to say that you want to turn off the buffering that is normally done in the standard I/O library. Some just try to guess what you probably want. Every command is a little different and if the command you want doesn't have the flag you need, you are basically out of luck.

Maybe I should explain the putative use case here. You have some command (or pipeline) X that will produce dribbles of data at uncertain intervals. If you run it at the terminal, you see each dribble timely, as it appears. But if you put X into a pipeline, say with

    X | tee ...

or

    X | grep ...

then the dribbles are buffered and only come out of X when an entire block is ready to be written, and the dribbles could be very old before the downstream part of the pipeline, including yourself, sees them. Because this is happening in user space inside of X, there is not a damn thing anyone farther downstream can do about it. The only escape is if X has some mode in which it turns off standard I/O buffering. Since standard I/O buffering is on by default, there is a good chance that the author of X did not think to affirmatively add this feature.

Note that adding the --unbuffered flag to the downstream grep does not solve the problem; grep will produce its own output timely, but it's still getting its input from X after a long delay.

One could imagine a program which would interpose a pseudo-tty, and make X think it is writing to a terminal, and then the standard I/O library would stay in line-buffered mode by default. Instead of running

    X | tee some-file | ...

or whatever, one would do

    pseudo-tty-pipe -c X | tee some-file | ...

which allocates a pseudo-tty device, attaches standard output to it, and forks. The child runs X, which dribbles timely into the pseudo-tty while the parent runs a read loop to remove dribbles from the master end of the TTY and copy them timely into the pipe. This would work. Although tee itself also has no --unbuffered flag so you might even have to:

    pseudo-tty-pipe -c X | pseudo-tty-pipe -c 'tee some-file' | ...

I don't think such a program exists, and anyway, this is all ridiculous, a ridiculous abuse of the standard I/O library's buffering behavior: we want line buffering, the library will only give it to us if the process is attached to a TTY device, so we fake up a TTY just to fool stdio into giving us what we want. And why? Simply because stdio has no way to explicitly say what we want.

But it could easily expose this behavior as a controllable feature. Currently there is a branch in the library that says how to set up a buffering mode when a stream is opened for the first time:

  • if the stream is for writing, and is attached to descriptor 2, it should be unbuffered; otherwise …

  • if the stream is for writing, and connects descriptor 1 to a terminal device, it should be line-buffered; otherwise …

  • if the moon is waxing …

  • otherwise, the stream should be block-buffered

To this, I propose a simple change, to be inserted right at the beginning:

If the environment variable STDIO_BUF is set to "line", streams default to ine buffering. If it's set to "none", streams default to no buffering. If it's set to "block", streams default to block buffered. If it's anything else, or unset, it is ignored.

Now instead of this:

    pseudo-tty-pipe --from X | tee some-file | ...

you write this:

    STDIO_BUF=line X | tee some-file | ...

Problem solved.

Or maybe you would like to do this:

    export STDIO_BUF=line

which then it affects every program in every pipeline in the rest of the session:

    X | tee some-file | ...

Control is global if you want it, and per-process if you want it.

This feature would cost around 20 lines of C code in the standard I/O library and would impose only an insigificant run-time cost. It would effectively add an --unbuffered flag to every program in the universe, retroactively, and the flag would be the same for every program. You would not have to remember that in mysql the magic option is -n and that in GNU grep it is --line-buffered and that for jq is is --unbuffered and that Python scripts can be unbuffered by supplying the -u flag and that in tee you are just SOL, etc. Setting STDIO_BUF=line would Just Work.

Programming languages would all get this for free also. Python already has PYTHONUNBUFFERED but in other languages you have to do something or other; in Perl you use some horrible Perl-4-ism like

    { my $ofh = select OUTPUT; $|++; select $ofh }

This proposal would fix every programming language everywhere. The Perl code would become:

    $ENV{STDIO_BUF} = 'line';

and every other language would be similarly simple:

    /* In C */
    putenv("STDIO_BUF=line");

[ Addendum 20180521: Mariusz Ceier corrects me, pointing out that this will not work for the process’ own standard streams, as they are pre-opened before the process gets a chance to set the variable. ]

It's easy to think of elaborations on this: STDIO_BUF=1:line might mean that only standard output gets line-buffering by default, everything else is up to the library.

This is an easy thing to do. I have wanted this for twenty years. How is it possible that it hasn't been in the GNU/Linux standard library for that long?

[ Addendum 20180521: it turns out there is quite a lot to say about the state of the art here. In particular, NetBSD has the feature very much as I described it. ]


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Mon, 07 May 2018

Katara constructs finite projective planes

This weekend I got a very exciting text message from Katara:

I have a math question for you

Oh boy! I hope it's one I can answer.

Okay

there's this game called spot it where you have cards with 8 symbols on them like so

The back of the 'Spot It' box shows four white circular cards,
each with 8 little pictures on it. A red double-headed arrow between
the first two cards shows that both cards have a picture of a
ladybug.  Another arrow between the second and third cards shows that
both cards have a picture of a red heart.  The next arrow connects the
trees on cards 2 and 3, and a fourth arrow shows that cards 1 and 4
both have apurple cat.  Even though part of card 3 is cut off, we can
see that they share a pair of lips. Cards 2 and 4 both have a gray
padlock picture, even though there's no arrow pointing it out.

and the goal is to find the one matching symbol on your card and the one in the middle

how is it possible that given any pair of cards, there is exactly one matching symbol

Well, whatever my failings as a dad, this is one problem I can solve. I went a little of overboard in my reply:

You need a particular kind of structure called a projective plane.

They only exist for certain numbers of symbols

A simpler example has 7 cards with 3 symbols each.

One thing that's cool about it is that the symbols and the cards are "dual": say you give each round card a name. Then make up a new deck of square cards. There's one square card for each symbol. So there's a square"Ladybug" card. The Ladybug card has on it the names of the round cards that have the Ladybug. Now you can play Spot with the square cards instead of the round ones: each two square cards have exactly one name in common.

In a geometric plane any two points lie on exactly one common line and any two lines intersect in exactly one common point. This is a sort of finite model of that, with cards playing the role of lines and symbols playing the role of points. Or vice versa, it doesn't matter.

More than you wanted to know 😂

ah thank you, I'm pretty sure I understand, sorry for not responding, my phone was charging

I still couldn't shut up about the finite projective planes:

No problem! No response necessary.

It is known that all finite projective planes have n²+n+1 points for some n. So I guess the Spot deck has either 31, 57, or 73 cards and the same number of symbols. Is 57 correct?

Must be 57 because I see from your picture that each card has 8 symbols.

Katara was very patient:

I guess, I would like to talk about this some more when i get home if that's okay

Any time.

(The game costs $13.)

Anyway this evening I cut up some index cards, and found a bunch of stickers in a drawer, and made Katara a projective plane of order 3. This has 13 cards, each with 4 different stickers, and again, every two cards share exactly one sticker. She was very pleased and wanted to know how to make them herself.

Each set of cards has an order, which is a non-negative integer. Then there must be !!n^2 + n + 1!! cards, each with !!n+1!! stickers or symbols. When !!n!! is a prime power, you can use field theory to construct a set of cards from the structure of the (unique) field of order !!n!!.

Fields to projective planes

Order 2

I'll describe the procedure using the plane of order !!n=2!!, which is unusually simple. There will be !!2^2+2+1 = 7!! cards, each with !!3!! of the !!7!! symbols.

Here is the finite field of order 2, called !!GF(2)!!:

+ 0 1
0 0 1
1 1 0
× 0 1
0 0 0
1 0 1
  • The stickers correspond to ordered triples of elements of !!GF(2)!!, except that !!\langle 0,0,0\rangle!! is always omitted. So they are:

    $$\require{cancel}\begin{array}{cc} \cancel{\langle 0,0,0\rangle} & \langle 1,0,0\rangle \\ \langle 0,0,1\rangle & \langle 1,0,1\rangle \\ \langle 0,1,0\rangle & \langle 1,1,0\rangle \\ \langle 0,1,1\rangle & \langle 1,1,1\rangle \\ \end{array} $$

    Of course, you probably do not want to use these symbols exactly. You might decide that !!\langle 1,0,0\rangle!! is a sticker with a picture of a fish, and !!\langle 0,1,0\rangle!! is a sticker with a ladybug.

  • Each card will have !!n+1 = 3!! stickers. To generate a card, pick any two stickers that haven't appeared together before and put them on the card. Say these stickers correspond to the triples !!\langle a,b,c\rangle!! and !!\langle x,y,z\rangle!!. To find the triple for the third sticker on the card, just add the first two triples componentwise, obtaining !!\langle a+x,b+y,c+z\rangle!!. Remember that the addition must be done according to the !!GF(2)!! addition table above! So for example if a card has !!\langle 1,0,1\rangle!! and !!\langle 0,1,1\rangle!!, its third triple will be

    $$\begin{align} \langle 1,0,1 \rangle + \langle 0,1,1 \rangle & = \\ \langle 1+0,0+1,1+1 \rangle & = \\ \langle 1,1,0 \rangle \end{align} $$

    Observe that it doesn't matter which two triples you add; you always get the third one!

Okay, well, that was simple.

Larger order

After Katara did the order 2 case, which has 7 cards, each with 3 of the 7 kinds of stickers, she was ready to move on to something bigger. I had already done the order 3 deck so she decided to do order 4. This has !!4^2+4+1 = 21!! cards each with 5 of the 21 kinds of stickers. The arithmetic is more complicated too; it's !!GF(2^2)!! instead of !!GF(2)!!:

+ 0 1 2 3
0 0 1 2 3
1 1 0 3 2
2 2 3 0 1
3 3 2 1 0
× 0 1 2 3
0 0 0 0 0
1 0 1 2 3
2 0 2 3 1
3 0 3 1 2

When the order !!n!! is larger than 2, there is another wrinkle. There are !!4^3 = 64!! possible triples, and we are throwing away !!\langle 0,0,0\rangle!! as usual, so we have 63. But we need !!4^2+4+1 = 21!!, not !!63!!.

Each sticker is represented not by one triple, but by three. The triples !!\langle a,b,c\rangle, \langle 2a,2b,2c\rangle,!! and !!\langle 3a,3b,3c\rangle!! must be understood to represent the same sticker, all the multiplications being done according to the table above. Then each group of three triples corresponds to a sticker, and we have 21 as we wanted.

Each triple must have a leftmost non-zero entry, and in each group of three similar triples, there will be one where this leftmost non-zero entry is a !!1!!; we will take this as the canonical representative of its class, and it can wear a costume or a disguise that makes it appear to begin with a !!2!! or a !!3!!.

We might assign stickers to triples like this:

$$ \begin{array}{rl} \cancel{\langle 0,0,0\rangle} & \\ \langle 0,0,1 \rangle & \text{apple} \\ \hline \langle 0,1,0 \rangle & \text{bicycle} \\ \langle 0,1,1 \rangle & \text{carrot} \\ \langle 0,1,2 \rangle & \text{dice} \\ \langle 0,1,3 \rangle & \text{elephant} \\ \hline \langle 1,0,0 \rangle & \text{frog} \\ \langle 1,0,1 \rangle & \text{goat} \\ \langle 1,0,2 \rangle & \text{hat} \\ \langle 1,0,3 \rangle & \text{igloo} \\ \langle 1,1,0 \rangle & \text{jellyfish} \\ \langle 1,1,1 \rangle & \text{kite} \\ \langle 1,1,2 \rangle & \text{ladybug} \\ \langle 1,1,3 \rangle & \text{mermaid} \\ \langle 1,2,0 \rangle & \text{nose} \\ \langle 1,2,1 \rangle & \text{octopus} \\ \langle 1,2,2 \rangle & \text{piano} \\ \langle 1,2,3 \rangle & \text{queen} \\ \langle 1,3,0 \rangle & \text{rainbow} \\ \langle 1,3,1 \rangle & \text{shoe} \\ \langle 1,3,2 \rangle & \text{trombone} \\ \langle 1,3,3 \rangle & \text{umbrella} \\ \end{array} $$

We can stop there, because everything after !!\langle 1,3,3 \rangle!! begins with a !!2!! or a !!3!!, and so is some other triple in disguise. For example what sticker goes with !!\langle 0,2,3 \rangle!!? That's actually !!\langle 0,1,2 \rangle!! in disguise, it's !!2·\langle 0,1,2 \rangle!!, which is “dice”. Okay, how about !!\langle 3,3,1 \rangle!!? That's the same as !!3\cdot\langle 1,1,2 \rangle!!, which is “ladybug”. There are !!21!!, as we wanted. Note that the !!21!! naturally breaks down as !!1+4+4^2!!, depending on how many zeroes are at the beginning; that's where that comes from.

Now, just like before, to make a card, we pick two triples that have not yet gone together, say !!\langle 0,0,1 \rangle!! and !!\langle 0,1,0 \rangle!!. We start adding these together as before, obtaining !!\langle 0,1,1 \rangle!!. But we must also add together the disguised versions of these triples, !!\langle 0,0,2 \rangle!! and !!\langle 0,0,3 \rangle!! for the first, and !!\langle 0,2,0 \rangle!! and !! \langle 0,3,0 \rangle!! for the second. This gets us two additional sums, !!\langle 0,2,3 \rangle!!, which is !!\langle 0,1,2 \rangle!! in disguise, and !!\langle 0,3,2 \rangle!!, which is !!\langle 0,1,3 \rangle!! in disguise.

It might seem like it also gets us !!\langle 0,2,2 \rangle!! and !!\langle 0,3,3 \rangle!!, but these are just !!\langle 0,1,1 \rangle!! again, in disguise. Since there are three disguises for !!\langle 0,0,1 \rangle!! and three for !!\langle 0,1,0 \rangle!!, we have nine possible sums, but it turns out the the nine sums are only three different triples, each in three different disguises. So our nine sums get us three additional triples, and, including the two we started with, that makes five, which is exactly how many we need for the first card. The first card gets the stickers for triples !!\langle 0,0,1 \rangle, \langle 0,1,0 \rangle \langle 0,1,1 \rangle \langle 0,1,2 \rangle,!! and !!\langle 0,1,3 \rangle,!! which are apple, bicycle, carrot, dice, and elephant.

That was anticlimactic. Let's do one more. We don't have a card yet with ladybug and trombone. These are !!\langle 1,1,2 \rangle!! and !!\langle 1,3,2 \rangle!!, and we must add them together, and also the disguised versions:

$$\begin{array}{c|ccc} & \langle 1,1,2 \rangle & \langle 2,2,3 \rangle & \langle 3,3,1 \rangle \\ \hline \langle 1,3,2 \rangle & \langle 0,2,0 \rangle & \langle 3,1,1 \rangle & \langle 2,0,3 \rangle \\ \langle 2,1,3 \rangle & \langle 3,0,1 \rangle & \langle 0,3,0 \rangle & \langle 1,2,2 \rangle \\ \langle 3,2,1 \rangle & \langle 2,3,3 \rangle & \langle 1,0,2 \rangle & \langle 0,1,0 \rangle \\ \end{array}$$

These nine results do indeed pick out three triples in three disguises each, and it's easy to select the three of these that are canonical: they have a 1 in the leftmost nonzero position, so the three sums are !!\langle 0,1,0 \rangle,!! !!\langle 1,0,2 \rangle,!! and !!\langle 1,2,2 \rangle!!, which are bicycle, hat, and piano. So the one card that has a ladybug and a trombone also has a bicycle, a hat, and a piano, which should not seem obvious. Note that this card does have the required single overlap with the other card we constructed: both have bicycles.

Well, that was fun. Katara did hers with colored dots instead of stickers:

Katara's set of 21 cards.  Each
has five colored dots of 21 total different colors, and each colored
dot is labeled with a letter of the alphabet in the same color.  There
are some mistakes that are crossed out and corrected, and a couple of
mistakes that are not yet crossed out or corrected.

The ABCDE card is in the upper left; the bicycle-hat-ladybug-piano-trombone one is the second row from the bottom, second column from the left. The colors look bad in this picture; the light is too yellow and so all the blues and purples look black.x

After I took this picture, we checked these cards and found a couple of calculation errors, which we corrected. A correct set of cards is:

$$ \begin{array}{ccc} \text{abcde} & \text{bhlpt} & \text{dgmpr} \\ \text{afghi} & \text{bimqu} & \text{dhjou} \\ \text{ajklm} & \text{cfkpu} & \text{diknt} \\ \text{anopq} & \text{cgjqt} & \text{efmot} \\ \text{arstu} & \text{chmns} & \text{eglnu} \\ \text{bfjnr} & \text{cilor} & \text{ehkqr} \\ \text{bgkos} & \text{dflqs} & \text{eijps} \\ \end{array} $$

Fun facts about finite projective planes:

  • This construction always turns a finite field of order !!n!! into a finite projective plane of order !!n!!.

  • A finite field of order !!n!! exists exactly when !!n!! is a prime power and then there is exactly one finite field. So this construction gives us finite projective planes of orders !!1,2,3,4,5,7,8,9,11,13,16!!, but not of orders !!6,10,12,14,15!!. Do finite projective planes of those latter orders exist?

  • Is this method the only way to construct a finite projective plane? Yes, when !!n<9!!. But there are four non-isomorphic projective planes of order !!9!!, and this only constructs one of them.

    What about for !!n≥11!!? Nobody knows.


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Sun, 06 May 2018

Shitpost roundup, 2018-04

Last month I regretted making only 22 posts but I promised:

April will be better; I'm on pace to break the previous volume record, and I've also been doing a good job of promoting better posts to the major leagues.

I blew it! I tied the previous volume record. But I also think I did do a decent job promoting the better posts. Usually I look over the previous month's posts and pick out two or three that seem to be of more interest than the others. Not this month! They are all shit, except the one ghostwritten by Anette Gordon-Reed. If this keeps up, I will stop doing these monthly roundup posts.


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Wed, 02 May 2018

Addenda to recent articles 201804

  • Andrew Rodland and Adam Vartanian explained ramp metering. Here's M. Rodland's explanation:

    ramp metering is the practice of installing signals on freeway onramps that only allow one car every few seconds, so that cars enter the freeway at evenly-spaced intervals instead of in bunches and don't cause as many problems merging.

    He added that it was widely used in California. McCain is headquartered in California, and mentions frequently on their web site that their equipment conforms to Caltrans standards.

  • M. Vartanian and Richard Soderberg also suggested an explanation for why the traffic control system might also control sprinklers and pumps. M. Soderberg says:

    DOTs in California and presumably elsewhere often have a need for erosion control on the steep inclines of earth surrounding their highway ramps. So any time you see a 45-degree incline covered in greenery, chances are it has a sprinkler system attached and carefully maintained by the DOT. Those same sprinklers are often within a few feet of the ramp's metering lights…

    That makes perfect sense! I had been imagining fire sprinklers, and then I was puzzled: why would you need fire sprinklers at an intersection?

  • Several readers suggested explanations for why soldier fly larvae are more expensive than pork chops. I rejected several explanations:

    • Hogs are kept in poor and inhumane conditions (often true, but their accommodations must still be much more expensive than the flies’)

    • Hog farmers are exempted from paying for the negative externalities of their occupation such as environmental degradation and antibiotic resistance (often true, but the fly farmers cannot be paying that much to offset externalities)

    • Slaughterhouse waste and rotten fruit are more expensive than the corn and soy on which hogs are fed (I think slaughterhouse waste and waste fruit are available essentially free to anyone who wants to haul them away)

    • The drying process is difficult and expensive (but the listed price for undried maggots is twice as high)

    But I find Marcel Fourné's suggestion much more plausible: the pork price is artificially depressed by enormous government subsidies.

    I started looking into the numbers on this, and got completely sidetracked on something only peripherally related:

    According to the USDA Census of Agriculture for 2012, in 2012 U.S. farms reported an inventory of 66 million pigs and hogs, and sales of 193 million pigs and hogs. (Table 20, page 22.)

    When I first saw this, I thought I must be misunderstanding the numbers. I said to myself:

    !!\frac{193}{66}\approx 3!!, so the inventory must be turning over three times a year. But that means that the average hog goes to market when it is four months old. That can't be right.

    Of course it isn't right, it isn't even close, it's complete nonsense. I wrote up my mistake but did not publish it, and while I was doing that I forgot to finish working on the subsidy numbers.

  • James Kushner directed my attention to the MUTCD news feed and in particular this amusing item:

    the FHWA issued Official Interpretation 4(09)-64 to clarify that the flash rate for traffic control signals and beacons is a single repetitive flash rate of approximately once per second, and that a combination of faster and slower flash rates that result in 50 to 60 flashes per minute is not compliant…

    James writes:

    I imagined a beacon that flashed once every ten seconds; after five such iterations, there was one iteration where the beacon would flash fifty times in a second. "But it flashes 55 times every minute, so, you know, it, uh, conforms to the standard..."

    But the Official Interpretation also says

    You asked whether the FHWA would be willing to consider experimentation with alternative flash rates for warning beacons. Any requests for experimentation would be evaluated on their merits and would be addressed separately from this official ruling.

    so there is still hope for James’ scheme.

  • Two readers suggested musical jokes. Jordan Fultz asks:

    Q: How does Lady Gaga like her steak?
    A: Raw, raw, raw-raw-raw!

    (This is Bad Romance)

    And betaveros asks:

    Q. What kind of overalls does Mario wear?
    A. Denim denim denim.

    (This is the Super Mario Bros. Underworld Theme)

    I feel like we might be hitting the bottom of the barrel.

Thanks to all readers who wrote to me, and also to all readers who did not write to me.


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Tue, 01 May 2018

What's in those mysterious cabinets?

Last Monday some folks were working on this thing on Walnut Street. I didn't remember having seen the inside of one before, so I took some pictures of it to look at later.

A brown metal cabinet,
about six feet high, filled with some sort of electronic equipment.
The equipment is modular.  Some modules have data sockets, some with
cables plugged in; others look like switchboards with colored
wires. The other side of the same
cabinet.  More modules, labeled “McCain”.  One has a 16-button keypad
with keys numberd 0–9 and A–F, and a set of ten numbered LEDs.
Another has an ordinary-looking electric power socket and labels such
as “SIGNAL BUS”, “FLASHER BUS”, “SB 1–6 MUST BE ON FOR NORMAL
OPERATION”.

Thanks to the Wonders of the Internet, it didn't take long to figure out what it is for. It is a controller for the traffic lights at the intersection.

In particular, the top module in the right-hand picture is a Model 170 ATC HC11 Controller manufactured by McCain Inc, a thirty-year old manufacturer of traffic control devices. The controller runs software developed and supported by McCain, and the cabinet is also made by McCain.

The descriptions of the controllers are written in a dense traffic control jargon that I find fascinating but opaque. For example, the 170 controller's product description reads:

The McCain 170E, 170E HC11, and 170 ATC HC11 controllers’ primary design function is to operate eight-phase dual ring intersections. Based on the software control package utilized, the 170’s control applications can expand to include: ramp metering, variable message signs, sprinklers, pumps, and changeable lane control.

I think I understand what variable message signs are, and I can guess at changeable lane control, but what are the sprinklers and pumps for? What is ramp metering?

[ Addendum 20180502: readers explain ]

The eight-phase dual ring intersection, which I had never heard of before, is an important topic in the traffic control world. I gather that it is a four-way intersection with a four-way traffic light that also has a left-turn-only green arrow portion. The eight “phases” refer to different traffic paths through the intersections that must be separately controlled: even numbers for the four paths through the intersection, and odd numbers 1,3,5,7 for the left-turn-only paths that do not pass through. Some phases conflict; for example phase 5 (left-turning in some direction, say from south to east), conflicts with phase 6 (through-passing heading in the opposite direction) but not with phase 1 (left-turning from north to west).

This is a diagram of the
intersection of two two-way streets, as seen from above.  Each of the
four incoming roads has an arrow showing the direction of through
traffic and another showing the direction of left-turning traffic.
Clockwise in order the through-traffic arrows are labeled with
Φ2,Φ4,Φ6,Φ8, and the corresponding left-turn arrows are labeled
Φ5,Φ7,Φ1,Φ3.  Arrows Φ5 and Φ1 are accompanied by green traffic
signals, and Φ4 and Φ6 by red signals.

There's plenty of detailed information about this available. For example, the U.S. Federal Highway Administration publishes their Traffic Signal Timing Manual. (Published in 2008, it has since been superseded.) Unfortunately, this seems to be too advanced for me! Section 4.2.1 (“Definitions and Terminology”) is the first place in the document that mentions the dual-ring layout, and it does so without explanation — apparently this is so elementary that anyone reading the Traffic Signal Timing Manual will already be familiar with it:

Over the years, the description of the “individual movements” of the dual-ring 8-movement controller as “phases” has blurred into common communicated terminology of “movement number” being synonymous as “phase number”.

But these helpful notes explain in more detail: a “ring” is “a sequence of phases that are not compatible and that must time sequentially”.

Then we measure the demand for each phase, and there is an interesting and complex design problem: how long should each phase last to optimize traffic flow through the intersection for safety and efficiency? See chapter 3a for more details of how this is done.

I love when I discover there is an entire technical domain that I never even suspected existed. If you like this kind of thing, you may enjoy geeking out over the Manual of Uniform Traffic Control Devices, which explains what traffic signs should look like and what each one means. Have you ever noticed that the green guide signs on the highway have up-pointing and down-pointing arrows that are totally different shapes?

Extract of the MUTCD “Figure 2D-2.
Arrows for Use on Guide Signs”.  The figure depicts
five types of up-pointing “Directional Arrows” and a down-pointing
“Down Arrow”.  The up arrows have long
shafts and similar heads.  The down arrow has a very short shaft and a
very wide head.

That's because they have different meanings: the up-pointing arrows mean “go this way” and the down-pointing arrows mean “use this lane”. The MUTCD says what the arrows should look like, how big they should be, and when each one should be used.

The MUTCD is the source of one of my favorite quotations:

Regulatory and warning signs should be used conservatively because these signs, if used to excess, tend to lose their effectiveness.

Words to live by! Programmers in particular should keep this in mind when designing error messages. You could spend your life studying this 864-page manual, and I think some people do.

Related geekery: Geometric highway design: how sharply can the Interstate curve and still be safe, and how much do the curves need to be banked? How do you design an interchange between two major highways? How about a highway exit?

Here's a highway off-ramp, exit 346A on Pennsylvania I-76 West:

Satellite view of a highway exit,
labeled “Schuylkill Expy”.  Traffic flow is from lower left to upper
right, with one road splitting to become two.  The entering road is
three lanes, and the left lane diverges and goes up a ramp while the
other two lanes continue in the same direction. The space between the
two lanes, before the roadway actually splits, is a long, narrow
triangle shape.

Did you know that the long pointy triangle thing is called a “gore”?

What happens if you can't make up your mind whether to stay on the highway or take the exit, you drive over the gore, and then smack into the thing beyond it where the roads divides? Well, you might survive, because there is a thing there that is designed to crush when you hit it. It might be a QuadGuard Elite Crash Cushion System, manufactured by Energy Absorption Systems, Inc..

It's such a big world out there, so much to know.


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Sun, 29 Apr 2018

Lipogrammatic math posts

In August 2011, on a particular famous discussion forum (brought up on this blog again and again) an individual A, notorious for such acts, posts a quasi-philosophical inquiry, incurring unpopularity, antagonism, and many bad marks, although also a surprising quantity of rational discussion, including a thoughtful solution or two.

Many months forward, a distinct party B puts up a substantial bounty on this inquiry, saying:

I would like a complete answer to this question which does not use the letter "e" at any point.

(My apology for any anguish you may go through at this point in my story on account of this quotation and its obvious and blatant faults. My wrongdoing was involuntary, but I had no way to avoid it and still maintain full accuracy.)

By and by, a valiant third individual constructs a brilliant disquisition satisfying this surprising condition and thus obtains B's award.

Now, this month, in our group's accompanying policy board, a fourth collaborator, a guy (or gal, for all I know) I shall call D, and who I think may lack a minimal inclination for fun, finds fault with A's original post and particularly with C's bounty, and complains as follows:

Should we discourage bounties that encourage “clever” but unclear answers?

(Again, I must ask you for absolution. This is a word-for-word quotation.)

A thorough dismissal of OP's complaint, from a fifth author, adds a fully satisfactory finish to our affair.


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Fri, 20 Apr 2018

Shitpost roundup, 2018-03

Here is a list of March's shitposts. I don't recall what my excuse was for there being only 22, but in my defense, I will add that they were almost all terrible. There was one decent math post I maybe should have promoted.

(And also Nancy and Squid, which was awesome, and also 100% Grade A shitpost. I thought when I posted it a crowd of people would burst into the room and carry me off on their shoulders. Instead, nobody seems to have noticed.)

April will be better; I'm on pace to break the previous volume record, and I've also been doing a good job of promoting better posts to the major leagues.


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Thu, 19 Apr 2018

Soldier fly protein: why so expensive?

Montgomery Burns, from ‘The Simpsons’, in his characteristic
pose: scowling, hunched, with his fingertipes pressed together nefariously.

There have been recurring news stories about the use of dried maggots as protein supplement in animal feed. For example Insects could feed the animals of tomorrow’s meat industry (maggots fed on slaughterhouse waste, particularly blood) or Insect farms gear up to feed soaring global protein demand (maggots fed on rotten fruit). Then they dry the larvae and either use them whole or grind them into meal. In particular the fly meal can be used as a replacement for fish meal, which is ground dried fish that is used as feed for fish in fish farms. (Yep, we grind up fish we don't like, to feed to other, better fish.)

I was referred to that second article by Metafilter and The Google, in its infinite wisdom, decided to show me an advertisement for dried fly larvae. The ad was for NaturesPeck, which sells bagged fly larvae and fly meal for use as poultry feed or wild bird feed. They have a special “value pack” that contains 16 pounds of dried larvae for $88. That is $5.50 a pound! Holy cow, WTF? How can that even be possible when my local grocery store is selling boneless center cut pork chops for $2.50 per pound?

Okay, I thought maybe NaturesPeck was some sort of boutique operation, charging a high markup for small quantities, maybe they claim to have sustainably-harvested fly meal from free-range organically-fed flies or something. So I went looking for an industrial wholesaler of bulk fly meal and quickly found Haocheng Mealworms Inc. in Xiangtan, China. This is definitely what I was looking for; they will be glad to sell you a standard 40-foot shipping container full of dried maggots or other larvae. The quoted price for dried mealworm larvae is $8400 per metric tonne, plus shipping ($170–200 per tonne).

Prices, converted to U.S. dollars per pound, are as follows:

Dried soldier fly maggots $2.49
Powdered soldier fly maggots 2.81
Dried mealworms 3.81
Powdered mealworms 4.22
Live mealworms 5.67
Canned fresh mealworms 7.08
Dried superworms 4.76
Live superworms 8.30
Canned fresh superworms 7.71

So it wasn't just that NaturesPeck was marking them up. Even the least expensive product costs as much as retail pork chops.

I don't get it. There must be some important aspect of this that I am missing, because a market failure of this magnitude is impossible.

BTW, Haocheng recommends that:

As a source of high protein additive, put [mealworms] into the bread, flour, instant noodle, pastry, biscuit, candy, condiment, and direct into the dishes on the dining table like the foodstuff, and process into the health care nourishment to improve the human body immunity ability.

Not at those prices, buddy.

[ Addendum 20180502: Some possible explanations ]


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Wed, 18 Apr 2018

Lower mathematics solves an easy problem

[ Warning: this article is mathematically uninteresting. ]

I woke up in the middle of the night last night and while I was waiting to go back to sleep, I browsed math Stack Exchange. At four in the morning I am not at my best, but sometimes I can learn something and sometimes I can even contribute.

The question that grabbed my attention this time was Arithmetic sequence where every term is prime?. OP wants to know if the arithmetic sequence $$d\mapsto a+d b$$ contains composite elements for every fixed positive integers !!a,b!!.

Now of course the answer is yes, or the counterexample would give us a quick and simple method for constructing prime numbers, and finding such has been an open problem for thousands of years. OP was certainly aware of this, but had not been able to find a simple proof. Their searching was confounded by more advanced matters relating to the Green-Tao theorem and such like, which, being more interesting, are much more widely discussed.

There are a couple of remarkable things about the answers that were given. First, even though the problem is easy, the first two answers posted were actually wrong, and another (quickly deleted) was so complicated that I couldn't tell if it was right or not.

One user immediately commented:

What happens if !!d=a!!?

which is very much to the point; when !!d=a!! then the element of the sequence is !!a+ab!! which is necessarily composite…

…unless !!a=1!!. So the comment does not quite take care of the whole question. A second user posted an answer with this same omission, and had to correct it later.

I might not have picked up on this case either, during the daytime. But at 4 AM I was not immediately certain that !!a+ab!! was composite and I think about it. I factored it to get !!a(1+b)!! and then I saw that if !!a=1!! or !!1+b=1!! then we lose. (!!1+b=1!! is impossible. !!a+ab!! might of course be composite even if !!a=1!!, but further argumentation is needed.)

So I did pick up on this, and gave a complete answer, of which the important part is:

Just take !!d=kb+k+a!! for any !!k!! whatever. Then the element is $$kb^2+(k+a)b+a = (kb+a)(b+a)$$ which is composite.

Okay, fine. But OP asked how I came up with that and if it was pure “insight”, so I thought I'd try to reconstruct how I got there at 4 AM. The problem is simple enough that I think I can remember most of how I got to the answer.

As I've mentioned before, I am not a pure insight kind of person. While better mathematicians are flitting swiftly from peak to peak, I plod along in the dark and gloomy valleys. I did not get !!d=kb+k+a!! in a brilliant flash of inspiration.

Instead, my thought process, as well as I can remember it, went like this:

  • “That doesn't work when !!a=1!!. But there must be composite numbers in sequences of the form !!d b+1!!. I wonder what they look like?”

  • “I guess I'll try an example. How about !!4d+1!!?”

  • “Hmm, it starts at 5, 5 is composite… No it isn't.”

  • “But it's increasing by 4 each time so it must hit each mod-5 residue class in some order.”

I should cut in at this point to add that my thinking was nowhere near this articulate or even verbal. The thing about the sequence hitting all the residue classes was more like a feeling in my body, like when I am recognizing a familiar place. When !!a!! and !!b!! are relatively prime, that means that when you are taking steps of size !!b!!, you hit all the !!a!!'s and don't skip any; that's what relatively prime is all about.

So maybe that counts as “insight”? Or “intuition about relative primality”? I think that description makes it sound much more impressive than it really is. I do not want a lot of credit for this. Maybe a better way to describe it is that I had been in this familiar place many times before, and I recognized it again.

Anyway I continued something like this:

  • “If it hits every residue class over and over in sequence it must hit residue class !!0!! infinitely often. How does that go? It hits !!5!!, so it hits !!5+4·5 = 25!! also.”

That was good enough for me; I did not even consider the next hit, !!45!!, perhaps because that number was too big for me to calculate at that moment.

I didn't use the phrase “residue class” either. That's just my verbal translation of my 4AM nonverbal thinking. At the time it was more like: there are some good things to hit and some bad ones, and the good ones are evenly spaced out, so if we hit each position in the even spacing-ness we must periodically hit some of the good things.

  • “Okay, then the first element of !!1+d b!! is !!1+b!!, and then every !!1+b!! steps after that it hits another multiple of !!1+b!!, so we want every !!1+b!! elements, so take !!d = 1 + k(1+b)!!.”

Then I posted the answer, saying that when !!a=1!! you take !!d=1+k(1+b)!! and the sequence element is !!1+b+kb+kb^2 = (1+b)(1+kb)!!.

Then I realized that I had the same feeling in my body even when !!a≠1!!, because it only depended on the way the residue classes repeated, and changing !!a!! doesn't affect that, it just slides everything left or right by a constant amount. So I went back to edit the !!1+k(1+b)!! to be !!a+k(1+b)!! instead.

I have no particular conclusion to draw about this.


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Mon, 16 Apr 2018

A familiar set with an unexpected order type

I dreamed this one up in high school and I recommend it as an exercise for kids at an appropriate level.

Consider the set of all Roman numerals

$${ \text{I}, \text{II}, \text{III}, \text{IV}, \text{V}, \ldots, \text{XIII}, \text{XIV}, \text{XV}, \text{XVI}, \ldots, \\ \text{XXXVIII}, \text{XXXIX}, \text{XL}, \text{XLI}, \ldots, \text{XLIX}, \text{L},\ldots,\\ \text{C}, \ldots , \text{D}, \ldots, \text{M}, \ldots, \text{MM}, \ldots, \text{MMM}, \ldots, \text{MMMM}, \ldots, \text{MMMMM}, \ldots }$$

where we allow an arbitrarily large number of M's on the front, so that every number has a unique representation. For example the number 10,067 is represented by !!\text{MMMMMMMMMMLXVII}!!.

Now sort the list into alphabetical order.

It is easy to show that it begins with !!\text{C}, \text{CC}, \text{CCC}, \text{CCCD}, \ldots!! and ends !!\text{XXXVII}, \text{XXXVIII}!!. But it's still an infinite list! Instead of being infinite at one end or the other, or even both, like most infinite lists, it's infinite in the middle.

Of course once you have the idea it's easy to think of more examples (!!\left\{ \frac1n\mid n\in\Bbb Z, n\ne 0\right\}!! for instance) but I hadn't seen anything like this before and I was quite pleased.


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