The Universe of Discourse

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 pseuto-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.

Finally, 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. 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.

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 ...


    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 */

[ 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.


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.

[Other articles in category /math] permanent link

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.

[Other articles in category /meta/shitpost] permanent link

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.

[Other articles in category /addenda] permanent link

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

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.

[Other articles in category /tech] permanent link

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.

[Other articles in category /meta/shitpost] permanent link

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 ]

[Other articles in category /food] permanent link

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.

[Other articles in category /math] permanent link

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.

[Other articles in category /math] permanent link

Sun, 15 Apr 2018

On the smallest natural number

The earliest known mathematics book printed in Europe is an untitled arithmetic text published in Treviso in 1478, Originally written in Venetian dialect.

The Treviso Arithmetic states unequivocally:

Number is a multitude brought together or assembled from several units, and always from two at least, as in the case of 2, which is the first and the smallest number.

(original Venetian)

And a little later:

Of [the digits] the first figure, 1, is not called a number but the source of number.

(original Venetian)

(English translations are from David Eugene Smith, A Source Book in Mathematics (1959). A complete translation appears in Frank J. Swetz, Capitalism and Arithmetic The New Math of the Fifteenth Century (1987).)

Complete text (in Venetian)

By the way, today is the 311th birthday of Leonhard Euler.

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Sat, 14 Apr 2018

Colored blobs on electric wires

The high-voltage power lines run along the New Jersey Turnpike for a long way, and there is this one short stretch of road where the wires have red, white, and yellow blobs on them. Google's Street View shot shows them quite clearly.

A thousand feet or so farther down the road, no more blobs.

I did Internet searches to find out what the blobs were about, and everyone seemed to agree that they were to make the wires more visible to low-flying aircraft. Which seemed reasonable, but puzzling, because as far as I knew there was no airport in the vicinity. And anyway, why blobs only on that one short stretch of wire?

Last week I drove Katara up to New York and when I saw the blobs coming I asked her to photograph them and email me the pictures. She did, and as I hoped, there in the EXIF data in the images was the exact location at which the pictures had been taken: !!(40.2106, -74.57726675)!!. I handed the coordinates to Google and it gave me the answer to my question:

Screengrab of a Google map showing
the section of the Turnpike under discussion.  The turnpike runs from
southwest to northeast.  A red 'location' marker shows the point at
which the photograph was taken.  About half a mile east of the marker,
with its main runway pointing almost directly at the marker, is an
area labeled 'Trenton Robbinsville Airport'

The wires with blobs are exactly in line with the runway of nearby Trenton-Robbinsville Airport. Mystery solved!

(It is not surprising that I didn't guess this. I had no idea there was a nearby airport. Trenton itself is about ten miles west of there, and its main airport, Trenton-Mercer Airport, is another five miles beyond that.)

I have been wondering for years why those blobs were in that exact place, and I am really glad to have it cleared up. Thank you, Google!

Dear vision-impaired readers: I wanted to add a description of the view in the iframed Google Street View picture above. Iframes do not support an alt attribute, and MDN says that longdesc is “not helpful for non-visual browsers”. I was not sure what to do.

(The image is a wide-angle shot of a view of the right-hand shoulder of a highway. There is a low chain-link fence in the foreground, and an autumnal landscape behind. The sky is blue but partly obscured by clouds. A high-voltage power pylon is visible at far left and several sets of wires go from it rightward across the whole top of the picture, reaching the top right-hand corner. On the upper sets of wires are evenly-spaced colored balls in orange-red, yellow, and white. Rotating the street view reveals more colored balls, stretching into the distance, but only to the north. To the south there is an overpass, and beyond the overpass the wires continue with no balls.)

In the future, is there a better place to put a description of an iframed image? Thanks.

[Other articles in category /tech] permanent link

Fri, 13 Apr 2018

Exciting homology group news!

As an undergraduate I wondered and wondered about how manifolds and things are classified in algebraic topology, but I couldn't find any way into the subject. All the presentations I found were too abstract and I never came out of it with any concrete idea of how you would actually calculate any specific fundamental groups. I knew that the fundamental group of the circle was !!\Bbb Z!! and the group of the torus was !!\Bbb Z^2!! and I understood basically why, but I didn't know how you would figure this out without geometric intuition.

This was fixed for me in the very last undergrad math class I took, at Columbia University with Johan Tysk. That was the lowest point of my adult life, but the algebraic topology was the one bright spot in it. I don't know what might have happened to me if I hadn't had that class to sustain my spirit. And I learned how to calculate homotopy groups!

(We used Professor Tysk's course notes, supplemented by William Massey's introduction to algebraic topology. I didn't buy a copy of Massey and I haven't read it all, but I think I can recommend it for this purpose. The parts I have read seemed clear and direct.)

Anyway there things stood for a long time. Over the next few decades I made a couple of superficial attempts to find out about homology groups, but again the presentations were too abstract. I had been told that the homology approach was preferred to the homotopy approach because the groups were easier to actually calculate. But none of the sources I found seemed to tell me how to actually calculate anything concrete.

Then a few days ago I was in the coffee shop working on a geometry problem involving an icosi-dodecahedron, and the woman next to me asked me what I was doing. Usually when someone asks me this in a coffee shop, they do not want to hear the answer, and I do not want to give it, because if I do their eyes will glaze over and then they will make some comment that I have heard before and do not want to hear again. But it transpired that this woman was a math postdoc at Penn, and an algebraic topologist, so I could launch into an explanation of what I was doing, comfortable in the knowledge that if I said something she didn't understand she would just stop me and ask a question. Yay, fun!

Her research is in “persistent homology”, which I had never heard of. So I looked that up and didn't get very far, also because I still didn't know anything about homology. (Also, as she says, the Wikipedia article is kinda crappy.) But I ran into her again a couple of days later and she explained the persistent part, and I know enough about what homology is that the explanation made sense.

Her research involves actually calculating actual homology groups of actual manifolds on an actual computer, so I was inspired to take another crack at understanding homology groups. I did a couple of web searches and when I searched for “betti number tutorial” I hit paydirt: these notes titled “persistent homology tutorial” by Xiaojin Zhu of the University of Wisconsin at Madison. They're only 37 slides long, and I could skip the first 15. Then slide 23 gives the magic key. Okay! I have not yet calculated any actual homology groups, so this post might be premature, but I expect I'll finish the slides in a couple of days and try my hand at the calculations and be more or less successful. And the instructions seem clear enough that I can imagine implementing a computer algorithm to calculate the homology groups for a big ugly complex, as this math postdoc does.

I had heard before that the advantage of the homology approach over the homotopy approach is that the homologies are easier to actually calculate with, and now I see why. I could have programmed a computer to do homotopy group calculations, but the output would in general have been some quotient of a free group given by a group presentation, and this is basically useless as far as further computation goes. For example the question of whether two differently-presented groups are isomorphic is undecidable, and I think similar sorts of questions, such as whether the group is abelian, or whether it is infinite, are similarly undecidable.

Sometimes you get a nice group, but usually you don't. For example the homotopy group of the Klein bottle is the quotient of the free group on two generators under the smallest equivalence relation in which !!aba = b!!; that is:

$$\langle a, b\mid abab^{-1}\rangle$$

which is not anything I have seen in any other context. Even the question of whether two given group elements are equal is in general undecidable. So you get an answer, but then you can't actually do anything with it once you have it. (“You're in a balloon!”) The homology approach throws away a lot of information, enough to render the results comprehensible, but it also leaves enough to do something with.

[Other articles in category /math] permanent link

Tue, 10 Apr 2018

Inconceivable things and non-contingent facts

Philosophy makes a distinguish between necessary and contingent facts, but I'm not exactly sure what it is. I think they would say that the election of Al Gore in 2000 is contingent because it's easy to imagine a universe in which it went the other way and the other guy won. But that seems to depend on our powers of imagination, which doesn't seem very rigorous. Is the mass of the electron necessary or contingent? What about the fine-structure constant?

What facts are necessary? Often in this context people fall back on mathematical truths, for example !!1+1=2!!, which does seem hard to assail. But I recently thought of something even farther down the scale, which seems to me even harder to argue.

Mathematics deals with many sorts of objects which are more or less like the ordinary numbers. Some are more complicated, and ordinary numbers are special cases, for example functions and matrices. Some are simpler, and are special cases themselves. Mathematicians can and do define !!2!! in many different ways. There are mathematical systems with !!1!! and !!+!! in which there is no !!2!!, and instead of !!1+1=2!! we have !!1+1=0!!. Well, not quite; there is !!2!!, but !!2=0!!. So one can say that !!1+1=2!! still, but the !!2!! is not very much like the !!2!! that we usually mean when we say !!1+1=2!!. Anyway certainly there is such a system, and I can certainly conceive of it, so there might be a philosophical argument that could be made that !!1+1=2!! is a contingent fact about how numbers happen to work in the universe in which we happen to find ourselves: we are not living in a universe where numbers form a field of characteristic 2.

But here's a fact that I think is unassailably necessary: rubies are red. Why? By definition! A ruby is a kind of gemstone, a type of aluminum oxide called a corundum, that has a deep red color. There are non-red corundums, but they are sapphires, not rubies, because a ruby is a red corundum. There is no such thing as a blue or a green ruby; a blue or green ruby is not a ruby at all, but a sapphire.

How about over in Narnia, where rubies are blue? Well, maybe the Narnians people call hats “avocadoes”, but whether those things are hats or avocadoes depends not on what the Narnians call them but on their properties. If those things are made of felt and the Narnians wear them on their heads, they are hats, regardless of what the Narnians call them; they are avocadoes only if they are globular and can be eaten on toast. Narnians might put actual avocadoes on their heads and then there might be an argument that these things were hats, but if the avocado is a hat it is only because it is customarily worn on the head.

And so too the Narnians can call !!2!! an avocado and say that !!1+1=\text{avocado}!! but that doesn't mean that !!1+1!! is an avocado, even in Narnia. Maybe the Narnians call avocadoes “rubies”, but they're still avocadoes, not rubies. And maybe the Narnians call blue corundums “rubies”, but they're still sapphires, not rubies, because rubies are red.

So I think it might be conceivable that !!1+1=2!! is contingent, and it's certainly easy to conceive of a universe with no rubies at all, but I can't conceive of any way that a ruby could be other than red.

[Other articles in category /misc/philosophy] permanent link

Wed, 04 Apr 2018

Genealogy of the Saudi royal family

[ Note: None of this is a joke, nothing here is intended humorously, and certainly none of it should be taken as mockery or disparagement. The naming conventions of Saudi royalty are not for me to judge or criticize, and if they cause problems for me, the problems are my own. It is, however, a serious lament. ]

The following innocuous claim appears in Wikipedia's article on Abdullah bin Abdul-Rahman:

He was the seventh son of the Emir of the Second Saudi State, Abdul Rahman bin Faisal.

Yesterday I tried to verify this claim and I was not able to do it.

Somewhere there must be a complete and authoritative pedigree of the entire Saudi royal family, but I could not find it online, perhaps because it is very big. There is a Saudi royal family official web site, and when I found that it does have a page about the family tree, I rejoiced, thinking my search was over. But the tree only lists the descendants of King Abdulaziz Ibn Saud, founder of the modern Saudi state. Abdullah was his half-brother and does not appear there.

Well, no problem, just Google the name, right? Ha!

Problem 1: These princes all have at least twenty kids each. No, seriously. The Wikipedia article on Ibn Saud himself lists twenty-one wives and then gives up, ending with an exhausted “Possibly other wives”. There is a separate article on his descendants that lists 72 children of various sexes, and the following section on grandchildren begins:

Due to the Islamic traditions of polygyny and easy divorce (on the male side), King Abdul Aziz [Ibn Saud] has approximately a thousand grandchildren.

Problem 2: They reuse many of the names. Because of course they do; if wife #12 wants to name her first son the same as the sixth son of wife #2, why not? They don't live in the same house. So among the children of Ibn Saud there are two Abdullahs (“servant of God”), two Badrs (“full moon”), two Fahds (“leopard”), two each of Majid (“majestic”), Mishari (I dunno), Talal (dunno), and Turki (“handsome”). There are three sons named Khalid (“eternal”). There is a Sa'ad and a Saad, which I think are the exact same name (“success”) as spelled by two different Wikipedia editors.

And then they reuse the names intergenerationally. Among Ibn Saud's numerous patrilineal grandsons there are at least six more Fahds, the sons respectively of Mohammed, Badr (the second one), Sultan, Turki (also the second one), Muqrin, and Salman. Abdulaziz Ibn Saud has a grandson also named Abdulaziz, whose name is therefore Abdulaziz bin Talal bin Abdulaziz Al Saud. (The “bin” means “son of”; the feminine form is “bint”.) It appears that the House of Saud does not name sons after their fathers, for which I am grateful.

Ibn Saud's father was Abdul Rahman (this is the Abdul Rahman of Abdullah bin Abdul-Rahman, who is the subject of this article. Remember him?) One of Ibn Saud's sons is also Abdul Rahman, I think probably the first one to be born after the death of his grandfather, and at least two of his patrilineal grandsons are also.

Problem 3: Romanization of Arabic names is done very inconsistently. I mentioned “Saad” and “Sa'ad” before. I find the name Abdul Rahman spelled variously “Abdul Rahman”, “Abdulrahman”, “Abdul-Rahman”, and “Abd al-Rahman”. This makes text searches difficult and unreliable. (The name, by the way, means "Servant of the gracious one”, referring to God.)

Problem 4: None of these people has a surname. Instead they are all patronymics. Ibn Saud has six grandsons named Fahd; how do you tell them apart? No problem, their fathers all have different names, so they are Fahd bin Mohammed, Fahd bin Badr, Fahd bin Sultan, Fahd bin Turki, Fahd bin Muqrin, and Fahd bin Salman. But again this confuses text searches terribly.

You can search for “Abdullah bin Abdul-Rahman” but many of the results will be about his descendants Fahd bin Abdullah bin Abdul Rahman, Fahd bin Khalid bin Abdullah bin Abdul Rahman, Fahd bin Muhammad bin Abdullah bin Abdul Rahman, Abdullah bin Bandar bin Abdullah bin Abdul Rahman, Faisal bin Abdullah bin Abdul Rahman, Faisal bin Abdul Rahman bin Abdullah bin Abdul Rahman, etc.

In combination with the reuse of the same few names, the result is even more confusing. There is Bandar bin Khalid, and Khalid bin Bandar; Fahad bin Khalid and Khalid bin Fahd.

There is Mohammed al Saud (Mohammed of (the house of) Saud) and Mohammed bin Saud (Mohammed the son of Saud).

There are grandsons named Saad bin Faisal, Faisal bin Bandar, Bandar bin Sultan, Sultan bin Fahd, Fahd bin Turki, Turki bin Talal, Talal bin Mansour, Mansour bin Mutaib, Mutaib bin Abdullah, and Abdullah bin Saad. I swear I am not making this up.

Perhaps Abdullah was the seventh son of Abdul Rahman.

Perhaps not.

I surrender.

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Sun, 25 Mar 2018

The Death of Stalin

Today I went to see The Death of Stalin. If someone is going to go to the trouble of making a comedy about the death of Stalin, that seems like a worthy attempt, and I will do them the courtesy of going to watch it. At least I can be sure it will not be the same old shit.

I was interested to see if it was possible to make a comedy about the death of Stalin, and if so, would it would be funny? I got my answer: no, you can't, and it isn't.

It was worth a shot, I guess, and I give the writers and director top marks for audacity. The cast was great. The acting was great. I thought Jason Isaacs as Marshal Zhukov stole every scene he was in. But yeah, it's hard to be funny when Lavrenty Beria is raping a bunch of fourteen-year-old girls, and the movie didn't work for me.

There's a long and solid tradition of comedy about completely loathsome people, but I think most of it follows pretty much the same pattern: terrible stuff happens to the loathsome people and it is funny because the people are so loathsome and because they so richly deserve all the terrible stuff that happens to them. It can be fun to see a horrible person sabotage themselves with their own horribleness.

(Examples off the top of my head: Fawlty Towers. Otto in A Fish Called Wanda. Jack Vance's Cugel books. Married With Children. I think this might have been the main attraction of Seinfeld, although if it is I didn't get the joke until after the series was over.)

Unfortunately this movie, being historical fiction, has to stick to the history: Malenkov gets swept under a rug. Khrushchev seizes power. Molotov keeps on doing what he does. Beria is murdered, but there is nothing funny about it, and I found it unsatisfying. Indeed, all of these horrible people are suffering because of the horrible world they have created for themselves, but I found no fun in it because there were another 170 million people suffering much worse from the same horrible crap. The coyote's look of dismay as he falls of the cliff loses all its savor if he has the road runner's broken body in his jaws when it happens.

So, eh. Sorry, Iannucci. I wanted to like your movie.

[ Odd trivium: I started writing articles in the “movies” section of this blog back in 2007, but this is the first one that has seen publication. ]

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Sat, 24 Mar 2018

Addenda to recent articles 201803

It's been a while since we had one of these. But gosh, people have sent me quite a lot of really interesting mail lately.

The Smith chart

  • I related my childhood disappointment at the limited number of cool coordinate systems. Norman Yarvin directed me to prolate spheroidal coordinates which are themselves a three-dimensional version of elliptic coordinates which are a system of exactly the sort I escribed in the article, this time parametrized by a family of ellipses and a family of hyperbolas, all of which share the same two foci; this article links in turn to parabolic coordinates in which the two families are curves are up-facing and down-facing parabolas that all share a focus. (Hmm, this seems like a special case of the ellipses, where one focus goes to infinity.)

    Walt Mankowski also referred me to the Smith chart, shown at right, which is definitely relevant. It is a sort of nomogram, and parametrizes certain points by their position on circles from two families

    $$\begin{align} F_1(c): && (x- c)^2 & + y^2 & =c^2 \\ F_2(c): && x^2 & + (y- c)^2 & =c^2 \\ \end{align} $$

    Electrical engineers use this for some sort of electrical engineer calculation. They use the letter !!j!! instead of !!i!! for the imaginary unit because they had already used !!i!! to stand for electrical current, which is totally reasonable because “electrical current” does after all start with the letter !!i!!. (In French! The French word is courant. Now do you understand? Stop asking questions!)

  • Regarding what part of the body Skaði was looking at when the Norse text says fótr, which is probably something like the foot, Alexander Gurney and Brent Yorgey reminded me that Biblical Hebrew often uses the foot as a euphemism for the genitals. One example that comes immediately to mind is important in the book of Ruth:

    And when Boaz had eaten and drunk, and his heart was merry, he went to lie down at the end of the heap of corn: and she came softly, and uncovered his feet, and laid her down. (Ruth 3:7)

    M. Gurney suggested Isaiah 6:2. (“Above him were seraphim, each with six wings: With two wings they covered their faces, with two they covered their feet, and with two they were flying.”) I think Ezekiel 16:25 is also of this type.

    I mentioned to Brent that I don't think Skaði was looking at the Æsir's genitals, because it wouldn't fit the tone of the story.

  • Alexander Gurney sent me a lot of other interesting material. I had translated the Old Icelandic hreðjar as “scrotum”, following Zoëga. But M. Gurney pointed out that the modern Icelandic for “radish” is hreðka. Coincidence? Or was hreðjar a euphemism even then? Zoëga doesn't mention it, but he doesn't say what word was used for “radish”, so I don't know.

    He also pointed me to Parts of the body in older Germanic and Scandinavian by Torild Washington Arnoldson. As in English, there are many words for the scrotum and testicles; some related to bags, some to balls, etc. Arnoldson does mention hreðjar in the section about words that are bag-derived but doesn't say why. Still if Arnoldson is right it is not about radishes.

    I should add that the Skáldskaparmál itself has a section about parts of the body listing suitable words and phrases for use by skálds:

    Hönd, fótr.

    … Á fæti heitir lær, kné, kálfi, bein, leggr, rist, jarki, il, tá. …

    (… The parts of the legs are called thigh, knee, calf, lower leg, upper leg, instep, arch, sole, toe … [ Brodeur ])

    I think Brodeur's phrase “of the legs” here is an interpolation. Then he glosses lær as “thigh”, kné as “knee”, kálfi as “calf”, and so on. This passage is what I was thinking of when I said

    Many of the words seem to match, which is sometimes helpful but also can be misleading, because many don't.

    I could disappear down this rabbit hole for a long time.

  • Regarding mental estimation of the number of primes less than 1,000, which the Prime Number Theorem says is approximately !!\frac{1000}{\ln 1000}!!, several people pointed out that if I had memorized !!\ln 10\approx 2.3!! then I would have had that there are around !!\frac{1000}{3·2.3}!! primes under 1,000.

    Now it happens that I do have memorized !!\ln 10\approx 2.3!! and although I didn't happen come up with it while driving that day, I did come up with it a couple of days later in the parking lot of a Wawa where I stopped to get coffee before my piano lesson. The next step, if you are in a parking lot, is to approximate the division as !!\frac{1000}{6.9} \approx \frac{1000}7 = 142.857\ldots!! (because you have !!\frac17=0.\overline{142857}!! memorized, don't you?) and that gives you an estimate of around 145 primes.

    Which, perhaps surprisingly, is worse than what I did the first time around; it is 14% too low instead of 8% too high. (The right answer is 168 and my original estimate was 182.)

    The explanation is that for small !!n!!, the approximation !!\pi(N)\sim\frac{N}{\ln N}!! is not actually very good, and I think the interpolation I did, using actual low-value counts, takes better account of the low-value error.

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Fri, 23 Mar 2018

Shitpost roundup, 2018-02

Here is a list of February's shitposts, later than usual, but who cares? Boldface indicates the articles that may (may) be of more general interest (ha). I think that I did a better job of noticing when a post wasn't shitty enough and promoting it, pre-publication, to this blog, so you will have seen all the better stuff already.

I'm pleased, volume over January is slightly up, and quality is definitely down, especially in the last half of the month. But I posted on only 21 of 28 days; I'll have to work on that.

[Other articles in category /meta/shitpost] permanent link

Thu, 22 Mar 2018

Does Skaði choose the husband with the best butt?

(Warning: I do not know anything about Old Norse, so everything I say about it should be understood as ill-informed speculation. I welcome corrections.)

In one of my favorite episodes from Norse mythology, the Æsir owe a payment to the Jötunn Skaði in compensation for killing her father. But they know she is very wealthy, and offer her an alternative compensation: one of their men in marriage.

Skaði wants to marry Baldr, because he is extremely handsome. But Baldr is already married. Odin proposes a compromise: the Æsir will line up behind a short curtain, and Skaði will choose her husband. She will marry whomever she picks; if she can pick out Baldr by his legs, she can have him. Skaði agrees, assuming that the beautiful Baldr will have the best legs.

(She chooses wrong. Njörðr has the best legs.)

Thinking on this as an adult, I said to myself “Aha, this is like that horn full of milk that was actually mead. I bet this was also cleaned up in the version I read, and that in the original material, Skaði was actually choosing the husband with the best butt.”

I went to check, and I was wrong. The sources say she was looking only at their feet.

I was going to just quote this:

she should choose for herself a husband from among the Æsir and choose by the feet only, seeing no more of him.

But then I got worried. This is of course not the original source but an English translation; what if it is inaccurate?

Well, there was nothing else to do but ask Snorri about it. He says:

En æsir buðu henni sætt ok yfirbætr ok it fyrsta, at hon skal kjósa sér mann af ásum ok kjósa at fótum ok sjá ekki fleira af.

(Sætt is recompense or settlement; yfirbætr similarly. (Bætr is a cure, as in “I was sick, but I got better”.) The first (fyrsta) part of the settlement is that she “shall choose a man for herself” (skal kjósa sér mann) but choose by the feet (kjósa at fótum) seeing nothing else (sjá ekki fleira af).)

The crucial word here is fótum, which certainly looks like “foot”. (It is the dative form of fótr.) Could it possibly mean the buttocks? I don't think so. It's hard to be 100% certain, because it could be a euphemism — anything could be a euphemism for the buttocks if you paused before saying it and raised one eyebrow. (Did the Norse bards ever do this?) Also the Norse seem to have divided up the leg differently than we do. Many of the words seem to match, which is sometimes helpful but also can be misleading, because many don't. For example, I think leggr, despite its appearance, means just the shank. And I think fótum may not be just the foot itself, but some part of the leg that includes the foot.

But I'm pretty sure fótum is not the butt, at least not canonically. To do this right I would look at all the other instances of fótr to see what I could glean from the usage, but I have other work to do today. So anyway, Skaði probably was looking at their feet, and not at their butts. Oh well.

However! the other part of Skaði's settlement is that the Æsir must make her laugh. In the version I first read, Loki achieves this by tying his beard to a goat's. Nope!

Þá gerði Loki þat, at hann batt um skegg geitar nökkurrar ok öðrum enda um hreðjar sér, ok létu þau ýmsi eftir ok skrækði hvárt tveggja hátt.

Skegg geitar nökkurar is indeed some goat's beard. But hann batt … ok öðrum enda um hreðjar sér is “he tied … the other end to his own scrotum”.

Useful resources:

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Wed, 21 Mar 2018

The 1943 Bengal famine

A couple of years ago I was reading Wikipedia's article about the the 1943 Bengal famine, and I was startled by the following claim:

"If food is so scarce, why hasn’t Gandhi died yet?"

Winston Churchill's response to an urgent request to release food stocks for India.

It was cited, but also marked with the “not in citation” tag, which is supposed to mean that someone checked the reference and found that it did not actually support the claim.

It sounded like it might be the sort of scurrilous lie that is widely repeated but not actually supportable, so I went to follow it up. It turned out that although the quotation was not quite exact, it was not misleadingly altered, and not a scurrilous lie at all. The attributed source (Tharoor, Shashi "The Ugly Briton". Time, (29 November 2010).) claimed:

Churchill's only response to a telegram from the government in Delhi about people perishing in the famine was to ask why Gandhi hadn't died yet.

I removed the “not in citation” tag, which I felt was very misleading.

Still, I felt that anything this shocking should be as well-supported as possible. It cited Tharoor, but Tharoor could have been mistaken. So I put in some effort and dug up the original source. It is from the journal entry of Archibald Wavell, then Viceroy of India, of 5 July 1944:

Winston sent me a peevish telegram to ask why Gandhi hadn't died yet! He has never answered my telegram about food.

This appears in the published version of Lord Wavell's journals. (Wavell, Archibald Percival. Wavell: The Viceroy's journal, p. 78. Moon, Penderel, ed. Oxford University Press, 1973.) This is the most reliable testimony one could hope for. The 1973 edition is available from the Internet Archive.

A few months later, the entire article was massively overhauled by a group of anglophiles and Churchill-rehabilitators. Having failed to remove the quotation for being uncited, and then having failed to mendaciously discredit the cited source, they removed the quotation in a typical episode of Wikipedia chicanery. In a 5,000-word article, one sentence quoting the views of the then-current British Prime Minister was deemed “undue weight”, and a failure to “fairly represent all significant viewpoints that have been published by reliable sources”.

Further reading: In Winston Churchill, Hollywood rewards a mass murderer. (Tharoor again, in last week's Washington Post.)

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Tue, 20 Mar 2018

English's -en suffix

In English we can sometimes turn an adjective into a verb by suffixing “-en”. For example:

black → blacken
red → redden
white → whiten
wide → widen

But not

blue → bluen*
green → greenen*
yellow → yellowen*
long → longen*

(Note that I am only looking at -en verbs that are adjective-derived present tenses. This post is not concerned with the many -en verbs that are past participles, such as “smitten” (past participle of “smite”), “spoken” (“speak”), “molten” (“melt”), “sodden” (“seethe”), etc.)

I asked some linguist about this once and they were sure it was purely morphological, something like: black, red, and white end in stop consonants, and blue, green, and yellow don't.

Well, let's see:

Stop Blacken
Open (?)
Fricative Coarsen
Nasal   Cleanen
Vowel   Angrien
Glide   Betteren

There are some fine points:

  • “Biggen” used to exist but has fallen out of use
  • Perhaps I should have ommitted “strengthen” and “hasten”, which are derived from nouns, not from adjectives
  • I'm not sure whether “closen”, “hotten” and “wetten” are good or bad so I left them off
  • “moisten” and “soften” might belong with the stops instead of the fricatives
  • etc.

but clearly the morphological explanation wins. I'm convinced.

[ Addendum: Wiktionary discusses this suffix, distinguishing it from the etymologically distinct participial “-en”, and says “it is not currently very productive in forming new words, being mostly restricted to monosyllabic bases which end in an obstruent”. ]

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Mon, 19 Mar 2018

Funky coordinate systems

I had a fun idea this morning. As a kid I was really interested in polar coordinates and kind of disappointed that there didn't seem to be any other coordinate systems to tinker with. But this morning I realized there were a lot.

Let !!F(c)!! be some parametrized family of curves that partition the plane, or almost all of the plane, say except for a finite number of exceptions. If you have two such families !!F_1(c)!! and !!F_2(c)!!, and if each curve in !!F_1!! intersects each curve in !!F_2!! in exactly one point (again with maybe a few exceptions) then you have a coordinate system: almost every point !!P!! lies on !!F_1(a)!! and !!F_2(b)!! for some unique choice of !!\langle a, b\rangle!!, and these are its coordinates in the !!F_1–F_2!! system.

For example, when !!F_1(c)!! is the family of lines !!x=c!! and !!F_2(c)!! is the family of lines !!y=c!! then you get ordinary Cartesian coordinates, and when !!F_1(c)!! is the family of circles !!x^2+y^2=c!! and !!F_2(c)!! is the family !!y=cx!! (plus also !!x=0!!) you get standard polar coordinates, which don't quite work because the origin is in every member of !!F_2!!, but it's the only weird exception.

But there are many other families that work. To take a particularly simple example you can pick some constant !!k!! and then take

$$\begin{align} F_1(c): && x & =c \\ F_2(c): && y & =kx+c. \end{align} $$

This is like Cartesian coordinates except the axes are skewed. I did know about this when I was a kid but I considered it not sufficiently interesting.

For a more interesting example, try

$$\begin{align} F_1(c): && x^2-y^2 & =c \\ F_2(c): && xy & =c \end{align} $$

which looks like this:

Two families of hyperbolas, as in the displayed equation
immediately preceding. The hyperbolas !!x^2-y^2 = c!! (in blue) and !!xy=c!! (in red)

I've seen that illustration before but I don't think I thought of using it as a coordinate system. Well, okay, every pair of hyperbolas intersects in two points, not one. So it's a parametrization of the boundary of real projective space or something, fine. Still fun!

In the very nice cases (such as the hyperbolas) each pair of curves is orthogonal at their point of intersection, but that's not a requirement, as with the skew Cartesian system. I'm pretty sure that if you have one family !!F!! you can construct a dual family !!F'!! that is orthogonal to it everywhere by letting !!F'!! be the paths of gradient descent or something. I'm not sure what the orthogonality is going to be important for but I bet it's sometimes useful.

You can also mix and match families, so for example take:

$$\begin{align} F_1(c): && x & =c \\ F_2(c): && xy & =c \end{align} $$

Some examples work better than others. The !!xy=c!! hyperbolas are kind of a mess when !!c=0!!, and they don't go together with the !!x^2+y^2=c!! circles in the right way at all: each circle intersects each hyperbola in four points. But it occurs to me that as with the projective plane thingy, we don't have to let that be a problem. Take !!S!! to be the quotient space of the plane where two points are identified if their !!F_1–F_2!!-coordinates are the same and then investigate !!S!!. Or maybe go more directly and take !!S = F_1 \times F_2!! (literally the Cartesian product), and then topologize !!S!! in some reasonably natural way. Maybe just give it the product topology. I dunno, I have to think about it.

(I was a bit worried about how to draw the hyperbola picture, but I tried Google Image search for “families of orthogonal hyperbolas”, and got just what I needed. Truly, we live in an age of marvels!)

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Mon, 12 Mar 2018

Quick and dirty prime counting

I've been thinking for a while that I probably ought to get around to memorizing all the prime numbers under 1,000, so that I don't have to wonder about things like 893 all the time, and last night in the car I started thinking about it again, and wondered how hard it would be. There are 25 primes under 100, so presumably fewer than 250 under 1,000, which is not excessive. But I wondered if I could get a better estimate.

The prime number theorem tells us that the number of primes less than !!n!! is !!O(\frac n{\log n})!! and I think the logarithm is a natural one, but maybe there is some constant factor in there or something, I forget and I did not want to think about it too hard because I was driving. Anyway I cannot do natural logarithms in my head.

Be we don't need to do any actual logarithms. Let's estimate the fraction of primes up to !!n!! as !!\frac 1{c\log n}!! where !!c!! is unknown and the base of the logarithm is then unimportant. The denominator scales linearly with the power of !!n!!, so the difference between the denominators for !!n=10!! and !!n=100!! is the same as the difference between the denominators for !!n=100!! and !!n=1000!!.

There are 4 primes less than 10, or !!\frac25!!, so the denominator is 2.5. And there are 25 primes less than 100, so the denominator here is 4. The difference is 1.5, so the denominator for !!n=1000!! ought to be around 5.5, and that means that about !!\frac2{11}!! of the numbers up to 1000 are prime. This yields an estimate of 182.

I found out later that the correct number is 186, so I felt pretty good about that.

[ Addendum: The correct number is 168, not 186, so I wasn't as close as I thought. ]

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Tue, 20 Feb 2018

Composition of utility pole ID tags

In a recent article discussing utility poles, and the metal ID plates they carry, I wondered what the plates were made of:

Steel would rust; and I thought even stainless steel wouldn't last as long as these tags need to. Aluminum is expensive. Tin degrades at low temperatures. … I will go test the tags with a magnet to see if they are ferrous.

They are not ferrous. Probably they are aluminum. My idea that aluminum is too expensive to use for the plates was ridiculous. The pole itself costs a lot of money. The sophisticated electrical equipment on the pole costs thousands of dollars. The insulated wire strung from the pole is made of copper. Compared with all this, a ten-centimeter oval of stamped aluminum is not a big deal.

1.8mm aluminum sheet costs $100 per square meter even if you don't buy it in great quantity. Those aluminum tags probably cost no more than fifty cents each.

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Wed, 14 Feb 2018

Utility poles

I am almost always interested in utility infrastructure. I see it every day, and often don't think about it. The electric power distribution grid is a gigantic machine, one of the biggest devices ever built, and people spend their whole lives becoming experts on just one part of it. What is it all for, how does it work? What goes wrong, and how do you fix it? Who makes the parts, and how much do they cost? Every day I go outside and see things like these big cylinders:

A wooden power utility pole,
including cobra-head street light,
with three large gray cylinders mounted on it under the main

and I wonder what they are. In this case from clues in the environment I was able to guess they were electrical power transformers. Power is distributed on these poles at about seven thousand volts, which is called “medium voltage”. But you do not want 7000-volt power in your house because it would come squirting out of the electric outlets in awesome lightnings and burn everything up. Also most household uses do not want three-phase power, they want single-phase power. So between the pole and the house there is a transformer to change the shape of the electricity to 120V, and that's what these things are. They turn out to be called “distribution transformers” and they are manufactured by — guess who? — General Electric, and they cost a few thousand bucks each. And because of the Wonders of the Internet, I can find out quite a lot about them. The cans are full of mineral oil, or sometimes vegetable oil! (Why are they full of oil? I don't know; I guess for insulation. But I could probably find out.) There are three because that is one way to change the three-phase power to single-phase, something I wish I understood better. Truly, we live in an age of marvels.

Anyway, I was having dinner with a friend recently and for some reason we got to talking about the ID plates on utility poles. The poles around here all carry ID numbers, and I imagine that back at the electric company there are giant books listing, for each pole ID number, where the pole is. Probably they computerized this back in the seventies, and the books are moldering in a closet somewhere.

As I discussed recently, some of those poles are a hundred years old, and the style of the ID tags has changed over that time:

An old, stamped-metal identification plate nailed to a wooden
utility pole.  The plate is elliptical, and says 'PHILA ELEC. Cº 79558 B' This wooden pole has the following letters burned into it: 'BWR
CPT 51017 SPSK6 250 PECO'

It looks to me like the original style was those oval plates that you see on the left, and that at some point some of the plates started to wear out and were replaced by the yellow digit tags in the middle picture. The most recent poles don't have tags: the identifier is burnt into the pole.

Poles in my neighborhood tend to have consecutive numbers. I don't think this was carefully planned. I guess how this happened is: when they sent the poles out on the truck to be installed, they also sent out a bunch of ID plates, perhaps already attached to the poles, or perhaps to be attached onsite. The plates would already have the numbers on them, and when you grab a bunch of them out of the stack they will naturally tend to have consecutive numbers, as in the pictures above, because that's how they were manufactured. So the poles in a vicinity will tend to have numbers that are close together, until they don't, because at that point the truck had to go back for more poles. So although you might find poles 79518–79604 in my neighborhood, poles 79605–79923 might be in a completely different part of the city.

Later on someone was inspecting pole 79557 (middle picture) and noticed that the number plate was wearing out. So they pried it off and replaced it with the yellow digit tag, which is much newer than the pole itself. The inspector will have a bunch of empty frames and a box full of digits, so they put up a new tag with the old ID number.

But sometime more recently they switched to these new-style poles with numbers burnt into them at the factory, in a different format than before. I have tried to imagine what the number-burning device looks like, but I'm not at all sure. Is it like a heated printing press, or perhaps a sort of configurable branding iron? Or is it more like a big soldering iron that is on a computer-controlled axis and writes the numbers on like a pen?

I wonder what the old plates are made of. They have to last a long time. For a while I was puzzled. Steel would rust; and I thought even stainless steel wouldn't last as long as these tags need to. Aluminum is expensive. Tin degrades at low temperatures. But thanks to the Wonders of the Internet, I have learned that, properly made, stainless steel tags can indeed last long enough; the web site of the British Stainless Steel Association advises me that even in rough conditions, stainless steel with the right composition can last 85 years outdoors. I will do what I should have done in the first place, and go test the tags with a magnet to see if they are ferrous.

Here's where some knucklehead in the Streets Department decided to nail a No Parking sign right over the ID tag:

A close-up of an old oval tag just peeking out from behind the
corner of the metal regulation sign that was nailed to the same pole

Another thing you can see on these poles is inspection tags:

A very old pole. Three groups of tags are nailed to it.  The
bottom two groups contains an oval tag stamped with OSMOSE and an
inspection year (2001 or 2013),
and a quarter-circle tag stamped with MITC-FUME.  The top group is
missing its oval tag, and has only a rather rusty quarter-circle that

Without the Internet I would just have to wonder what these were and what OSMOSE meant. It is the name of the company that PECO has hired to inspect and maintain the poles. They specialize in this kind of work. This old pole was inspected in 2001 and again in 2013. The dated inspection tag from the previous inspection is lost but we can see a pie-shaped tag that says WOODFUME. You may recall from my previous article that the main killer of wood poles is fungal infection. Woodfume is an inexpensive fumigant that retards pole decay. It propagates into the pole and decomposes into MITC (methyl isothiocyanate). By 2001 PECO had switched to using MITC-FUME, which impregnates the pole directly with MITC. Osmose will be glad to tell you all about it.

(Warning: Probably at least 30% of the surmise in this article is wrong.)

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Tue, 13 Feb 2018

Weighted Reservoir Sampling

(If you already know about reservoir sampling, just skip to the good part.)

The basic reservoir sampling algorithm asks us to select a random item from a list, easy peasy, except:

  1. Each item must be selected with equal probability
  2. We don't know ahead of time how big the list is
  3. We may only make one pass over the list
  4. We may use only constant memory

Maybe the items are being read from a pipe or some other lazy data structure. There might be zillions of them, so we can't simply load them into an array. Obviously something like this doesn't work:

# Python
from random import random
selected =
for item in inputs:
    if random() < 0.5:
        selected = item

because it doesn't select the items with equal probability. Far from it! The last item is selected as often as all the preceding items put together.

The requirements may seem at first impossible to satisfy, but it can be done and it's not even difficult:

from random import random
n = 0
selected = None

for item in inputs:
    n += 1
    if random() < 1/n:
        selected = item

The inputs here is some sort of generator that presents the list of items, one at a time. After the loop completes, the selected item is in selected. A proof that this selects each item equiprobably is left as an easy exercise, or see this math StackExchange post. A variation for selecting !!k!! items instead of only one is quite easy.

The good part

Last week I thought of a different simple variation. Suppose each item !!s_i!! is presented along with an arbitrary non-negative weight !!w_i!!, measuring the relative likelihood of its being selected for the output. For example, an item with weight 6 should be selected twice as often as an item with weight 3, and three times as often as an item with weight 2.

The total weight is !!W = \sum w_i!! and at the end, whenever that is, we want to have selected each item !!s_i!! with probability !!\frac{w_i}{W}!!:

total_weight = 0
selected = None

for item, weight in inputs:
    if weight == 0: continue
    total += weight
    if random() < weight/total:
        selected = item

The correctness proof is almost the same. Clearly this reduces to the standard algorithm when all the weights are equal.

This isn't a major change, but it seems useful and I hadn't seen it before.

[Other articles in category /prog] permanent link

Mon, 12 Feb 2018

Philadelphia sports fans behaving badly

Philadelphia sports fans have a bad reputation. For example, we are famous for booing Santa Claus and hitting him with snowballs. I wasn't around for that; it happened in 1968. When the Santa died in 2015, he got an obituary in the Phildelphia Inquirer:

Frank Olivo, the Santa Claus who got pelted with snowballs at the Eagles game that winter day in 1968, died Thursday, April 30…

The most famous story of this type is about Ed Rendell (after he was Philadelphia District Attorney, but before he was Mayor) betting a Eagles fan that they could not throw snowballs all the way from their upper-deck seat onto the field. This was originally reported in 1989 by Steve Lopez in the Inquirer.

(Lopez's story is a blast. He called up Rendell, who denied the claim, and referred Lopez to a friend who had been there with him. Lopez left a message for the friend. Then Rendell called back to confess. Later Rendell's friend called back to deny the story. Lopez wrote:

Was former D.A. Ed Rendell's worst mistake to (A) bet a drunken hooligan he couldn't reach the field, (B) lie about it, (C) confess, or (D) take his friend down with him?

My vote is C. Too honest. Why do you think he can't win an election?

A few years later Rendell was elected Mayor of Philadelphia, and later, Governor of Pennsylvania. Anyway, I digress.)

I don't attend football games, and baseball games are not held in snowy weather, so we have to find other things to throw on the field. I am too young to remember Bat Day, where each attending ticket-holder was presented with a miniature souvenir baseball bat; that was eliminated long ago because too many bats were thrown at the visiting players. (I do remember when those bats stopped being sold at the concession stands, for the same reason.) Over the years, all the larger and harder premiums were eliminated, one by one, but we are an adaptable people and once, to protest a bad call by the umpire, we delayed the game by wadding up our free promotional sport socks and throwing them onto the field. That was the end of Sock Day.

On one memorable occasion, two very fat gentlemen down by the third-base line ran out of patience during an excessively long rain delay and climbed over the fence, ran out and belly-flopped onto the infield, sliding on the wet tarpaulin all the way to the first-base side. Confronted there by security, they evaded capture by turning around and sliding back. These heroes were eventually run down, but only after livening up what had been a very trying evening.

The main point of this note is to shore up a less well-known story of this type. I have seen it reported that Phillies fans once booed Miss Pennsylvania, and I have also seen people suggest that this never really happened. On my honor, it did happen. We not only booed Miss Pennsylvania, we booed her for singing the national anthem. I was at that game, in 1993. The Star-Spangled Banner has a lot of problems that the singer must solve one way or another, and there are a lot of ways to interpret it. But it has a melody, and the singer's interpretation is not permitted to stray so far from the standard that they are singing a different song that happens to have the same words. I booed too, and I'm not ashamed to admit it.

[Other articles in category /games] permanent link

Wed, 07 Feb 2018

The many faces of the Petersen graph

(Actually the Petersen graph cannot really be said to have faces, as it is nonplanar. HA! HA! I MAKE JOKE!​!1!)

This article was going to be about how GraphViz renders the Petersen graph, but instead it turned out to be about how GraphViz doesn't render the Petersen graph. The GraphViz stuff will be along later.

Here we have the Petersen graph, which, according to Donald Knuth, “serves as a counterexample to many optimistic predictions about what might be true for graphs in general.” It is not that the Petersen graph is stubborn! But it marches to the beat of a different drummer. If you have not met it before, prepare to be delighted.

The Petersen
graph has two sets of five vertices each.  Each set is connected into
a pentagonal ring.  There are five more edges between vertices in
opposite rings, but instead of being connected 0–0 1–1 2–2 3–3 4–4,
they are connected 0–0 1–2 2–4 3–1 4–3.

This is the basic structure: a blue 5-cycle, and a red 5-cycle. Corresponding vertices in the two cycles are connected by five purple edges. But there is a twist! Notice that the vertices in the red cycle are connected in the order 1–3–5–2–4.

There are different ways to lay out the Petersen graph that showcase its many interesting properties. For example, the standard presentation, above, demonstrates that the Petersen graph is nonplanar, since it obviously contracts to !!K_5!!. The presentation below obscures this, but it is good for seeing that the graph has diameter only 2:

Wait, what? Where did the pentagons go?

Try this instead:

The Petersen
graph laid out as a tree, with a root attached to three level-1 nodes,
each attached to 2 level-2 nodes.  The six level-2 nodes are then
connected into a ring so that each level-2 node is at distance 1 or
distance 2 from each other level-2 node.

Again the red vertices are connected in the order 1–3–5–2–4.

Okay, that is indeed the Petersen graph, but how does it help us see that the graph has diameter 2? Color the nodes by how far down they are from the root:

  • Obviously, the root node (black) has distance at most 2 to every other node, because the tree has only depth 2.

  • Each of the three second-level nodes (red) is distance 2 from the other two, via a path through the root.

  • The six third-level nodes (blue) are linked in a 6-cycle (dotted lines), so that each third-level node is at most two steps away along the cycle from the others, except for the one furthest away, but that is its sibling in the tree, and it has a path of length 2 through their common parent.

  • And since each third-level node (say, the one with the red ring) is connected by a dotted edge (orange) to cousins in both of the other branches of the tree, it's only distance 2 from both of its red uncle nodes.

Looking at the pentagonal version, you would not suspect the Petersen graph of also having a sixfold symmetry, but it does. We'll get there in two steps. Again, here's a version where it's not so easy to see that it's actually the Petersen graph, but whatever it is, it is at least clear that it has an automorphism of order six (give it a one-sixth turn):


The represents three vertices, one in each color. In the picture they are superimposed, but in the actual graph, no pair of the three is connected by an edge. Instead, each of the three is connected not to the others but to a tenth vertex that I omitted from the diagram entirely.

Let's pull apart the three vertices and reveal the hidden tenth vertex and its three edges:


Here is the same drawing, recolored to match the tree diagram from before; the outer hexagon is just the 6-cycle formed by the six blue leaf nodes:


But maybe it's easier to see if we look for red and blue pentagons. There are a couple of ways to do that:

narf   narf

As always, the red vertices are connected in the order 1–3–5–2–4.

Finally, here's a presentation you don't often see. It demonstrates that the Petersen graph also has fourfold symmetry:


Again, and represent single vertices stretched out into dumbbell shapes. The diagram only shows 14 of the 15 edges; the fifteenth connects the two dumbbells.

The pentagons are deeply hidden here. Can you find them? (Spoiler)

Even though this article was supposed to be about GraphViz, I found it impossible to get it to render the diagrams I wanted it to, and I had to fall back on Inkscape. Fortunately Inkscape is a ton of fun.

[Other articles in category /math] permanent link

Thu, 01 Feb 2018

Shitpost roundup, 2018-01

As I may have mentioned, I have started another blog, called Content-type: text/shitpost.

Last month I said:

The shitposts have been suffering quality creep and I am making an effort to lower my standards. I will keep you posted about how this develops.

I think I am doing better. I will continue my efforts to emphasize quantity over quality, with a multi-pronged approach:

  • Faster production with lower production standards
  • Less filtering of possible topics for relevance, general interest, or almost anything else
  • Promote insufficiently shitty posts to The Universe of Discourse

It will be a struggle, but I resolve to do my best!

Here is a list of January's shitposts. Boldface indicates the articles that may (may) be of more general interest (ha). There are fewer of these than last month because I promoted several of the better ones, so you have seen them already.

[Other articles in category /meta/shitpost] permanent link

Tue, 30 Jan 2018

Sapporo has closed

Sapporo restaurant, on West 49th Street in New York, closed yesterday. There are 24,000 restaurants in New York, and for many, many years, Sapporo was my favorite.

Sapporo was a ramen restaurant, probably the first in New York. I remember first hearing the word “ramen” in the early 1980s, when the Larmen Dosanko appeared near Lincoln Center. But Sapporo opened in 1975. I started going there around 1984. I didn't discover it on my own; I think my dad and I happened in one day when we were in the neighborhood. But it made a big impression on me, and I would regularly stop in whenever I was nearby, and sometimes I would walk downtown (about 45 minutes) just to eat there.

When I was fifteen years old, I did somethinng fifteen-year-old boys often do: I grew six inches and added thirty pounds in one year. I ate all the time. I spent so much time eating that it wasn't enjoyable any more, and I complained that I was tired of it and didn't have enough time to do anything else. I would come home from school and eat a double-decker sandwich (sliced muenster with mayonnaise was my favorite), half a pound of feta cheese, three yogurts, and whatever leftovers I could find in the fridge, and then two hours later when my parents got home I would ask “What's for dinner? I'm starving.” I fell in love with Sapporo because it was the only restaurant where I could afford to order as much food as I could eat. I would come out of Sapporo full. Sometimes when I left Sapporo there was still some rice or noodles in my bowl, and I would stop thinking about food for an hour or two.

On the table were three intriguing bottles, one brown, one pale yellow, and one bright red. The brown one was obviously soy sauce, but what were the others? I learned that the pale yellow one was vinegar and the bright red one was hot oil and had great fun trying them out in different combinations. I had never thought of using vinegar as a condiment, and I loved it.

Sapporo was where I first had agedashi dofu, which is delectable cubes of soft tofu, dusted in flour or starch, fried brown and crisp, and served in savory broth. It is ephemeral: you have to eat it right away, before it gets cold and soggy.

My favorite dish was the pork cutlet donburi. (They didn't call it “katsudon”; I didn't learn that until later.) The cutlet was embedded, with onions, in a fried egg that coverered and adhered to the top of the rice, and I can still remember how it tasted with the soy sauce and vinegar, and the texture of it. It was tricky to pick up the cutlet, with its attached fried egg, with chopsticks. I now use chopsticks as well as I use a fork, automatically and unconsciously, and I think Sapporo was probably where I learned to do it.

Because of Sapporo, I became so enamored of vinegar that I started to put it on everything. I didn't like mustard, I thought, but one day I learned that the principal ingrediant in mustard, other than the mustard itself, was vinegar. This put a new light on things and I immediately decided I had better give mustard another try. I discovered that I did indeed like mustard (like vinegar, but with extra flavor!) and I have liked it ever since.

(Around the same time of my life, I was learning about beer, and I eagerly tried Sapporo beer, which they feature, hoping that it would be as wonderful as Sapporo restaurant. I was disappointed.)

I deeply regret that I missed my last chance to eat there. I was staying in Midtown with Toph last summer and suggested that we eat dinner there, but she chose to go somewhere else. It didn't occur to me that Sapporo might not always be there, or I would have insisted.

Goodbye, Sapporo. I love you and I will miss you.

[Other articles in category /food] permanent link

Mon, 29 Jan 2018

The rubber duck explains coherence spaces

I've spent a chunk of the past week, at least, trying to understand the idea of a coherence space (or coherent space). This appears in Jean-Yves Girard's Proofs and Types, and it's a model of a data type. For example, the type of integers and the type of booleans can be modeled as coherence spaces.

The definition is one of those simple but bafflingly abstract ones that you often meet in mathematics: There is a set !!\lvert \mathcal{A}\rvert!! of tokens, and the points of the coherence space !!\mathcal{A}!! (“cliques”) are sets of tokens. The cliques are required to satisfy two properties:

  1. If !!a!! is a clique, and !!a'\subset a!!, then !!a'!! is also a clique.
  2. Suppose !!\mathcal M!! is some family of cliques such that !!a\cup a'!! is a clique for each !!a, a'\in \mathcal M!!. Then !!\bigcup {\mathcal M}!! is also a clique.

To beginning math students it often seems like the sorts of definitions are generated at random. Okay, today we're going to study Eulerian preorders with no maximum element that are closed under finite unions; tomorrow we're going to study semispatulated coalgebras with countably infinite signatures and the weak Cosell property. Whatever, man.

I have a long article about this in progress, but I'll summarize: they are never generated at random. The properties are carefully chosen because we have something in mind that we are trying to model, and until you understand what that thing is, and why someone thinks those properties are the important ones, you are not going to get anywhere.

So when I see something like this I must stop immediately and ask ‘wat’. I can try to come up with the explanation myself, or I can read on hoping for an explanation, or I can do some of each, but I am not going to progress very far until I understand what it is about. And I'm not sure anyone short of Alexander Grothendieck would have any more success trying to move on with the definition itself and nothing else.

Girard explains shortly after:

The aim is to interpret a type by a coherence space !!\mathcal{A}!!, and a term of this type as a point [clique] of !!\mathcal{A}!!, infinite in general…

Okay, fine. I understand the point of the project, although not why the definition is what it is. I know a fair amount about types. And Girard has given two examples: booleans and integers. But these examples are unusually simple, because none of the cliques has more than one element, and so the examples are not as illuminating as they might be.

Some of the ways I tried to press onward were:

  1. Read ahead and see if there is more explanation. I tried this but I still wasn't getting it. The next section seemed clear: the cliques define a “coherence” relation on the tokens, from which the cliques can be recovered. Consider a graph, where the vertices are tokens and there is an edge !!a—b!! exactly when !!\{a, a'\}!! is a clique; we say that !!a!! and !!a'!! are coherent. Then the cliques of the coherence space are exactly the cliques of the graph; hence the name. The graph is called the web of the space, and from the web one can recover the original space.

    But after that part came stable functions, which I couldn't figure out, and I got stuck again.

  2. Read ahead and see if there is a more complicated specific example. There wasn't.

  3. Read ahead and see if any of the derived concepts are familiar, and if so then work backward. For instance, if I had been able to recognize that I already knew what stable functions were, I might have been able to leverage tha tinto an understanding of what was going on with the coherence spaces. But for me they were just another problem of the same sort: what is a stable function supposed to be modeling?

  4. Read someone else's explanation instead. I tried several without much success. They all seemed to be written for someone who already had a clue what was going on. (That is a large part of the reason I have written up this long and clueless explanation.)

  5. Try to construct some examples and see if they make sense in the context of what comes later. For example, I know what the coherence space of booleans looks like because Girard showed me. Can I figure out the structure of the coherence space for the type of “wrapped booleans”?

    -- (Haskell)
    data WrappedBoolean = W Bool

    Can I figure it out for the type of pairs of booleans?

    -- (Haskell)
    type BooleanPair = (Bool, Bool)

None of this was working. I had several different ideas about what the coherence spaces might look like for other types, but none of them seemed to fit with what Girard was doing. I couldn't come up with any consistent story.

So I prepared to ask on StackExchange, and I spent about an hour writing up my question, explaining all the things I had tried and what the problems were with each one. And as I drew near to the end of this, the clouds parted! I never had to post the question. I was in the middle of composing this paragraph:

In section 8.4 Girard defines a direct product of coherence spaces, but it doesn't look like the direct product I need to get a product type; it looks more like a disjoint union type. If the coherence space for Pairbool is the square of the coherence space for !!{{\mathcal B}ool}!!, how? It has 4 2-cliques, but if those are the total elements of !!{{\mathcal B}ool}^2!!, then what are do the 1-cliques mean?

I decided I hadn't made enough of an effort to understand the direct product. So even though I couldn't see how it could possibly give me anything like what I wanted, I followed its definition for !!{{\mathcal B}ool}^2!! — and the light came on.

Here's the puzzling coproduct-like definition of the product of two coherence spaces, from page 61:

If !!{\mathcal A}!! and !!{\mathcal B}!! are two coherence spaces, we define !!{\mathcal A}\&{\mathcal B}!! by:

!!|{\mathcal A}\&{\mathcal B}| = |{\mathcal A}| + |{\mathcal B}| = \{1\}×|{\mathcal A}| \cup \{2\}×|{\mathcal B}|!!

That is, the tokens in the product space are literally the disjoint union of the tokens in the component spaces.

And the edges in the product's web are whatever they were in !!{\mathcal A}!!'s web (except lifted from !!|{\mathcal A}|!! to !!\{1\}×|{\mathcal A}|!!), whatever they were in !!{\mathcal B}!!'s web (similarly), and also there is an edge between every !!\langle1, {\mathcal A}\rangle!! and each !!\langle2, {\mathcal B}\rangle!!. For !!{{\mathcal B}ool}^2!! the web looks like this:

There is no edge between !!\langle 1, \text{True}\rangle!! and !!\langle 1, \text{False}\rangle!! because in !!{{\mathcal B}ool}!! there is no edge between !!\text{True}!! and !!\text{False}!!.

This graph has nine cliques. Here they are ordered by set inclusion:

(In this second diagram I have abbreviated the pair !!\langle1, \text{True}\rangle!! to just !!1T!!. The top nodes in the diagram are each labeled with a set of two ordered pairs.)

What does this mean? The ordered pairs of booleans are being represented by functions. The boolean pair !!\langle x, y\rangle!! is represented by the function that takes as its argument a number, either 1 or 2, and then returns the corresponding component of the pair: the first component !!x!! if the argument was 1, and the second component !!y!! if the argument was 2.

The nodes in the bottom diagram represent functions. The top row are fully-defined functions. For example, !!\{1F, 2T\}!! is the function with !!f(1) = \text{False}!! and !!f(2) = \text{True}!!, representing the boolean pair !!\langle\text{False}, \text{True}\rangle!!. Similarly if we were looking at a space of infinite lists, we could consider it a function from !!\Bbb N!! to whatever the type of the lists elements was. Then the top row of nodes in the coherence space would be infinite sets of pairs of the form !!\langle n, \text{(list element)}\rangle!!.

The lower nodes are still functions, but they are functions about which we have only incomplete information. The node !!\{2T\}!! is a function for which !!f(2) = \text{True}!!. But we don't yet know what !!f(1)!! is because we haven't yet tried to compute it. And the bottommost node !!\varnothing!! is a function where we don't know anything at all — yet. As we test the function on various arguments, we move up the graph, always following the edges. The lower nodes are approximations to the upper ones, made on the basis of incomplete information about what is higher up.

Now the importance of finite approximants on page 56 becomes clearer. !!{{\mathcal B}ool}^2!! is already finite. But in general the space is infinite because the type is functions on an infinite domain, or infinite lists, or something of that sort. In such a space we can't get all the way to the top row of nodes because to do that we would have to call the function on all its possible arguments, or examine every element of the list, which is impossible. Girard says “Above all, there are enough finite approximants to a.” I didn't understand what he meant by “enough”. But what he means is that each clique !!a!! is the union of its finite approximants: each bit of information in the function !!a!! is obtainable from some finite approximation of !!a!!. The “stable functions” of section 8.3 start to become less nebulous also.

I had been thinking that the !!\varnothing!! node was somehow like the !!\bot!! element in a Scott domain, and then I struggled to identify anything like !!\langle \text{False}, \bot\rangle!!. It looks at first like you can do it somehow, because there are the right number of nodes at the middle level. Trouble arises in other coherence spaces.
For the WrappedBoolean type, for example, the type has four elements: !! W\ \text{True}, W\ \text{False}, W\ \bot,!! and !!\bot!!. I think the coherence space for WrappedBoolean is just like the one for !!{{\mathcal B}ool}!!:

Presented with a value from WrappedBoolean, you don't initially know what it is. Then you examine it, and you know whether it is !!W\ \text{True}!! or !!W\ \text{False}!!. You are now done.

I think there isn't anything like !!\bot!! or !!W\ \bot!! in the coherence space. Or maybe they they are there but sharing the !!\varnothing!! node. But I think more likely partial objects will appear in some other way.

Whew! Now I can move along.

(If you don't understand why “rubber duck”, Wikipedia explains:

Many programmers have had the experience of explaining a problem to someone else, possibly even to someone who knows nothing about programming, and then hitting upon the solution in the process of explaining the problem.

[“Rubber duck”] is a reference to a story in the book The Pragmatic Programmer in which a programmer would carry around a rubber duck and debug their code by forcing themselves to explain it, line-by-line, to the duck.

I spent a week on this but didn't figure it out until I tried formulating my question for StackExchange. The draft question, never completed, is here if for some reason you want to see what it looked like.)

[Other articles in category /math/logic] permanent link

Thu, 25 Jan 2018

Samuel Johnson and Ossian

For me, a little of Samuel Johnson goes a long way, because he was a tremendous asshole, and the draught is too strong for me to take much at once. But he is at his best when he is in opposition to someone who is an even bigger asshole, in this case James Macpherson.

Macpherson was a Scottish poet who perpetrated a major hoax for his own literary benefit. He claimed to have discovered and translated a collection of 3rd-century Gaelic manuscripts, written by a bard named Ossian, which he then published, with great commercial and critical success.

Thomas Bailey Saunders, in The Life and Letters of James Macpherson (1894), writes:

[Macpherson] was six-and-twenty, and flattered to the top of his bent. What happened with him was only what happens with most literary adventurers whose success is immediate and greater than their strict deserts; he contracted the foolish temper in which a man regards all criticism as ignorant and impertinent, and declines to take advice from any one.

Ossian and Macpherson did receive a great deal of criticism. Much of it was rooted in anti-Scottish bigotry, but many people at the time correctly suspected that Macpherson had written the "translations” himself, from scratch or nearly so. There was a great controversy, in which nobody participated more forcefully (or impolitely) than Johnson, a noted anti-Scottish bigot, who said that not only was the poems’ claimed history fraudulent, but that the poems themselves were rubbish.

The argument raged for some time. Johnson took up the matter in Journey to the Western Islands of Scotland (1775), in which he said:

I believe [the poems] never existed in any other form than that which we have seen. The editor, or author, never could shew the original; nor can it be shewn by any other; to revenge reasonable incredulity, by refusing evidence, is a degree of insolence, with which the world is not yet acquainted; and stubborn audacity is the last refuge of guilt. It would be easy to shew it if he had it; but whence could it be had?

Macpherson was furious to learn, before it was published, what Journey to the Western Islands would say, and attempted to prevent the passage from appearing in the book at all. When he discovered he was too late for this, he suggested that a slip of paper be inserted into the printed copies, apologizing and withdrawing the paragraph. This suggestion was ignored, and Johnson's book was published with no alteration.

Macpherson then challenged Johnson to a duel, and then, Johnson having declined, sent him a final letter, now lost, which a contemporary described as telling Johnson:

as he had declined to withdraw from his book the injurious expressions reflecting on Mr. Macpherson's private character, his age and infirmities alone protected him from the treatment due to an infamous liar and traducer.

(John Clark, An Answer to Mr. [William] Shaw's Inquiry, p. 51. Reprinted in Works of Ossian, vol. 1, 1783.)

Johnson's famous reply to this letter, quoted by Boswell, was:


I RECEIVED your foolish and impudent letter. Any violence offered me I shall do my best to repel; and what I cannot do for myself, the law shall do for me. I hope I shall never be deterred from detecting what I think a cheat, by the menaces of a ruffian.

“What would you have me retract? I thought your book an imposture; I think it an imposture still. For this opinion I have given my reasons to the publick, which I here dare you to refute. Your rage I defy. Your abilities, since your Homer, are not so formidable; and what I hear of your morals inclines me to pay regard not to what you shall say, but to what you shall prove. You may print this if you will.

“SAM. Johnson.”

Both Macpherson and Johnson are buried in Westminster Abbey. Some say Macpherson bought his way in.

[Other articles in category /book] permanent link

Mon, 22 Jan 2018

The gods of Stackexchange karma are fickle

A few years ago an se.math user asked why their undistinguished answer to some humdrum question had gotten so many upvotes. I replied:

It's the gods of Stackexchange Karma evening your score for that really clever answer you posted two weeks ago that still has 0 upvotes.

If you're going to do Stack Exchange, I think it's important not to stress out about the whys of the votes, and particularly important not to take them personally. The karma gods do not always show the most refined taste. As Brandon Tartikoff once said, “All hits are flukes”.

Today I got an unusually flukey hit. But first, here are some nice examples in the opposite direction, posts that I put a lot of trouble and effort into, which were clear and useful, helpful to the querent, and which received no upvotes at all.

Here the querent asked, suppose I have several nonstandard dice with various labeling on the faces. Player A rolls some of the dice, and Player B rolls a different set of dice. How do I calculate things like “Player A has an X% chance of rolling a higher total”?

Mathematicians are not really the right people to ask this to, because many of them will reply obtusely, informing you that it depends on how many dice are rolled and on how their faces are actually labeled, and that the question did not specify these, but that if it had, the problem would be trivial. (I thought there were comments to that effect on this question, but if there were they have been deleted. In any case nobody else answered.) But this person was writing a computer game and wanted to understand how to implement a computer algorithm for doing the calculation. There is a lot one can do to help this person. I posted an answer that I thought was very nice. The querent liked it, but it got zero upvotes. This happens quite often. Which is fine, because the fun of doing it and the satisfaction of helping someone are reward enough.

Like that one, many of these questions are ignored because they aren't mathematically interesting. Or sometimes the mathematics is simple but the computer implementation is not.

Sometimes I do them just because I want to know the particular answer. (How much of an advantage does Player A get from being allowed to add 1 to their total?)

Sometimes there's an interesting pedagogical problem, such as: how do I give a hint that will point in the right direction without giving away the whole secret? Or: how do I wade through this person's confused explanation and understand what they are really asking, or what they are really confused about? That last person was given a proof that glossed over six or seven steps in the reasoning, focusing instead on the technically interesting induction proof in the middle. The six or seven steps are straightforward, if you already have enough practice with logical reasoning about quantified statements. But this querent didn't follow the reasoning that led up to the induction, so they didn't understand why the induction was useful or what it was for.

Some people are just confused about what the question is. That person has a complicated-seeming homework exercise about the behavior of the logistic map, and need helps interpreting it from someone who has a better idea what is going on. The answer didn't require any research effort, and it's mathematically uninteresting, but it did require attention from someone who has a better idea of what is going on. The querent was happy, I was happy, and nobody else noticed.

I never take the voting personally, because the gods of Stackexchange karma are so fickle, and today I got a reminder of that. Today's runaway hit was for a complete triviality that I knocked off in two minutes. It currently has 94 upvotes and seems likely to get a gold medal. (That gold medal, plus $4.25, will get me a free latte at Starbuck's!)

The question asks how to find a prime factor of 7,999,973 without using a calculator.

It's one of those easy-if-you-happen-to-know-the-trick things, and I just happened to get there before one of the (many) other people who happens to know the trick.

As we used to say in the system administration biz, some days you're an idiot if you can't explain how to do real-time robot arm control in Unix, other days you're a genius if you fix their terminal by plugging it in.

[ Addendum 20180123: I got the gold medal. Woo-hoo, free latte! ]

[Other articles in category /math] permanent link

Sat, 20 Jan 2018

I forgot there are party conventions

Yesterday I made a huge mistake in my article about California's bill requiring presidential candidates to disclose their tax returns. I asked:

Did I miss anything?

Yes, yes I did. I forgot that party nominees are picked not by per-state primary elections, but by national conventions. Even had Ronnie won the California Republican primary election, rather than Trump, that would not be enough the get him on the ballot in the general election.

In the corrected version of my scenario, California sends many Ronnie supporters to the Republican National Convention, where the other delegates overwhelmingly nominate Trump anyway. Trump then becomes the Republican party candidate and appears on the California general election ballot in November. Whoops.

I had concluded that conceding California couldn't hurt Trump, and it could actually a huge benefit to him. After correcting my mistake, most of the benefit evaporates.

I wonder if Trump might blow off California in 2020 anyway. The upside would be that he could spend the resources in places that might give him some electoral votes. (One reader pointed out that Trump didn't blow off California in the 2016 election, but of course the situation was completely different. In 2016 he was not the incumbent, he was in a crowded field, and he did not yet know that he was going to lose California by 30%.)

Traditionally, candidates campaign even in states they expect to lose. One reason is to show support for candidates in local elections. I can imagine many California Republican candidates would prefer that Trump didn't show up. Another reason is to preserve at least a pretense that they are representatives of the whole people. Newly-elected Presidents have often upheld the dignity of the office by stating the need for unity after a national election and promising (however implausibly) to work for all Americans. I don't care to write the rest of this paragraph.

The major downside that I can think of (for Trump) is that Republican voters in California would be infuriated, and while they can't directly affect the outcome of the presidential election, they can make it politically impossible for their congressional representatives to work with Trump once he is elected. A California-led anti-Trump bloc in Congress would probably cause huge problems for him.

Wild times we live in.

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Fri, 19 Jan 2018

Presidential tax return disclosure

The California state legislature passed a bill that would require presidential candidates to disclose their past five tax returns in order to qualify for California primary elections. The bill was vetoed by Governor Brown, but what if it had become law?

Suppose Donald Trump ran for re-election in 2020, as seems likely, barring his death or expulsion. And suppose he declined once again to disclose his tax returns, and was excluded from the California Republican primary election. I don't see how this could possibly hurt Trump, and it could benefit him.

It doesn't matter to Trump whether he enters the primary or wins the primary. Trump lost California by 30% in 2016. Either way he would be just as certain to get the same number of electors: zero. So he would have no incentive to comply with the law by releasing his tax returns.

Most candidates would do it anyway, because they try to maintain a pretense of representing the entire country they are campaigning to lead, but Trump is really different in this way. I can easily imagine he might simply refuse to campaign in California, instead dismissing the entire state with some vulgar comment. If there is a downside for Trump, I don't see what it could be.

Someone else (call them “Ronnie”) would then win the California Republican primary. Certainly Ronnie is better-qualified and more competent than Trump, and most likely Ronnie is much more attractive to the California electorate.

Ronnie might even be more attractive than the Democratic candidate, and might defeat them in the general election, depriving Trump's challenger of 55 electoral votes and swinging the election heavily in Trump's favor.

Did I miss anything?

[ Addendum 20180120: Yeah, I forgot that after the primary there is a convention that nominates a national party candidate. Whooops. Further discussion. ]

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Thu, 18 Jan 2018

England in a pother in 1533

In autumn 2014 I paid for something and got $15.33 in change. I thought I'd take the hint from the Universe and read Wikipedia's article on the year 1533. This turned out unexpectedly exciting. 1533 was a big year in English history. Here are the highlights:

  • January 25: King Henry VIII marries Anne Boleyn
  • March 30: Thomas Cranmer becomes Archbishop of Canterbury
  • May 23: Cranmer annuls Henry's marriage to Catherine of Aragon
  • June 1: Boleyn crowned queen consort by Cranmer
  • July 11: Henry excommunicated by Pope Clement VII
  • September 7: Princess Elizabeth, later Queen Elizabeth I, is born to Boleyn

A story clearly emerges here, the story of Henry's frantic response to Anne Boleyn's surprise pregnancy.

The first thing to notice is that Elizabeth was born only seven months after Henry married Boleyn. The next thing to notice is that Henry was still married to Catherine when he married Boleyn. He had to get Cranmer to annul the marriage, issuing a retroactive decree that not only was Henry not married to Catherine, but he had never been married to her.

In 2014 I imagined that Henry appointed Cranmer to be Archbishop on condition that he get the annulment, and eventually decided that was not the case. Looking at it now, I'm not sure why I decided that.

Wikipedia says:

While Cranmer was following Charles through Italy, he received a royal letter dated 1 October 1532 informing him that he had been appointed the new Archbishop of Canterbury, following the death of archbishop William Warham [on 22 August]. Cranmer was ordered to return to England. The appointment had been secured by the family of Anne Boleyn, who was being courted by Henry.

Cranmer had been working on that annulment since at least 1527. In 1532 he was ambassador to Charles V, the Holy Roman Emperor, who was the nephew of Henry's current wife Catherine. I suppose a large part of Cranmer's job was trying to persuade Charles to support the annulment. (He was unsuccessful.) When Charles conveniently went to Rome (what for? Wikipedia doesn't say) Cranmer followed him and tried to drum up support there for the annulment. (He was unsuccessful in that too.)

Fortunately there was a convenient vacancy, and Henry called him back to fill it, and got his annulment that way. In 2014 I thought Warham's death was just a little too convenient, but I decided that he died too early for it to have been arranged by Henry. Now I'm less sure — Henry was certainly capable of such cold-blooded planning — but I can't find any mention of foul play, and The Divorce of Henry VIII: The Untold Story from Inside the Vatican describes the death as “convenient though entirely natural”.

[ Addendum: This article used to say that Elizabeth was born “less than seven months” after Henry and Boleyn's marriage. Daniel Holtz has pointed out that this was wrong. The exact amount is 225 days, or 32 weeks plus one day. The management regrets the error. ]

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Tue, 16 Jan 2018

Plutonium collection

In an earlier article, I said:

If I were in charge of keeping plutonium out of the wrong hands, I would still worry about [people extracting it from plutonium-fueled pacemakers].

This turns out to be no worry at all. The isotope in the pacemaker batteries is Pu-238, which is entirely unsuitable for making weapons. Pu-238 is very dangerous, being both radioactive and highly poisonous, but it is not fissile. In a fission chain reaction, an already-unstable atomic nucleus is hit by a high-energy neutron, which causes it to fragment into two lighter nuclei. This releases a large amount of nuclear binding energy, and more neutrons which continue the reaction. The only nuclei that are unstable enough for this to work have an odd number of neutrons (for reasons I do not understand), and Pu-238 does not fit the bill (Z=94, N=144). Plutonium fission weapons are made from Pu-241 (N=147), and this must be carefully separated from the Pu-238, which tends to impede the chain reaction. Similarly, uranium weapons are made from U-235, and this must be painstakingly extracted from the vastly more common U-238 with high-powered centrifuges.

But I did not know this when I spent part of the weekend thinking about the difficulties of collecting plutonium from pacemakers, and discussing it with a correspondent. It was an interesting exercise, so I will publish it anyway.

While mulling it over I tried to identify the biggest real risks, and what would be the most effective defenses against them. An exercise one does when considering security problems is to switch hats: if I were the bad guy, what would I try? What problems would I have to overcome, and what measures would most effectively frustrate me? So I put on my Black Hat and tried to think about it from the viewpoint of someone, let's call him George, who wants to build a nuclear weapon from pacemaker batteries.

I calculated (I hope correctly) that a pacemaker had around 0.165 mg of plutonium, and learned online that one needs 4–6 kg to make a plutonium bomb. With skill and experience one can supposedly get this down to 2 kg, but let's take 25,000 pacemakers as the number George would need. How could he get this much plutonium?

(Please bear in mind that the following discussion is entirely theoretical, and takes place in an imaginary world in which plutonium-powered pacemakers are common. In the real world, they were never common, and the last ones were manufactured in 1974. And this imaginary world exists in an imaginary universe in which plutonium-238 can sustain a chain reaction.)

Obviously, George's top target would be the factory where the pacemakers are made. Best of all is to steal the plutonium before it is encapsulated, say just after it has been delivered to the factory. But equally obviously, this is where the security will be the most concentrated. The factory is not as juicy a target as it might seem at first. Plutonium is radioactive and toxic, so they do not want to have to store a lot of it on-site. They will have it delivered as late as possible, in amounts as small as possible, and use it up as quickly as possible. The chances of George getting a big haul of plutonium by hitting the factory seem poor.

Second-best is for George to steal the capsules in bulk before they are turned into pacemakers. Third-best is for him to steal cartons of pacemakers from the factory or from the hospitals they are delivered to. But bulk theft is not something George can pull off over and over. The authorities will quickly realize that someone is going after pacemakers. And after George's first heist, everyone will be looking for him.

If the project gets to the point of retrieving pacemakers after they are implanted, George's problems multiply enormously. It is impractical to remove a pacemaker from of a living subject. George would need to steal them from funeral homes or crematoria. These places are required to collect the capsules for return to Oak Ridge, and conceivably might sometimes have more than one on hand at a time, but probably not more than a few. It's going to be a long slog, and it beggars belief that George would be able to get enough pacemakers this way without someone noticing that something was up.

The last resort is for George to locate people with pacemakers, murder, and dissect them. Even if George somehow knows whom to kill, he'd have to be Merlin to arrange the murder of 25,000 people without getting caught. Merlin doesn't need plutonium; he can create nuclear fireballs just by waving his magic wand.

If George does manage to collect the 25,000 capsules, his problems get even worse. He has to open the titanium capsules, already difficult because they are carefully made to be hard to open — you wouldn't want the plutonium getting out, would you? He has to open them without spilling the plutonium, or inhaling it, or making any sort of mess while extracting it. He has to do this 25,000 times without messing up, and without ingesting the tiniest speck of plutonium, or he is dead.

He has to find a way to safely store the plutonium while he is accumulating it. He has to keep it hidden not only from people actively looking for him — and they will be, with great yearning — but also from every Joe Blow who happens to be checking background radiation levels in the vicinity.

And George cannot afford to take his time and be cautious. He is racing against the clock, because every 464 days, 1% of his accumulated stock, however much that is, will turn into U-234 and be useless. The more he accumulates, the harder it is to keep up. If George has 25,000 pacemakers in a warehouse, ready for processing, one pacemaker-worth of Pu-238 will be going bad every two days.

In connection with this, my correspondent brought up the famous case of the Radioactive Boy Scout, which I had had in mind. (The RBS gathered a recklessly large amount of americium-241 from common household smoke detectors.) Ignoring again the unsuitability of americium for fission weapons (an even number of neutrons again), the project is obviously much easier. At the very least, you can try calling up a manufacturer of smoke alarms, telling them you are building an apartment complex in Seoul, and that you need to bulk-order 2,000 units or whatever. You can rob the warehouse at Home Depot. You can even buy them online.

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Sun, 14 Jan 2018

How do plutonium-powered pacemakers work?

I woke up in the middle of the night wondering: Some people have implanted medical devices, such as pacemakers, that are plutonium-powered. How the hell does that work? The plutonium gets hot, but what then? You need electricity. Surely there is not a tiny turbine generator in there!

There is not, and the answer turns out to be really interesting, and to involve a bunch of physics I didn't know.

If one end of a wire is hotter than the other, electrons tend to diffuse from the hot end to the cold end; the amount of diffusion depends on the material and the temperature. Take two wires of different metals and join them into a loop. (This is called a thermocouple.) Make one of the joints hotter than the other. Electrons will diffuse from the hot joint to the cold joint. If there were only one metal, this would not be very interesting. But the electrons diffuse better through one wire (say wire A) than through the other (B), and this means that there will be net electron flow from the hot side down through wire A and then back up through B, creating an electric current. This is called the Seebeck effect. The potential difference between the joints is proportional to the temperature difference, on the order of a few hundred microvolts per kelvin. Because of this simple proportionality, the main use of the thermocouple is to measure the temperature difference by measuring the voltage or current induced in the wire. But if you don't need a lot of power, the thermocouple can be used as a current source.

In practice they don't use a single loop, but rather a long loop of alternating metals, with many junctions:

A long row of conductors alternately of two different materials,
  each joined in series to the end of the next, snaking back and forth
  so that all the A-B junctions are on top and all the B-A junctions
  on the bottom.  Between the top and bottom is an insulating layer.
  The top set of junctions is heated.  Heat flows from top to bottom
  and creates a current in the series of

This is called a thermopile; when the heat source is radioactive material, as here, the device is called a radioisotope thermoelectric generator (RTG). The illustration shows the thermocouples strung out in a long line, but in an actual RTG you put the plutonium in a capsule and put the thermocouples in the wall of the capsule, with the outside joints attached to heat sinks. The plutonium heats up the inside joints to generate the current.

RTGs are more commonly used to power spacecraft, but there are a few dozen people still in the U.S. with plutonium-powered thermopile batteries in their pacemakers.

In pacemakers, the plutonium was sealed inside a titanium capsule, which was strong enough to survive an accident (such as a bullet impact or auto collision) or cremation. But Wikipedia says the technique was abandoned because of worries that the capsule wouldn't be absolutely certain to survive a cremation. (Cremation temperatures go up to around 1000°C; titanium melts at 1668°C.) Loose plutonium in the environment would be Very Bad.

(I wondered if there wasn't also worry about plutonium being recovered for weapons use, but the risk seems much smaller: you need several kilograms of plutonium to make a bomb, and a pacemaker has only around 135 mg, if I did the conversion from curies correctly. Even so, if I were in charge of keeping plutonium out of the wrong hands, I would still worry about this. It does not seem totally out of the realm of possibility that someone could collect 25,000 pacemakers. Opening 25,000 titanium capsules does sound rather tedious.)

Earlier a completely different nuclear-powered pacemaker was tried, based on promethium-powered betavoltaics. This is not a heat-conversion process. Instead, a semiconductor does some quantum physics magic with the electrons produced by radioactive beta decay. This was first demonstrated by Henry Moseley in 1913. Moseley is better-known for discovering that atoms have an atomic number, thus explaining the periodic table. The periodic table had previously been formulated in terms of atomic mass, which put some of the elements in the wrong order. Scientists guessed they were in the wrong order, because the periodicity didn't work, but they weren't sure why. Moseley was able to compute the electric charge of the atomic nucleus from spectrographic observations. I have often wondered what else Moseley would have done if he had not been killed in the European war at the age of 27.

It took a while to gather the information about this. Several of Wikipedia's articles on the topic are not up to their usual standards. The one about the radioisotope thermoelectric generator is excellent, though.

Thermopile illustration is by FluxTeq (Own work) CC BY-SA 4.0, via Wikimedia Commons.

[ Addendum 20180115: Commenters on Hacker News have pointed out that my concern about the use of plutonium in fission weapons is easily satisfied: the fuel in the batteries is Pu-238, which is not fissile. The plutonium to use in bombs is Pu-241, and indeed, when building a plutonium bomb you need to remove as much Pu-238 as possible, to prevent its non-fissile nuclei from interfering with the chain reaction. Interestingly, you can tell this from looking at the numbers: atomic nuclei with an odd number of neutrons are much more fissile than those with an even number. Plutonium is atomic number 94, so Pu-238 has an even number of neutrons and is not usable. The other isotope commonly used in fission is U-235, with 143 neutrons. I had planned to publish a long article today detailing the difficulties of gathering enough plutonium from pacemakers to make a bomb, but now I think I might have to rewrite it as a comedy. ]

[ Addendum 20170116: I published it anyway, with some editing. ]

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Sun, 07 Jan 2018

The horn of mead

Well, yesterday I wrote an article about the drinking contest in the Gylfaginning and specifically about what was in the horn. I was very pleased with it. In the article, I said several times:

Obviously, the horn was full of mead.

A couple of my Gentle Readers have gently pointed out that I was wrong, wrong, wrong. I am deeply embarrassed.

The punch line of the story is that the end of the horn is attached to the ocean, and Thor cannot empty it, because he is trying to drink the ocean. The horn is therefore not filled with mead; it is filled with seawater.

How could I make such a dumb mistake? As I mentioned, the version I read first as a child stated that the horn was full of milk. And as a child I wondered: how could the horn be full of milk if it was attached to the sea? I decided that whatever enchantment connected the horn to the sea also changed the water to milk as it came into the horn. Later, when I realized that the milk was a falsehood, I retained my idea that there was an enchantment turning the seawater into something else.

But there is nothing in the text to support this. The jötunns don't tell Thor that the horn is full of mead. Adam Sjøgren pointed out that if they had, Thor would have known immediately that something was wrong. But as the story is, they bring the horn, they say that even wimps can empty it in three draughts, and they leave it at that. Wouldn't Thor notice that he is not drinking mead (or milk)? I think certainly, and perhaps he is initially surprised. But he is in a drinking contest and this is what they have brought him to drink, so he drinks it. The alternative is to put down the horn and complain, which would be completely out of character.

And the narrator doesn't say, and mustn't, that the horn was full of mead, because it wasn't; that would be in impermissible deceit of the audience. (“Hey, wait, you told us before that the horn was full of mead!”)

I wrote:

The story does not say what was in the horn. Because why would they bother to say what was in the horn? It was obviously mead.

No, it's not. It's because the narrator wants us to assume it is obviously mead, and then to spring the surprise on us as it was sprung on Thor: it was actually the ocean. The way it is told is a clever piece of misdirection. The two translators I quoted picked up on this, and I completely misunderstood it.

I have mixed feelings about Neil Gaiman, but Veit Heller pointed out that Gaiman understood this perfectly. In his Norse Mythology he tells the story this way:

He raised the brimming horn to his lips and began to drink. The mead of the giants was cold and salty…

In yesterday's article I presented a fantasy of Marion French, the author of my childhood “milk” version, hearing Snorri tell the story:

“Utgarða-Loki called his skutilsveinn, and requested him to bring the penalty-horn that his hirðmen were wont to drink from…”

“Excuse me! Excuse me, Mr. Sturluson! Just what were they wont to drink from it?”

“Eh, what's that?”

”What beverage was in the horn?”

“Why, mead, of course. What did you think it was, milk?”

But this couldn't have been how it went down. I now imagine it was more like this:

“Excuse me! Excuse me, Mr. Sturluson! Just what were they wont to drink from it?”

“Shhh! I'm getting to that! Stop interrupting!”

Thanks again to Adam Sjøgren and Veit Heller for pointing out my error, and especially for not wounding my pride any more than they had to.

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The horn of milk

When I was a kid I had a book of “Myths and Legends of the Ages”, by Marion N. French. One of the myths was the story of Thor's ill-fated visit to Utgard. The jötunns of Utgard challenge Thor and Loki to various contests and defeat them all through a combination of talent and guile. In one of these contests, Thor is given a drinking horn and told that even the wimpiest of the jötunns is able to empty it of its contents in three drinks. (The jötunns are lying. The pointy end of the horn has been invisibly connected to the ocean.)

The book specified that the horn was full of milk, and as a sweet and innocent kiddie I did not question this. Decades later it hit me suddenly: no way was the horn filled with milk. When the mighty jötunns of Utgard are sitting around in their hall, they do not hold contests to see who can drink the most milk. Obviously, the horn was full of mead.

The next sentence I wrote in the draft version of this article was:

   In the canonical source material (poetic edda maybe?) the horn is full
   of *mead*. Of course it is.

In my drafts, I often write this sort of bald statement of fact, intending to go back later and check it, and perhaps produce a citation. As the quotation above betrays, I was absolutely certain that when I hunted down the original source it would contradict Ms. French and say mead. But I have now hunted down the canonical source material (in the Prose Edda, it turns out, not the Poetic one) and as far as I can tell it does not say mead!

Here is an extract of an 1880 translation by Rasmus Björn Anderson, provided by WikiSource:

He went into the hall, called his cup-bearer, and requested him to take the sconce-horn that his thanes were wont to drink from. The cup-bearer immediately brought forward the horn and handed it to Thor. Said Utgard-Loke: From this horn it is thought to be well drunk if it is emptied in one draught, some men empty it in two draughts, but there is no drinker so wretched that he cannot exhaust it in three.

For comparison, here is the 1916 translation of Arthur Gilchrist Brodeur, provided by

He went into the hall and called his serving-boy, and bade him bring the sconce-horn which the henchmen were wont to drink off. Straightway the serving-lad came forward with the horn and put it into Thor's hand. Then said Útgarda-Loki: 'It is held that this horn is well drained if it is drunk off in one drink, but some drink it off in two; but no one is so poor a man at drinking that it fails to drain off in three.'

In both cases the following text details Thor's unsuccessful attempts to drain the horn, and Utgard-Loki's patronizing mockery of him after. But neither one mentions at any point what was in the horn.

I thought it would be fun to take a look at the original Old Norse to see if the translators had elided this detail, and if it would look interesting. It was fun and it did look interesting. Here it is, courtesy of Heimskringla.NO:

Útgarða-Loki segir, at þat má vel vera, ok gengr inn í höllina ok kallar skutilsvein sinn, biðr, at hann taki vítishorn þat, er hirðmenn eru vanir at drekka af. Því næst kemr fram skutilsveinn með horninu ok fær Þór í hönd. Þá mælti Útgarða-Loki: "Af horni þessu þykkir þá vel drukkit, ef í einum drykk gengr af, en sumir menn drekka af í tveim drykkjum, en engi er svá lítill drykkjumaðr, at eigi gangi af í þrimr."

This was written in Old Norse around 1220, and I was astounded at how much of it is recognizable, at least when you already know what it is going to say. However, the following examples are all ill-informed speculation, and at least one of my confident claims is likely to be wrong. I hope that some of my Gentle Readers are Icelanders and can correct my more ridiculous errors.

“Höllina” is the hall. “Kallar” is to call in. The horn appears three times, as ‘horninu’, ‘horni’, and in ‘vítishorn’, which is a compound that specifies what kind of horn it is. “Þór í hönd” is “in Thor's hand”. (The ‘Þ’ is pronounced like the /th/ of “Thor”.) “Drekka”, “drukkit”, “drykk”, “drykkjum”, and “drykkjumaðr” are about drinking or draughts; “vel drukkit” is “well-drunk”. You can see the one-two-three in there as “einum-tveim-þrimr”. (Remember that the “þ” is a /th/.) One can almost see English in:

sumir menn drekka af í tveim drykkjum

which says “some men drink it in two drinks”. And “lítill drykkjumaðr” is a little-drinking-person, which I translated above as “wimp”.

It might be tempting to guess that “með horninu” is a mead-horn, but I'm pretty sure it is not; mead is “mjað” or “mjöð”. I'm not sure, but I think “með” here is just “with”, akin to modern German “mit”, so that:

næst kemr fram skutilsveinn með horninu

is something like “next, the skutilsveinn came with the horn”. (The skutilsveinn is something we don't have in English; compare trying to translate “designated hitter” into Old Norse.)

For a laugh, I tried putting this into Google Translate, and I was impressed with the results. It makes a heroic effort, and produces something that does capture some of the sense of the passage. It identifies the language as Icelandic, which while not correct, isn't entirely incorrect either. (The author, Snorri Sturluson, was in fact Icelandic.) Google somehow mistakes the horn for a corner, and it completely fails to get the obsolete term “hirðmenn” (roughly, “henchmen”), mistaking it for herdsmen. The skutilsveinn is one of the hirðmenn.

Anyway there is no mead here, and none in the rest of the story, which details Thor's unsuccessful attempts to drink the ocean. Nor is there any milk, which would be “mjólk”.

So where does that leave us? The jötunns challenge Thor to a drinking contest, and bring him a horn, and even though it was obviously mead, the story does not say what was in the horn.

Because why would they bother to say what was in the horn? It was obviously mead. When the boys crack open a cold one, you do not have to specify what it was that was cold, and nobody should suppose that it was a cold bottle of milk.

I imagine Marion N. French sitting by the fire, listening while Snorri tells the story of Thor and the enchanted drinking horn of Utgard:

“Utgarða-Loki called his skutilsveinn, and requested him to bring the penalty-horn that his hirðmen were wont to drink from…”

“Excuse me! Excuse me, Mr. Sturluson! Just what were they wont to drink from it?”

“Eh, what's that?”

”What beverage was in the horn?”

“Why, mead, of course. What did you think it was, milk?”

(Merriment ensues, liberally seasoned with patronizing mockery.)

(In preparing this article, I found it helpful to consult Zoëga's Concise Dictionary of Old Icelandic of 1910.)

[ Addendum 2018-01-17: Holy cow, I was so wrong. It was so obviously not mead. I was so, so wrong. Amazingly, unbelievably wrong. ]

[ Addendum 2018-03-22: A followup in which I investigate what organs Skaði looked at when choosing her husband, and what two things Loki tied together to make her laugh. ]

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Fri, 05 Jan 2018

Hebrew John Doe

Last month I wrote about the Turkish analog of “Joe Blow”. I got email from Gaal Yahas, who said

I bet you'll get plenty of replies on your last post about translating "John Doe" to different languages.

Sadly no. But M. Yahas did tell me in detail about the Hebrew version, and I did a little additional research.

The Hebrew version of “Joe Blow” / “John Doe” is unequivocally “Ploni Almoni”. This usage goes back at least to the Book of Ruth, approximately 2500 years ago. Ruth's husband has died without leaving an heir, and custom demands that a close relative of her father-in-law should marry her, to keep the property in the family. Boaz takes on this duty, but first meets with another man, who is a closer relative than he:

Then went Boaz up to the gate, and sat him down there: and, behold, the kinsman of whom Boaz spake came by; unto whom he said, Ho, such a one! turn aside, sit down here. And he turned aside, and sat down.

(Ruth 4:1, KJV)

This other relative declines to marry Ruth. He is not named, and is referred to in the Hebrew version as Ploni Almoni, translated here as “such a one”. This article in The Jewish Chronicle discusses the possible etymology of these words, glossing “ploni” as akin to “covered” or “hidden” and “almoni” as akin to “silenced” or “muted”.

Ploni Almoni also appears in the book of Samuel, probably even older than Ruth:

David answered Ahimelek the priest, “The king sent me on a mission and said to me, 'No one is to know anything about the mission I am sending you on.' As for my men, I have told them to meet me at a certain place.”

(1 Samuel 21:2, NIV)

The mission is secret, so David does not reveal the meeting place to Ahimelek. Instead, he refers to it as Ploni Almoni. There is a similar usage at 2 Kings 6:8.

Apparently the use of “Ploni” in Hebrew to mean “some guy” continues through the Talmud and up to the present day. M. Yahas also alerted me to two small but storied streets in Tel Aviv. According to this article from Haaretz:

A wealthy American businessman was buying up chunks of real estate in Tel Aviv. He purchased the two alleyways with the intention of naming them after himself and his wife, even going so far as to put up temporary shingles with the streets’ new names. But he had christened the streets without official permission from the city council.

The mayor was so incensed by the businessman’s chutzpah that he decided to temporarily name the alleyways Simta Almonit and Simta Plonit.

And so they remain, 95 years later.

(M. Yahas explains that “Simta” means “alley” and is feminine, so that Ploni and Almoni take the feminine ‘-it’ ending to agree with it.)

Wikipedia has not one but many articles on this topic and related ones:

My own tiny contribution in this area: my in-laws live in a rather distant and undeveloped neighborhood on the periphery of Seoul, and I once referred to it as 아무데도동 (/amudedo-dong/), approximately “nowhereville”. This is not standard in Korean, but I believe the meaning is clear.

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Tue, 02 Jan 2018

Shitpost roundup, 2017-12

As I mentioned before, I have started another blog, called Content-type: text/shitpost.

The shitposts have been suffering quality creep and I am making an effort to lower my standards. I will keep you posted about how this develops. (I don't think the quality creep was the cause of lower volume this month; rather, I was on vacation for a week.)

Here is a list of last month's shitposts. I have added a short blurb to those that may be of more general interest.

I plan to continue to post monthly summaries here.

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Converting Google Docs to Markdown

I was on vacation last week and I didn't bring my computer, which has been a good choice in the past. But I did bring my phone, and I spent some quiet time writing various parts of around 20 blog posts on the phone. I composed these in my phone's Google Docs app, which seemed at the time like a reasonable choice.

But when I got back I found that it wasn't as easy as I had expected to get the documents back out. What I really wanted was Markdown. HTML would have been acceptable, since Blosxom accepts that also. I could download a single document in one of several formats, including HTML and ODF, but I had twenty and didn't want to do them one at a time. Google has a bulk download feature, to download a zip file of an entire folder, but upon unzipping I found that all twenty documents had been converted to Microsoft's docx format and I didn't know a good way to handle these. I could not find an option for a bulk download in any other format.

Several tools will compose in Markdown and then export to Google docs, but the only option I found for translating from Google docs to Markdown was Renato Mangini's Google Apps script. I would have had to add the script to each of the 20 files, then run it, and the output appears in email, so for this task, it was even less like what I wanted.

The right answer turned out to be: Accept Google's bulk download of docx files and then use Pandoc to convert the docx to Markdown:

for i in *.docx; do
    echo -n "$i ? ";
    read j; mv -i "$i" $j.docx;
    pandoc --extract-media . -t markdown -o "$(suf "$j" mkdn)" "$j.docx";

The read is because I had given the files Unix-unfriendly names like Polyominoes as orthogonal polygons.docx and I wanted to give them shorter names like orthogonal-polyominoes.docx.

The suf command is a little utility that performs the very common task of removing or changing the suffix of a filename. The suf "$j" mkdn command means that if $j is something like foo.docx it should turn into foo.mkdn. Here's the tiny source code:

    # Usage: suf FILENAME [suffix]
    # If filename ends with a suffix, the suffix is replaced with the given suffix
    # otheriswe, the given suffix is appended
    # For example:
    #   suf baz    => foo.baz
    #   suf foo     baz    => foo.baz
    #   suf        => foo
    #   suf foo            => foo

    @ARGV == 2 or @ARGV == 1 or usage();
    my ($file, $suf) = @ARGV;
    $file =~ s/\.[^.]*$//;
    if (defined $suf) {
      print "$file.$suf\n";
    } else {
      print "$file\n";

    sub usage {
      print STDERR "Usage: suf filename [newsuffix]\n";
      exit 1;

Often, I feel that I have written too much code, but not this time. Some people might be tempted to add bells and whistles to this: what if the suffix is not delimited by a dot character? What if I only want to change certain suffixes? What if my foot swells up? What if the moon falls out of the sky? Blah blah blah. No, for that we can break out sed.

Next time I go on vacation I will know better and I will not use Google Docs. I don't know yet what instead. StackEdit maybe.

[ Addendum 20180108: Eric Roode pointed out that the program above has a genuine bug: if given a filename like a.b/c.d it truncates the entire b/c.d instead of just the d. The current version fixes this. ]

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