# The Universe of Discourse

Fri, 28 Jan 2022

Yesterday I was thinking on these creepy Munchkins, and wondering what they were doing there:

It occurred to me that these guys are quite consistent with the look of the original illustrations, by W.W. Denslow. Here's Denslow's picture of three Munchkins greeting Dorothy:

(Click for complete illustration.)

Denslow and Frank Baum had a falling out after the publication of The Wonderful Wizard of Oz, and the illustrations for the thirteen sequels were done by John R. Neill, in a very different style. Dorothy aged up to eleven or twelve years old, and became a blonde with a fashionable bob.

Thu, 27 Jan 2022

I just randomly happened upon this recording of Pippa Evans singing “How Much is that Doggie in the Window” to the tune of “Cabaret”, and this reminded me of something I was surprised I hadn't mentioned before.

In the 1939 MGM production of The Wizard of Oz, there is a brief musical number, The Lollipop Guild, that has the same music as the refrain of Money, also from Cabaret. I am not aware of anyone else who has noticed this.

One has the lyrics “money makes the world go around” and the other has “We represent the lollipop guild”. And the two songs not only have the same rhythm, but the same melody and both are accompanied by the same twitchy, mechanical dance, performed by three creepy Munchkins in one case and by creepy Liza Minelli and Joel Grey in the other.

Surely the writers of Cabaret didn't do this on purpose? Did they? While it seems plausible that they might have forgotten the “Lollipop Guild” bit, I think it's impossible that they could both have missed it completely; they would have been 11 and 12 years old when The Wizard of Oz was first released.

(Now I want to recast The Wizard of Oz with Minelli as Dorothy and Grey as the Wizard. Bonus trivia, Liza Minelli is Judy Garland's daughter. Bonus bonus trivia, Joel Grey originated the role of the Wizard in the stage production of Wicked).

I have this nice little utility program called menupick. It's a filter that reads a list of items on standard input, prompts the user to select one or more of them, then prints the selected items on standard output. So for example:

    emacs $(ls *.blog | menupick)  displays a list of those files and a prompt:  0. Rocketeer.blog 1. Watchmen.blog 2. death-of-stalin.blog 3. old-ladies.blog 4. self-esteem.blog >  Then I can type 1 2 4 to select items 1, 2, and 4, or 1-4 !3 (“1 through 4, but not 3”) similarly. It has some other features I use less commonly. It's a useful component in other commands, such as this oneliner git-addq that I use every day:  git add$(git dirtyfiles "$@" | menupick -1)  (The -1 means that if the standard input contains only a single item, just select it without issuing a prompt.) The interactive prompting runs in a loop, so that if the menu is long I can browse it a page at a time, adding items, or maybe removing items that I have added before, adjusting the selection until I have what I want. Then entering a blank line terminates the interaction. This is useful when I want to ponder the choices, but for some of the most common use cases I wanted a way to tell menupick “I am only going to select a single item, so don't loop the interaction”. I have wanted that for a long time but never got around to implementing it until this week. I added a -s flag which tells it to terminate the interaction instantly, once a single item has been selected. I modified the copy in $HOME/bin/menupick, got it working the way I wanted, then copied the modified code to my utils git repository to commit and push the changes. And I got a very sad diff, shown here only in part:

diff --git a/bin/menupick b/bin/menupick
index bc3967b..b894652 100755
@@ -129,7 +129,7 @@ sub usage {
-1: if there is only one item, select it without prompting
-n pagesize: maximum number of items on each page of the menu
(default 30)
-    -q: quick mode: exit as soon as at least one item has been selected
+    -s: exit immediately once a single item has been selected

Commands:
Each line of input is a series of words of the form


I had already implemented almost the exact same feature, called it -q, and completely forgotten to use it, completely failed to install it, and then added the new -s feature to the old version of the program 18 months later.

(Now I'm asking myself: how could I avoid this in the future? And the clear answer is: many people have a program that downloads and installs their utiities and configuration from a central repository, and why don't I have one of those myself? Double oops.)

Mon, 24 Jan 2022

You sometimes read news articles that say that some object is 98.42 feet tall, and it is clear what happened was that the object was originally reported to be 30 meters tall …

As an expectant parent, I was warned that if crib slats are too far apart, the baby can get its head wedged in between them and die. How far is too far apart? According to everyone, 2⅜ inches is the maximum safe distance. Having been told this repeatedly, I asked in one training class if 2⅜ inches was really the maximum safe distance; had 2½ inches been determined to be unsafe? I was assured that 2⅜ inches was the maximum. And there's the opposite question: why not just say 2¼ inches, which is presumably safe and easier to measure accurately?

But sometime later I guessed what had happened: someone had determined that 6 cm was a safe separation, and 6cm is 2.362 inches. 2⅜ inches exceeds this by only !!\frac1{80}!! inch, about half a percent. 7cm would have been 2¾ in, and that probably is too big or they would have said so.

The 2⅜, I have learned, is actually codified in U.S. consumer product safety law. (Formerly it was at 16 CFR 1508; it has since moved and I don't know where it is now.) And looking at that document I see that it actually says:

The distance between components (such as slats, spindles, crib rods, and corner posts) shall not be greater than 6 centimeters (2⅜ inches) at any point.

Uh huh. Nailed it.

I still don't know where they got the 6cm from. I guess there is someone at the Commerce Department whose job is jamming babies’ heads between crib bars.

Sun, 23 Jan 2022

Recently I've been thinking that maybe the thing I really dislike about set theory might the power set axiom. I need to do a lot more research about this, so any blog articles about it will be in the distant future. But while looking into it I ran across an example of a mathematical notation that annoyed me.

This paper of Gitman, Hamkins, and Johnstone considers a subtheory of ZFC, which they call “!!ZFC-!!”, obtained by omitting the power set axiom. Fine so far. But the main point of the paper:

Nevertheless, these deficits of !!ZFC-!! are completely repaired by strengthening it to the theory !!ZFC^−!!, obtained by using collection rather than replacement in the axiomatization above.

Got that? They are comparing two theories that they call “!!ZFC-!!” and “!!ZFC^-!!”.

Sat, 22 Jan 2022

A couple of weeks ago I had this dumb game on my phone, there are these characters fighting monsters. Each character has a special power that charges up over time, and then when you push a button the character announces their catch phrase and the special power activates.

This one character with the biggest hat had the catch phrase

and I began to dread activating this character's power. Every time, I wanted to grab them by the shoulders and yell “That's what destiny is, you don't get a choice!” But they kept on saying it.

So I had to delete the whole thing.

Fri, 21 Jan 2022

Divisibility and modular residues are among the most important concepts in elementary number theory, but the terminology for them is clumsy and hard to pronounce.

• !!n!! is divisible by !!5!!
• !!n!! is a multiple of !!5!!
• !!5!! divides !!n!!

The first two are 8 syllables long. The last one is tolerably short but is backwards. Similarly:

• The mod-!!5!! residue of !!n!! is !!3!!

is awful. It can be abbreviated to

• !!n!! has the form !!5k+3!!

but that is also long, and introduces a dummy !!k!! that may be completely superfluous. You can say “!!n!! is !!3!! mod !!5!!” or “!!n!! mod !!5!! is !!3!!” but people find that confusing if there is a lot of it piled up.

Common terms should be short and clean. I wish there were a mathematical jargon term for “has the form !!5k+3!!” that was not so cumbersome. And I would like a term for “mod-5 residue” that is comparable in length and simplicity to “fifth root”.

For mod-!!2!! residues we have the special term “parity”. I wonder if something like “!!5!!-ity” could catch on? This doesn't seem too barbaric to me. It's quite similar to the terminology we already use for !!n!!-gons. What is the name for a polygon with !!33!! sides? Is it a triskadekawhatever? No, it's just a !!33!!-gon, simple.

Then one might say things like:

• “Primes larger than !!3!! have !!6!!-ity of !!±1!!”

• “The !!4!!-ity of a square is !!0!! or !!1!!” or “a perfect square always has !!4!!-ity of !!0!! or !!1!!”

• “A number is a sum of two squares if and only its prime factorization includes every prime with !!4!!-ity !!3!! an even number of times.”

• “For each !!n!!, the set of numbers of !!n!!-ity !!1!! is closed under multiplication”

For “multiple of !!n!!” I suggest that “even” and “odd” be extended so that "!!5!!-even" means a multiple of !!5!!, and "!!5!!-odd" means a nonmultiple of !!5!!. I think “!!n!! is 5-odd” is a clear improvement on “!!n!! is a nonmultiple of 5”:

• “The sum or product of two !!n!!-even numbers is !!n!!-even; the product of two !!n!!-odd numbers is !!n!!-odd, if !!n!! is prime, but the sum may not be. (!!n=2!! is a special case)”

• “If the sum of three squares is !!5!!-even, then at least one of the squares is !!5!!-even, because !!5!!-odd squares have !!5!!-ity !!±1!!, and you cannot add three !!±1's!! to get zero”

• “A number is !!9!!-even if the sum of its digits is !!9!!-even”

It's conceivable that “5-ity” could be mistaken for “five-eighty” but I don't think it will be a big problem in practice. The stress is different, the vowel is different, and also, numbers like !!380!! and !!580!! just do not come up that often.

The next mouth-full-of-marbles term I'd want to take on would be “is relatively prime to”. I'd want it to be short, punchy, and symmetric-sounding. I wonder if it would be enough to abbreviate “least common multiple” and “greatest common divsor” to “join” and “meet” respectively? Then “!!m!! and !!n!! are relatively prime” becomes “!!m!! meet !!n!! is !!1!!” and we get short phrasings like “If !!m!! is !!n!!-even, then !!m!! join !!n!! is just !!m!!”. We might abbreviate a little further: “!!m!! meet !!n!! is 1” becomes just “!!m!! meets !!n!!”.

[ Addendum: Eirikr Åsheim reminds me that “!!m!! and !!n!! are coprime” is already standard and is shorter than “!!m!! is relatively prime to !!n!!”. True, I had forgotten. ]

Thu, 20 Jan 2022

Instead of multiplying the total by 3 at each step, you can multiply it by 2, which gives you a (correct but useless) test for divisibility by 8.

But one reader was surprised that I called it “useless”, saying:

I only know of one test for divisibility by 8: if the last three digits of a number are divisible by 8, so is the original number. Fine … until the last three digits are something like 696.

Most of these divisibility tricks are of limited usefulness, because they are not less effort than short division, which takes care of the general problem. I discussed short division in the first article in this series with this example:

Suppose you want to see if 1234 is divisible by 7. It's 1200-something, so take away 700, which leaves 500-something. 500-what? 530-something. So take away 490, leaving 40-something. 40-what? 44. Now take away 42, leaving 2. That's not 0, so 1234 is not divisible by 7.

For a number like 696, take away 640, leaving 56. 56 is divisible by 8, so 696 is also. Suppose we were going 996 instead? From 996 take away 800 leaving 196, and then take away 160 leaving 36, which is not divisible by 8. For divisibility by 8 you can ignore all but the last 3 digits but it works quite well for other small divisors, even when the dividend is large.

This not not what I usually do myself, though. My own method is a bit hard to describe but I will try. The number has the form !!ABB!! where !!BB!! is a multiple of 4, or else we would not be checking it in the first place. The !!BB!! part has a ⸢parity⸣, it is either an even multiple of 4 (that is, a multiple of 8) or an odd multiple of 4 (otherwise). This ⸢parity⸣ must match the (ordinary) parity of !!A!!. !!ABB!! is divisible by 8 if and only if the parities match. For example, 104 is divisible by 8 because both parts are ⸢odd⸣. Similarly 696 where both parts are ⸢even⸣. But 852 is not divisible by 8, because the 8 is even but the 52 is ⸢odd⸣.

Wed, 19 Jan 2022

The news today contains the story “Italian Senate Accidentally Plays 30 Seconds Of NSFW Tifa Lockhart Video” although I have not been able to find any source I would consider reliable. TheGamer reports:

The conference was hosted Monday by Nobel Prize winner Giorgio Parisi and featured several Italian senators. At some point during the Zoom call, a user … broke into the call and started broadcasting hentai videos.

Assuming this is accurate, it is disappointing on so many levels. Most obviously because if this was going to happen at all one would hope that it was an embarrassing mistake on the part of someone who was invited to the call, perhaps even the Nobel laureate, and not just some juvenile vandal who ran into the room with a sock on his dick.

If someone was going to go to the trouble of pulling this prank at all, why some run-of-the mill computer-generated video? Why not something really offensive? Or thematically appropriate, such as a scene from one of Cicciolina's films?

I think the guy who did this should feel ashamed of his squandered opportunity, and try a little harder next time. The world is watching!

I got a cute little surprise today. I was thinking: suppose someone gives you a large square integer and asks you to find the next larger square. You can't really do any better than to extract the square root, add 1, and square the result. But if someone gives you two consecutive square numbers, you can find the next one with much less work. Say the two squares are !!b = n^2!! and !!a = n^2+2n+1!!, where !!n!! is unknown. Then you want to find !!n^2+4n+4!!, which is simply !!2a-b+2!!. No square rooting is required.

So the squares can be defined by the recurrence \begin{align} s_0 & = 0 \\ s_1 & = 1 \\ s_{n+1} & = 2s_n - s_{n-1} + 2\tag{\ast} \end{align}

This looks a great deal like the Fibonacci recurrence:

\begin{align} f_0 & = 0 \\ f_1 & = 1 \\ f_{n+1} & = f_n + f_{n-1} \end{align}

and I was a bit surprised because I thought all those Fibonacci-ish recurrences turned out to be approximately exponential. For example, !!f_n = O(\phi^n)!! where !!\phi=\frac12(1 + \sqrt 5)!!. And actually the !!f_0!! and !!f_1!! values don't matter, whatever you start with you get !!f_n = O(\phi^n)!!; the differences are small and are hidden in the Landau sign.

Similarly, if the recurrence is !!g_{n+1} = 2g_n + g_{n-1}!! you get !!g_n = O((1+\sqrt2)^n)!!, exponential again. So I was surprised that !!(\ast)!! produced squares instead of something exponential.

But as it turns out, it is producing something exponential. Sort of. Kind of. Not really.

!!\def\sm#1,#2,#3,#4{\left[\begin{smallmatrix}{#1}&{#2}\\{#3}&{#4}\end{smallmatrix}\right]}!!

There are a number of ways to explain the appearance of the !!\phi!! constant in the Fibonacci sequence. Feel free to replace this one with whatever you prefer: The Fibonacci recurrence can be written as $$\left[\matrix{1&1\\1&0}\right] \left[\matrix{f_n\\f_{n-1}}\right] = \left[\matrix{f_{n+1}\\f_n}\right]$$ so that $$\left[\matrix{1&1\\1&0}\right]^n \left[\matrix{1\\0}\right] = \left[\matrix{f_{n+1}\\f_n}\right]$$

and !!\phi!! appears because it is the positive eigenvalue of the square matrix !!\sm1,1,1,0!!. Similarly, !!1+\sqrt2!! is the positive eigenvalue of the matrix !!\sm 2,1,1,0!! that arises in connection with the !!g_n!! sequences that obey !!g_{n+1} = 2g_n + g_{n-1}!!.

For !!s_n!! the recurrence !!(\ast)!! is !!s_{n+1} = 2s_n - s_{n-1} + 2!!, Briefly disregarding the 2, we get the matrix form

$$\left[\matrix{2&-1\\1&0}\right]^n \left[\matrix{s_1\\s_0}\right] = \left[\matrix{s_{n+1}\\s_n}\right]$$

and the eigenvalues of !!\sm2,-1,1,0!! are exactly !!1!!. Where the Fibonacci sequence had !!f_n \approx k\cdot\phi^n!! we get instead !!s_n \approx k\cdot1^n!!, and instead of exploding, the exponential part remains well-behaved and the lower-order contributions remain significant.

If the two initial terms are !!t_0!! and !!t_1!!, then !!n!!th term of the sequence is simply !!t_0 + n(t_1-t_0)!!. That extra !!+2!! I temporarily disregarded in the previous paragraph is making all the interesting contributions: $$0, 0, 2, 6, 12, 20, \ldots, n(n-1) \ldots$$ and when you add the !!t_0 + n(t_1-t_0)!! and put !!t_0=0, t_1=1!! you get the squares.

So the squares can be considered a sort of Fibonacci-ish approximately exponential sequence, except that the exponential part doesn't matter because the base of the exponent is !!1!!.

Tue, 18 Jan 2022

This morning Katara and I were taking our vitamins, and Katara asked why vitamin K was letter “K”.

I said "It stands for ‘koagulation’.”

“No,” replied Katara.

“Yes,” I said.

“No.”

“Yes.”

By this time she must have known something was up, because she knows that I will make up lots of silly nonsense, but if challenged I will always recant immediately.

“It does in German.”

Lorrie says she discovered the secret to dealing with me, thirty years ago: always take everything I say at face value. The unlikely-seeming things are true more often than not, and the few that aren't I will quickly retract.

I started to write an addendum to last week's article about how Mike Wazowski is not scary:

I have to admit that if Mike Wazowski popped out of my closet one night, I would scream like a little boy.

And then I remembered something I haven't thought of for a long, long time.

My parents owned a copy of this poster, originally by an artist named Karl Smith:

When I was a small child, maybe three or four, I was terrified of the creature standing by the word “Night”:

One night after bedtime I was dangling my leg over the edge of the bed and something very much like this creature popped right up through the floor and growled at me to get back in bed. I didn't scream, but it scared the crap out of me.

I no longer remember why I was so frightened by this one creature in particular, rather than say the snail-bodied flamingo or the dimetrodon with the head of Shaggy Rogers. And while are obviously a lot of differences between this person and Mike Wazowski (most obviously, the wrong number of eyes) there are also some important similarities. If Mike himself had popped out of the floor I would probably have been similarly terrified.

So, Mike, if you're reading this, please know that I accept your non-scariness not as a truly held belief, but only as a conceit of the movie.

[ If any of my Gentle Readers knows anything more about Karl Smith or this poster in particular, I would be very interested to hear it. ]