Content-Type: text/shitpost
https://shitpost.plover.com
Mark Dominus ShitpostsenWell-ordering blah
https://shitpost.plover.com/2023/11/27#omega-omega
<p>I was going to write something about the ordinal number <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%cf%89%5e%cf%89%24">, but
then I got bogged down in a lot of blather about well-orders and
smaller ordinal numbers. I asked folks in Recurse Center if this
article was interesting and they very genrly and constructively said
it was not. So I am publishing it here.</p>
<p><em>Caveat lector</em></p>
<hr />
<p>Well-founded ordering is a fundamental idea in set theory, the basis of all
inductive arguments. The idea of a set with a well-founded ordering
is that if you start somewhere, and the moved to an element of the set
that is “smaller” in the ordering, and then do it again and again, you
must eventually get stuck at an element for which there is no
“smaller” element.</p>
<p>The prototypical example of a well-founded ordering is the ordinary <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%0a%5clt%20%24"> relation on the natural numbers. You can't keep passing from one
natural number to a smaller one without eventually getting stuck at
<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%240%24">. And the prototypical example of a <em>not</em> well-founded ordering
is the ordinary <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%20%5clt%20%24"> relation on the integers, because you can move
to <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%2d1%24">, then <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%2d2%24">, then <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%2d3%24">, and you never do get stuck. Or
for a slightmy more subtle non-example, the ordinary <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5clt%24"> relation on
the positive rational numbers, whre you can go <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%241%2c%20%5cfrac12%2c%20%5cfrac13%2c%0a%5cfrac14%2c%5cldots%24"> and you still never do get stuck.</p>
<p>To see the relationship with inductive arguments, think of induction
as working this way. In an inductive argument you say well, if the
claim were false for some large example it would also be false
for a smaller example, then also for an even smaller one, and you
could keep going like that until you got down to a trivial example,
but the claim is easy to verify for the trivial examples, so it must be
true for the large ones also. To work, the notion of "smaller"
has to guarantee to end at a trivial example after a finite number of
steps, and that's what "well-founded" gets you.</p>
<p>There are lots of examples of well-founded orders of the natural numbers that aren't the
usual <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5clt%20%24"> relation. The simplest nonstandard
one is:</p>
<ol>
<li><p>Exactly the same as the usual order, except…</p></li>
<li><p>Instead of <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%243%24"> being smallest, Every number is considered to be less than <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%243%24">.</p></li>
</ol>
<p>$$
0\prec 1\prec 2 \prec 4 \prec 5 \prec \ldots \prec 3
$$</p>
<p>We use the symbol <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5cprec%24"> instead of <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5clt%24"> to remind ourselves
that this is something like but not the same as the usual less-than
relation. It doesn't have to be <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%243%24"> that is out of line, it doesn't
matter. I just picked <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%243%24"> to emphasize that the choice was
arbitrary.</p>
<p>This is not just a simple renaming of the natural numbers, because in
the usual ordering there is no largest number, and here there <em>is</em> a
largest number, namely <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%243%24">. But the order is still well-founded. Even if
you start at <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%243%24">, the first time you move to a smaller number, it's
some other finite number and at that point you can be sure that the process can't
go on forever. You can move from <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%243%24"> down to <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%241000007%24">, but from
there you have at most <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%241000007%24"> moves before you are certain to get
stuck at <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%240%24">.</p>
<p>The way I presented this ordering is a <em>little</em> bit odd to set
theorists, because set theorists have a standard set of names and
notations for different well-founded orders. The ordinary natural
numbers one is called <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5comega%24">. The one above, which is like
<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5comega%24"> except it has one extra element on the end, larger than the
others, is called <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5comega%20%2b%201%24">. Instead of describing it the way I
did, with <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%243%24"> pulled out of line and stuck at the end, set theorists
usually call that largest element “<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5comega%24">” and write it like this:</p>
<p>$$
0\prec 1\prec 2 \prec 3 \prec 4 \prec 5 \prec \ldots \prec \omega
$$</p>
<p>It's the same thing, just with slightly less silly names. But it's
important to remember that something like <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5comega%20%2b%201%24"> describes a
perfectly well-defined ordering relation that could be put on the
ordinary natural numbers, not the usual ordering but no less
legitimate.</p>
<p>Of course you can add more than one big element on the right-hand end;
those orderings are <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5comega%2b1%2c%20%5comega%2b2%2c%24"> and so on.</p>
<p>You can even add an infinite number of elements on the right. Suppose
we take two copies of the natural numbers, one painted blue and one
painted green. Then we define the following order:</p>
<ol>
<li>If <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24"> and <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24m%24"> are the same color, then <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24"> is smaller than
<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24m%24"> in the usual way, just if <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%5clt%20m%24"> as ordinary unpainted numbers</li>
<li>If they are different colors then one is blue and one is green, and
the <span style="color: darkblue"><strong>blue</strong></span> one is smaller</li>
</ol>
<p>Now we have this ordering:</p>
<p>$$
\underbrace{
\color{darkblue}{0} \prec
\color{darkblue}{1} \prec
\color{darkblue}{2} \prec
\color{darkblue}{3} \prec \ldots
}_{\text{blue numbers}}
\prec
\underbrace{
\color{darkgreen}{0} \prec
\color{darkgreen}{1} \prec
\color{darkgreen}{2} \prec
\color{darkgreen}{3} \prec \ldots
}_{\text{green numbers}}
$$</p>
<p>If we don't like using paint, we could phrase it like this:</p>
<ol>
<li>If <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24"> and <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24m%24"> are the same parity, then <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24"> is smaller than
<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24m%24"> in the usual way, just if <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%5clt%20m%24"> as ordinary unpainted numbers</li>
<li>If they are different parities then one is even and one is odd, and
the <strong>even</strong> one is smaller</li>
</ol>
<p>$$
\underbrace{
0 \prec 2 \prec 4 \prec 6 \prec\ldots
}_{\text{even numbers}}
\prec
\underbrace{
1 \prec 3 \prec 5 \prec 7 \prec\ldots
}_{\text{odd numbers}}
$$</p>
<p>The standard name for this is <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5comega%20%2b%20%5comega%24"> or <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%242%5ccdot%5comega%20%24">
and the elements are usually written like this:</p>
<p>$$
0 \prec 1 \prec 2 \prec 3 \prec\ldots
\prec
\omega \prec \omega+1 \prec \omega+2 \prec\ldots
$$</p>
<p>I like to think of <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5comega%24"> as a game in which there is a track
of squares extending forever to the right. The leftmost square is
labeled <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%240%24">. One one square there is a penny. Two players
play a game in which they take turns moving the penny to the right some
number of squares. If a player moves the penny to <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%240%24">, they lose. Must
the game come to an end, or could it go on forever? Clearly it must
come to an end, even though the track itself is infinite. If the
penny starts on square 1000007, the game can't possibly last more than
1000007 moves. (As a game this is no fun at all, since the first
player can immediately move the penny to square <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%241%24">, but I'm only
interested in whether the game will end.)</p>
<p><img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5comega%2b1%24"> is the game where, instead of starting somewhere on the
track, the penny starts in player 1's pocket, and their first move is
to take it out and place it on one of the squares of the track. Again
the game must come to an end, although unlike in the previous case we
can't say ahead of time how long it will take. If we say “no more
than 1000007 moves”, player 1 can belie us by taking out the penny and
placing it on square 2061982 instead. But what we <em>can</em> say at the
beginning of the game is “I can't tell you now how long the game will
take, but I will once Player 1 makes her first move.”</p>
<p>For <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%242%5ccdot%5comega%24"> I like to think of two tracks, one above the
other. Now a player has two kinds of move:</p>
<ol>
<li>Move the penny to a square farther left in the same track, or</li>
<li>If the penny is in the upper track, move it to <em>any</em> square in the
lower track</li>
</ol>
<p>Again Player 1 begins the game by taking the penny from her pocket and
placing it on any square.</p>
<p>If the penny is on square 1000007 of the top track, I can't tell you
how long the game will take to end. But I can say “I will tell you
how much longer the game will take, not right now but after no more
than 1000008 moves from now.” Because after 1000007 moves, either
someone will have moved the penny to the lower track, and I can tell
how long the rest of the game will take, or the penny will have moved
left 1000007 times and be on the leftmost square of the top track and
the next move <em>must</em> take it to the lower track.</p>
<p>And if the game hasn't started yet, I can't <em>yet</em> tell you when I will
be able to say how much longer the game will take. But I will be able
to do that once Player 1 has made her first move and put the penny on
the board.</p>
<p>This reminds me of an anecdote I once heard from another programmer.
He told me his boss had come to him to ask him if he could do a certain
task; he had replied that he could, and the boss had asked him how long he
thought it would take.</p>
<blockquote>
<p>He said “I don't know.”</p>
<p>His boss, being a reasonable woman, asked him when he would be able to
tell her.</p>
<p>He said “I don't know.”</p>
<p>The boss, having dealt with this guy before, did not lose her
temper. Instead, she asked “How long will it take you to figure that
out?”</p>
<p>“Not more than two days,” he said at once.</p>
<p>“Okay, just to make sure there is no miscommunication, are you telling
me that in two days you will be able to tell me how long it will
take you to estimate how long the task will take?”</p>
<p>“That's right.”</p>
</blockquote>
<p>And they parted amicably, both parties satsified, at least for the
time being. Communication between management and engineering doesn't
always turn out so well!</p>
<p>In that programmer's game, there were three tracks, and the penny was
on the second space on the topmost track. He was playing the game
<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%242%5ccdot%5comega%20%2b%201%24">. At most two days later, the penny had moved to
<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5comega%20%2b%20n%24">, where <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24"> was how long it would take him produce his
estimate of the project timeline.</p>
<p>It's easy to add more tracks. Let's add an infinite stack of tracks,
one atop the other. Now when Player 1 takes the penny from her pocket
she can put it on any space on any track. Again, a legal move is to
move the penny left on the same track, or to any space on any lower
track.</p>
<p>How long before you can say how long the game ends? Human
language is not well-suited to this guarantee.</p>
<blockquote>
<p>Even once Player 1 was made her first move, I may not be able to tell you
how long the game will take, <br />
and I also may not be able to tell you
how long before I can tell you how long the game will take.</p>
<p>And I may not
be able to tell you how long before I can tell you how long
before I can tell you how long the game will take. </p>
<p>And I can't even
tell you now how many times I will have to stack up “I may not be able to tell
you”. </p>
<p><strong>BUT</strong> once Player 1 has made her first move, I <em>will</em> be
able to tell you how many times I have to stack it up.</p>
</blockquote>
<p>Is the game really guaranteed to end? Yes, it really is. After
Player 1's first move, the penny is on some square of some track, say
square <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24"> of track <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24m%24">.
After at most <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%2b1%24"> moves, the track
number <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24m%24"> must decrease. And then there can only be a finite
number of moves before it decreases again. And it can decrease at
most <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24m%24"> times before the penny is on the bottom track, and then
the game must end after a bounded number of moves.</p>
<p>This ordering is called <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5comega%5e2%24">. A penny on square <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24"> of
track <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24m%24"> is said to be at <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24m%c2%b7%5comega%20%2b%20n%24">. If we want to think
about a way to order the natural numbers with order type <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%cf%89%5e2%24">, we
can do it like this:</p>
<blockquote>
<p>Every number larger than <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%240%24"> can be put in the form <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%242%5ei%c2%b7j%24"> where <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24j%24"> is
odd. For example, <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24200%20%3d%202%5e3%c2%b725%24"> and <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%2437%20%3d%202%5e0%c2%b737%24">.</p>
<ol>
<li><img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%240%24">, as usual, is smaller than every other number.</li>
<li>Otherwise, the numbers can be thought of as <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%20%3d%202%5ei%c2%b7j%24"> and
<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%27%20%3d%202%5e%7bi%27%7d%c2%b7j%27%24">. Consider <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%5cprec%20n%27%24">:
<ol>
<li>if <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24i%20%5clt%20i%27%2c%20_or_%0a%3e%20%20%20%202%2e%20if%20%24">i = i'<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%2c%20_and_%20%24">j\lt j'!!</li>
</ol></li>
</ol>
</blockquote>
<p>Written out explicitly, the ordering looks like this:</p>
<p>$$\begin{array}{l}
0 \prec \\
2^0\cdot 1 \prec 2^0·3 \prec 2^0·5\prec 2^0·7 \prec\ldots \\
2^1\cdot 1 \prec 2^1·3 \prec 2^1·5\prec 2^1·7 \prec\ldots \\
2^2\cdot 1 \prec 2^2·3 \ldots
\end{array}
$$</p>
<p>or if you prefer</p>
<p>$$\begin{array}{l}
0 \\
\prec 1 \prec 3 \prec 5 \prec 7 \prec \ldots \\
\prec 2 \prec 6 \prec 12 \prec 24 \prec \ldots \\
\prec 4 \prec 12 \prec 20 \prec 28 \prec \ldots
\end{array}
$$</p>
<p>Well, none of that was <em>actually</em> what I planned to write about, but I
am going to stop here and continue tomorrow.</p>
The Sun's Wish
https://shitpost.plover.com/2023/11/25#sun-at-night
<p>The Sun loves looking down and seeing the mortals scurrying about like
germs, busy at our daily activities. But she's a little bit sad
because she doesn't know much about what we do at night. She has
never seen a late-night movie and has not even imagined sitting around a
campfire, toasting marshmallows and singing songs.</p>
<p>One day the Sun was granted her wish to spend a night on Eartha. Just at sundown she
was transformed into a woman. She had dinner at a jazz club, then
went out to a cocktail bar where she met a new friend. They went out
for midnight supper and then went back to the friend's apartment.</p>
<p>Just before dawn she kissed her new friend and returned to her work, content.</p>
Passing thought
https://shitpost.plover.com/2023/10/30#finite
<p>All numbers are finite, but some numbers are more finite than others.</p>
Sisters
https://shitpost.plover.com/2023/09/09#sisters
<p>Famous sisters <a href="https://en.wikipedia.org/wiki/Gloria_Steinem">Gloria Steinem</a>
and <a href="https://en.wikipedia.org/wiki/Mediastinum">Media Steinem</a>.</p>
Skinks
https://shitpost.plover.com/2023/05/31#skink
<p>How sure are we that the blue-tongued skink and the blue-tailed skink
aren't the same animal walking in different directions?</p>
Modok
https://shitpost.plover.com/2023/03/28#modok
<p>Does MODOK need to shave?</p>
<p>Does he blow his nose? How? He can't reach it. Now I picture the
hapless AIM scientist who has to attend MODOK with an enormous spotted hanky
when he catches cold.</p>
Croatian puzzle
https://shitpost.plover.com/2023/03/22#croatia
<p>In Serbian, Croatian, and other Slavic languages, <em>srp</em> (or ср̑п) means a sickle.
And <em>sȑpskī</em> (ср̏пскӣ) means the Serbs or the Serbian language.</p>
<p>But it's <em>Croatia</em>, not Serbia, that is actually sickle-shaped.</p>
I have had this conversation more than once
https://shitpost.plover.com/2023/02/27#judgmental
<p>Therapist: You're a very judgmental person.</p>
<p>Me: That's because is <em>good</em> to be judgmental </p>
<p>Me: Most people should be more judgmental actually</p>
<p>Me: I don't know what the fuck is wrong with them all</p>
Makes ya think, don't it?
https://shitpost.plover.com/2023/02/23#circuits-and-districts
<p>According to
<a href="https://thingofthings.substack.com/p/facts-i-learned-from-im-perfect-youre">Ozy Brennan</a>,
quoting from <em>I'm Perfect, You're Doomed</em>:</p>
<blockquote>
<p>The Assemblies are Jehovah’s Witness conventions. There are two
kinds: the Circuit Assembly, which occurs twice a year at the largest local
Kingdom Hall, and the District Assembly, which is the same thing but bigger.</p>
</blockquote>
<p>Isn't that curious? The U.S. federal court system does it the other
way around: there are 13 “circuits”, each of which is divided into
“districts”. For example, the Third Circuit comprises
the District of Delaware, the District of New Jersey, and three
districts in Pennsylvania.</p>
<p>What do you suppose it means? Coincidence?? Conspiracy??¿?</p>
<p>Other not-really-related mysteries: <a href="https://law.stackexchange.com/questions/86883/why-is-oklahoma-divided-into-three-districts">Why Oklahoma?</a>:</p>
More about the worst finite field
https://shitpost.plover.com/2023/02/13#worst-finite-field-2
<p>Going down to the basement today I realized that this could actually
be reasoned out, maybe. Every finite field has the form
<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24GF%28p%5en%29%24"> for some prime <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24p%24"> and some number <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24">. There is not
much to distinguish finite fields, they are all pretty much the same.
Except maybe you could distinguish the <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%3d1%24"> cases from the <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%3e1%24"> cases,
those are a <em>little</em> different.</p>
<p>Aha, but if you consider the <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%3d0%24"> case, the answer becomes clear.</p>
Worst finite field
https://shitpost.plover.com/2023/02/13#worst-finite-field
<p>I don't know why a voice in my head kept demanding to know what
was the worst finite field. But I couldn't stop thinking about it,
until I realized the answer: It's clearly <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24GF%281%29%24">.</p>
How to select uniform random points on a sphere
https://shitpost.plover.com/2023/02/09#random-sphere-points
<p>Select <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24x%2c%20y%2c%20%24"> and <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24z%24"> uniformly at random. Then discard the
selection and start over unless
<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24x%5e2%2by%5e2%2bz%5e2%3d1%24">.</p>
Happy new year!
https://shitpost.plover.com/2023/01/07#happy-new-year-2023
<p>It's 2023! Is Groupon dead yet?</p>
Ahem
https://shitpost.plover.com/2022/11/11#twitter
<p>I quit twitter before it was cool.</p>
Last night's mathematical dream
https://shitpost.plover.com/2022/07/06#math-dream
<p>Last night I dreamt that I was explaining to someone how to solve a certain math problem.
I said you shouldn't consider the situation as an ongoing process, but instead imagine it
as a fully-realized decision tree, and then apply the Davis-Putnam algorithm. In the dream
someone spoke up from across the room: “You mean the Davis-Putnam-<em>Brown</em> algorithm.”</p>
<p>“What??” I exclaimed in mock surprise. “Next you are going to tell me that Bunyakovsky
discovered the Cauchy-Schwarz inequality!”</p>
<p>Tell me, Dr. Freud, what does this mean?</p>
Aphorism
https://shitpost.plover.com/2022/04/29#aphorism
<p>It's not enough to make the coffee, you also have to drink it.</p>
Holy Saturday
https://shitpost.plover.com/2022/04/17#holy-saturday
<p>In many Christian communities, it is traditional to eat cheese on Holy
Saturday, to commemorate the day Jesus spent ripening in a cave.</p>
Octahedral cathedral
https://shitpost.plover.com/2022/03/28#octahedral-cathedral
<p>As far as I can tell there are no octahedral cathedrals. Why not?</p>
<p>Google search produces several <em>claimed</em> examples, such as
the Cathedral of San Flaviano in Giulianova. But photographs make clear that this
is actually an <em>octagonal</em> cathedral, actually an octagonal prism surmounted by a dome.
Similarly, St. Basil's cathedral in Moscow has a floor plan with a roughly eightfold symmetry,
but is not octahedral in any way.</p>
<p>Polyhedral buildings are common, but the space is dominated by
cuboids. Even the Kaaba in Mecca, despite its name (“cube” in Arabic)
and its enormous religious significance, is only approximately
cubical, visibly irregular.</p>
<p>The Egyptians famously made pyramids but they are all pentahedral.
none is a tetrahedron, much less a regular tetrahedron. A regular
tetrahedron is too steep for practical construction anyway;
the Egyptians had bad experiences with overly-steep pyramids.</p>
<p>It should not be too hard to make a building in the shape of a regular
octahedron, considered as a triangular antiprism. I would be
surprised if there weren't one somewhere. When I am King of the
World, there will be an octahedral cathedral or someone will have brief
but very uncomfortable conversation with me about their failure.</p>
<p>Also, where the hell is my Sonar Taxlaw fanfic?</p>
Pavillion
https://shitpost.plover.com/2022/03/28#pavillion
<p>[ This had been
sitting unpublished on the other blog for six months. It has now been weighed in the balanc e and found wanting. ]</p>
<p>If a billion is a thousand millions, and a trillion is a million
millions, how much is a pavillion?</p>
<table cellpadding="10em">
<tr><th>Numbers<th>Not numbers
<tr><td align=center>
bazillion<br>
billion<br>
gazillion<br>
million<br>
trillion<br>
zillion<br>
<td align=center>
Brazilian<br>
cotillion<br>
pavilion<br>
pillion<br>
vermillion<br>
</table>
<p>Surely someone must already have discussed the Brazilian cotillion in
the vermillion pavilion, it's too hard to miss.</p>
<p>(There is a joke about how George W. Bush, upon learning that four
Brazilian miners were trapped underground, asked how many a Brazilian
was.)</p>
<p>[ <a href="https://blog.plover.com/lang/squillions.html">Previously</a> ]</p>
Physics puzzle
https://shitpost.plover.com/2022/02/23#preheat
<p>If the laws of physics are time-symmetric, why do I always have to preheat the oven,
and never postheat it?</p>
Worst Bassist
https://shitpost.plover.com/2022/02/23#worst-bassist
<p>Sid Vicious, clearly.</p>
<p>In fact I think I'd be comfortable nominating the Sex Pistols as the
worst band, overall. The world is full of obnoxious and incompetent
bands. What gives the Pistols the prize is their insistence that they
represented authenticity — maybe they hated you, but at least you knew
it. This insistence, though, was itself nothing but a pose,
manufactured by them and Malcolm MacLaren.</p>
Happy new year!
https://shitpost.plover.com/2022/02/07#happy-new-year-2022
<p>It's 2022! Is Groupon dead yet?</p>
Today I learned…
https://shitpost.plover.com/2021/10/14#20211014
Spelling gloating
https://shitpost.plover.com/2021/03/14#spelling
<p>Pretty pleased with myself today for spelling “Aung San Suu Kyi”
without having to look it up.</p>
Worst Nobel Peace Prize recipient update
https://shitpost.plover.com/2021/03/14#worst-nobel-peace-prize-recipient-2
<p><a href="https://shitpost.plover.com/w/worst-nobel-peace-prize-recipient.html">Previously, I had some suggestions</a>.</p>
<p>Since then, Aung San Suu Kyi has been looking like a good choice.</p>
<p>Also, since they apparently gave Obama the prize for not being George
Bush, they should give Joe Biden <em>two</em> prizes for not being Donald
Trump.</p>