The Universe of Discourse
https://blog.plover.com
The Universe of Discourse (Mark Dominus Blog)enWas the Lollipop Guild inspired by W.W. Denslow?
https://blog.plover.com/2022/01/28#denslow
<p>Yesterday I was thinking on these creepy Munchkins, and wondering what
they were doing there:</p>
<p><a href="https://pic.blog.plover.com/art/denslow/lollipop.png"><img
src="https://pic.blog.plover.com/art/denslow/lollipop-th.png" class="center" alt="Still from the
1939 MGM film “The Wizard of Oz”. Three midgets, dressed respectively
in bright red, green, and blue suits, look simultaneously like
overgrown babies and like weirdly shrunken old men. They are bald
except for fancifully curled golden mohawks that resemble gilded
staircase bannisters, and are making contorted faces."/></a></p>
<p>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:</p>
<p><a href="https://pic.blog.plover.com/art/denslow/denslow.jpg"><img src="https://pic.blog.plover.com/art/denslow/denslow-crop.jpg"
class="center" alt="Dorothy and Toto stand before the Witch of the
North and three bowing Munchkins. The Munchkins are nearly bald, but
each has a little forelock on his forehead. Two have long beards and
mustaches, one has only a pencil mustache. All three have round eyes
and big puffy cheeks like overgrown babies."/></a></p>
<p>(Click for complete illustration.)</p>
<p>Denslow and Frank Baum had a falling out after the publication of <em>The
Wonderful Wizard of Oz</em>, 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.</p>
One song to the tune of another
https://blog.plover.com/2022/01/27#lollipop
<p>I just randomly happened upon
<a href="https://www.youtube.com/watch?v=NP1BfiwMeAI">this recording of Pippa Evans singing “How Much is that Doggie in the Window” to the tune of “Cabaret”</a>,
and this reminded me of something I was surprised I hadn't mentioned
before.</p>
<p>In the 1939 MGM production of <em>The Wizard of
Oz</em>, there is a brief musical number,
<a href="https://www.youtube.com/watch?v=XBsf8qsxs2M">The Lollipop Guild</a>,
that has the same music as the refrain of
<a href="https://www.youtube.com/watch?v=PIAXG_QcQNU">Money</a>, also from
<em>Cabaret</em>. I am not aware of anyone else who has noticed this.</p>
<p>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.</p>
<p>Surely <a href="https://en.wikipedia.org/wiki/Kander_and_Ebb">the writers of <em>Cabaret</em></a> didn't do this
on purpose? Did they? While it seems plausible that they might have
<em>forgotten</em> 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 <em>The Wizard of Oz</em> was first released.</p>
<p>(Now I want to recast <em>The Wizard of Oz</em> 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 <a href="https://en.wikipedia.org/wiki/Wicked_%28musical%29"><em>Wicked</em></a>).</p>
Yet another software archaeology failure
https://blog.plover.com/2022/01/27#lost-feature
<p>I have this nice little utility program called
<a href="https://github.com/mjdominus/util/blob/master/bin/menupick"><code>menupick</code></a>.
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:</p>
<pre><code> emacs $(ls *.blog | menupick)
</code></pre>
<p>displays a list of those files and a prompt:</p>
<pre><code> 0. Rocketeer.blog
1. Watchmen.blog
2. death-of-stalin.blog
3. old-ladies.blog
4. self-esteem.blog
>
</code></pre>
<p>Then I can type <code>1 2 4</code> to select items 1, 2, and 4, or <code>1-4 !3</code> (“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
<a href="https://github.com/mjdominus/git-util/blob/master/git-addq"><code>git-addq</code></a>
that I use every day:</p>
<pre><code> git add $(git dirtyfiles "$@" | menupick -1)
</code></pre>
<p>(The <code>-1</code> means that if the standard input contains only a single item,
just select it without issuing a prompt.)</p>
<p>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 <code>menupick</code> “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 <code>-s</code> flag which tells it to terminate the
interaction instantly, once a single item has been selected.</p>
<p>I modified the copy in <code>$HOME/bin/menupick</code>, got it working the way I
wanted, then copied the modified code to my <code>utils</code> git repository to commit
and push the changes. And I got a very sad diff, shown here only in part:</p>
<pre><code>diff --git a/bin/menupick b/bin/menupick
index bc3967b..b894652 100755
--- a/bin/menupick
+++ b/bin/menupick
@@ -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
</code></pre>
<p>I had already implemented almost the exact same feature, called it
<code>-q</code>, and completely forgotten to use it, completely failed to install
it, and then added the new <code>-s</code> feature to the <em>old</em> version of the
program 18 months later.</p>
<p>(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.)</p>
<p>[ <a href="https://blog.plover.com/oops/title.html">Previously</a>; <a href="https://shitpost.plover.com/s/shuffle.html">previouslier</a> ]</p>
Excessive precision in crib slat spacing?
https://blog.plover.com/2022/01/24#precision
<p><a href="https://blog.plover.com/physics/precision.html">A couple of years back I wrote</a>:</p>
<blockquote>
<p>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 …</p>
</blockquote>
<p>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?
<a href="https://www.google.com/search?channel=fs&client=ubuntu&q=crib+slat+spacing">According to everyone</a>,
2⅜ inches is the maximum safe distance. Having been told this
repeatedly, I asked in one training class if 2⅜ inches was <em>really</em>
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?</p>
<p>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 <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5cfrac1%7b80%7d%24"> inch, about half a percent. 7cm would
have been 2¾ in, and that probably <em>is</em> too big or they would have
said so.</p>
<p>The 2⅜, I have learned, is actually codified in U.S. consumer product safety law.
(Formerly it was at
<a href="https://www.govinfo.gov/content/pkg/CFR-2011-title16-vol2/pdf/CFR-2011-title16-vol2-part1508.pdf">16 CFR 1508</a>;
it has since moved and I don't know where it is now.) And looking at
that document I see that it <em>actually</em> says:</p>
<blockquote>
<p>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.</p>
</blockquote>
<p>Uh huh. Nailed it.</p>
<p>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.</p>
Annoying mathematical notation
https://blog.plover.com/2022/01/23#notation-2
<p>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.</p>
<p><a href="https://arxiv.org/abs/1110.2430">This paper of Gitman, Hamkins, and Johnstone</a> considers a subtheory of
ZFC, which they call “<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24ZFC%2d%24">”, obtained by omitting the power set
axiom. Fine so far. But the main point of the paper:</p>
<blockquote>
<p>Nevertheless, these deficits of <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24ZFC%2d%24"> are completely repaired by
strengthening it to the theory <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24ZFC%5e%e2%88%92%24">, obtained by using
collection rather than replacement in the axiomatization above.</p>
</blockquote>
<p>Got that? They are comparing two theories that they call “<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24ZFC%2d%24">” and “<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24ZFC%5e%2d%24">”.</p>
<p>(<a href="https://web.archive.org/web/20131114002529/https://boolesrings.org/victoriagitman/2011/10/09/what-is-the-theory-zfc-without-power-set/">Blog post by Gitman</a>)</p>
<p>[ <a href="https://blog.plover.com/math/notation.html">Previously</a> ]</p>
Bad writing
https://blog.plover.com/2022/01/22#destiny
<p>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.</p>
<p>This one character with the biggest hat had the catch phrase</p>
<blockquote>
<p>I follow my own destiny!</p>
</blockquote>
<p>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 <em>is</em>, you
don't get a choice!” But they kept on saying it.</p>
<p>So I had to delete the whole thing.</p>
A proposal for improved language around divisibility
https://blog.plover.com/2022/01/21#divisibility-terminology
<p>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.</p>
<ul>
<li><img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24"> is divisible by <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%245%24"></li>
<li><img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24"> is a multiple of <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%245%24"></li>
<li><img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%245%24"> divides <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24"></li>
</ul>
<p>The first two are 8 syllables long. The last one is tolerably short
but is backwards. Similarly:</p>
<ul>
<li>The mod-<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%245%24"> residue of <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24"> is <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%243%24"></li>
</ul>
<p>is awful. It can be abbreviated to</p>
<ul>
<li><img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24"> has the form <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%245k%2b3%24"></li>
</ul>
<p>but that is also long, and introduces a dummy <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24k%24"> that may be
completely superfluous. You can say “<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24"> is <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%243%24"> mod <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%245%24">” or “<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24"> mod
<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%245%24"> is <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%243%24">” but people
find that confusing if there is a lot of it piled up.</p>
<p>Common terms should be short and clean. I wish there were a
mathematical jargon term for “has the form <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%245k%2b3%24">” 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”.</p>
<p>For mod-<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%242%24"> residues we have the special term “parity”. I wonder if
something like “<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%245%24">-ity” could catch on? This doesn't seem too barbaric
to me. It's quite similar to the terminology we already use for
<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24">-gons. What is the name for a polygon with <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%2433%24"> sides? Is it a
triskadekawhatever? No, it's just a <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%2433%24">-gon, simple.</p>
<p>Then one might say things like:</p>
<ul>
<li><p>“Primes larger than <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%243%24"> have <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%246%24">-ity of <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%c2%b11%24">”</p></li>
<li><p>“The <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%244%24">-ity of a square is <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%240%24"> or <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%241%24">” or “a perfect square always has <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%244%24">-ity
of <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%240%24"> or <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%241%24">”</p></li>
<li><p>“A number is a sum of two squares if and only its prime factorization
includes every prime with <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%244%24">-ity <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%243%24"> an <em>even</em> number of times.”</p></li>
<li><p>“For each <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24">, the set of numbers of <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24">-ity <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%241%24"> is closed under multiplication”</p></li>
</ul>
<p>For “multiple of <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24">” I suggest that “even” and “odd” be extended so
that "<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%245%24">-even" means a multiple of <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%245%24">, and "<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%245%24">-odd" means a nonmultiple
of <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%245%24">. I think “<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24"> is 5-odd” is a clear improvement on “<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24">
is a nonmultiple of 5”:</p>
<ul>
<li><p>“The sum or product of two <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24">-even numbers is <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24">-even; the
product of two <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24">-odd numbers is <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24">-odd, if <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24"> is prime, but
the sum may not be. (<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%3d2%24"> is a special case)”</p></li>
<li><p>“If the sum of three squares is <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%245%24">-even, then at least one of the
squares is <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%245%24">-even, because <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%245%24">-odd squares have <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%245%24">-ity <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%c2%b11%24">, and you
cannot add three <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%c2%b11%27s%24"> to get zero”</p></li>
<li><p>“A number is <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%249%24">-even if the sum of its digits is <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%249%24">-even”</p></li>
</ul>
<p>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 <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24380%24"> and <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24580%24"> just do
not come up that often.</p>
<p>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 “<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24m%24"> and <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24"> are relatively prime”
becomes “<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24m%24"> meet <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24"> is <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%241%24">” and we get short phrasings like
“If <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24m%24"> is <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24">-even, then <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24m%24"> join <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24"> is just <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24m%24">”. We
might abbreviate a little further: “<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24m%24"> meet <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24"> is 1” becomes
just “<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24m%24"> meets <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24">”.</p>
<p>[ Addendum: Eirikr Åsheim reminds me that “<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24m%24"> and <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24"> are coprime” is already standard and is
shorter than “<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24m%24"> is relatively prime to <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24">”. True, I had forgotten. ]</p>
Testing for divisibility by 8
https://blog.plover.com/2022/01/20#divisibility-by-8
<p><a href="https://blog.plover.com/math/divisibility-by-7.html">I recently wrote</a>:</p>
<blockquote>
<p>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. </p>
</blockquote>
<p>But one reader was surprised that I called it “useless”, saying:</p>
<blockquote>
<p>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.</p>
</blockquote>
<p>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.
<a href="https://blog.plover.com/math/divisibility-by-7.html">I discussed short division in the first article in this series</a>
with this example:</p>
<blockquote>
<p>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.</p>
</blockquote>
<p>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.</p>
<p>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 <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24ABB%24">
where <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24BB%24"> is a multiple of 4, or else we would not be checking it
in the first place. The <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24BB%24"> 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 <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24A%24">.
<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24ABB%24"> 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⸣.</p>
Pranking the Italian Senate
https://blog.plover.com/2022/01/19#italian-senate
<p>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. <a href="https://www.thegamer.com/italian-senate-tifa-lockhart-hentai/">TheGamer reports</a>:</p>
<blockquote>
<p>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.</p>
</blockquote>
<p>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.</p>
<p>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 <a href="https://en.wikipedia.org/wiki/Cicciolina%23Political_life">Cicciolina's</a> films?</p>
<p>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!</p>
The squares are kinda Fibonacci-like
https://blog.plover.com/2022/01/19#fibonacci-squares
<p>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 <em>two</em>
consecutive square numbers, you can find the next one with much less
work. Say the two squares are <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24b%20%3d%20n%5e2%24"> and <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24a%20%3d%20n%5e2%2b2n%2b1%24">, where
<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24"> is unknown. Then
you want to find <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%5e2%2b4n%2b4%24">, which is simply <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%242a%2db%2b2%24">. No square
rooting is required.</p>
<p>So the squares can be defined by the recurrence $$\begin{align} s_0 &
= 0 \\ s_1 & = 1 \\ s_{n+1} & = 2s_n - s_{n-1} + 2\tag{$\ast$}
\end{align} $$</p>
<p>This looks a great deal like the Fibonacci recurrence:</p>
<p>$$\begin{align}
f_0 & = 0 \\
f_1 & = 1 \\
f_{n+1} & = f_n + f_{n-1}
\end{align}
$$</p>
<p>and I was a bit surprised because I thought all those Fibonacci-ish
recurrences turned out to be approximately exponential. For example,
<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24f_n%20%3d%20O%28%5cphi%5en%29%24"> where <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5cphi%3d%5cfrac12%281%20%2b%20%5csqrt%205%29%24">. And
actually the <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24f_0%24"> and <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24f_1%24"> values don't matter, whatever you
start with you get <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24f_n%20%3d%20O%28%5cphi%5en%29%24">; the differences are small
and are hidden in the Landau sign.</p>
<p>Similarly, if the recurrence is <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24g_%7bn%2b1%7d%20%3d%202g_n%20%2b%20g_%7bn%2d1%7d%24"> you get
<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24g_n%20%3d%20O%28%281%2b%5csqrt2%29%5en%29%24">, exponential again. So I was surprised
that <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%28%5cast%29%24"> produced squares instead of something exponential.</p>
<p>But as it turns out, it <em>is</em> producing something exponential. Sort
of. Kind of. Not really.</p>
<p><img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5cdef%5csm%231%2c%232%2c%233%2c%234%7b%5cleft%5b%5cbegin%7bsmallmatrix%7d%7b%231%7d%26%7b%232%7d%5c%5c%5c%5c%7b%233%7d%26%7b%234%7d%5cend%7bsmallmatrix%7d%5cright%5d%7d%24"></p>
<p>There are
a number of ways to explain the appearance of the
<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5cphi%24"> 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
$$\left[\matrix{1&1\\1&0}\right]^n
\left[\matrix{1\\0}\right] =
\left[\matrix{f_{n+1}\\f_n}\right]
$$</p>
<p>and <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5cphi%24"> appears because it is the positive eigenvalue of the square matrix
<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5csm1%2c1%2c1%2c0%24">.
Similarly, <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%241%2b%5csqrt2%24"> is the
positive eigenvalue of the matrix <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5csm%202%2c1%2c1%2c0%24"> that arises in
connection with the <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24g_n%24"> sequences
that obey <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24g_%7bn%2b1%7d%20%3d%202g_n%20%2b%20g_%7bn%2d1%7d%24">.</p>
<p>For <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24s_n%24"> the recurrence <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%28%5cast%29%24"> is <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24s_%7bn%2b1%7d%20%3d%202s_n%20%2d%0as_%7bn%2d1%7d%20%2b%202%24">,
Briefly disregarding the 2, we get the matrix form</p>
<p>$$\left[\matrix{2&-1\\1&0}\right]^n
\left[\matrix{s_1\\s_0}\right] =
\left[\matrix{s_{n+1}\\s_n}\right]
$$</p>
<p>and the eigenvalues of <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5csm2%2c%2d1%2c1%2c0%24"> are exactly <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%241%24">. Where the
Fibonacci sequence had
<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24f_n%20%5capprox%20k%5ccdot%5cphi%5en%24"> we get instead <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24s_n%20%5capprox%20k%5ccdot1%5en%24">, and
instead of exploding, the exponential part remains well-behaved and
the lower-order contributions remain significant.</p>
<p>If the two initial terms are <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24t_0%24"> and <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24t_1%24">, then <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24">th term of
the sequence is
simply <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24t_0%20%2b%20n%28t_1%2dt_0%29%24">. That extra <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%2b2%24"> 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 <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24t_0%20%2b%20n%28t_1%2dt_0%29%24"> and put <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24t_0%3d0%2c%20t_1%3d1%24"> you get the
squares.</p>
<p>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 <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%241%24">.</p>
<p>How about that.</p>
Life with Dominus
https://blog.plover.com/2022/01/18#vitamin-k
<p>This morning Katara and I were taking our vitamins, and Katara asked
why vitamin K was letter “K”.</p>
<p>I said "It stands for ‘koagulation’.”</p>
<p>“No,” replied Katara.</p>
<p>“Yes,” I said.</p>
<p>“No.”</p>
<p>“Yes.”</p>
<p>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.</p>
<p>“‘Coagulation’ doesn't start with a ‘K’.”</p>
<p><a href="https://en.wikipedia.org/wiki/Vitamin_K#History">“It does in German.”</a></p>
<p>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.</p>
Mike Wazowski's prevenge
https://blog.plover.com/2022/01/18#mike-wazowski
<p>I started to write an addendum to <a href="https://blog.plover.com/movie/self-esteem.html">last week's article about how Mike
Wazowski is not scary</a>:</p>
<blockquote>
<p>I have to admit that if Mike Wazowski popped out of
my closet one night, I would scream like a little boy.</p>
</blockquote>
<p>And then I remembered something I haven't thought of for a long, long
time.</p>
<p>My parents owned a copy of this poster, originally by an artist named
Karl Smith:</p>
<p><img src="https://pic.blog.plover.com/brain/mike-wazowski/gg.png" alt="This is a print of an old Scottish
prayer done up as an illuminated manuscript. The text is in
old-fashioned black letters with a red capital letter at the start of
each word. Around the text is scrollwork in sea-green, and a number
of monsters and fanciful beasts in red, blue, black, and yellow. The
poem reads “From Ghoulies And Ghosties Long Leggitie Beasties And
Things That Go Bump In The Night Good Lord Deliver Us”." class="center" /></p>
<p>When I was a small child, maybe three or four, I was terrified of the
creature standing by the word “Night”:</p>
<p><a href="https://pic.blog.plover.com/brain/mike-wazowski/not-mike.jpg"><img
src="https://pic.blog.plover.com/brain/mike-wazowski/not-mike-th.jpg" alt="Closeup of one of the assorted
monsters. This creature has a round blue body with two eyes, a lage
flat nose, and a mouth that goes up at one corner and down at the
other. It has yellow legs and a fishlike tail, and appears to be
wearing red high-heeled shoes. Red hands (or hands wearing red
gloves) are attached to the sides of its head/body where the ears
might be. There are two yellow horns or anntennae on top of its
head." class="center" border=0/></a></p>
<p>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.</p>
<p>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.</p>
<p>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.</p>
<p>[ 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. ]</p>