The Universe of Discourse


Thu, 14 Nov 2019

Soondubu victory

Katara and I are in a virtuous cycle where she thinks of some food she wants to eat and then asks me to cook it, I scratch my head and say "Well, I never did, but I could try", and then I do, and she tells me it was really good. This time she suggested I should make soondubu jjigae (순두부찌개), which is a Korean stew that features very soft tofu. (“Dubu” (두부) is tofu, and “soon dubu” is soft tofu.)

I did it and it came out good, everyone was happy and told me how great I am. Cooking for my family makes me feel like a good dad and husband. Hey, look, I am doing my job! I love when I do my job.

Soondubu in the
bot on the range.  The  pot is dark-glazed heavy earthenware.  The
soondubu is a dark-red soup (from the chili paste) full of floating
pieces of white dubu and green-edged zucchini.

I thought maybe soondubu would be one of those things where you can make it at home with endless toil but in the end you have a product that is almost as good as the $6.95 version you could get from the place downstairs. But it was actually pretty easy. Korean cuisine is often very straightforward and this was one of those times. I approximately followed this recipe but with several changes. (One of these days I'll write a blog article about how so many people stress out about the details of recipes.) The overall method is:

Dice up some onion and garlic finely and put them in the pot with sesame oil and kochujang. Then you add the meat if you are doing that, and then about a half a cup of chopped up kimchi. Then you put in broth and you add the soondubu. Easy-peasy.

The recipe on that page called for beef but I used chicken meat in cubes because that was what Katara asked for. All the soondubu recipes I found call for kochugaru (red pepper flakes) instead of kochujang (red pepper soybean paste) but I didn't have any handy and so what?

Somewhere in the world there is some food snob who will sneer and say that Real Soondubu is always made with kochugaru, and using kochujang is totally inauthentic. But somewhere else there is someone who will say “well, my grandmother always liked to use kojujang instead”, and Grandma outranks the food snob. Also I decided this year that the whole idea of “authentic” recipes is bullshit and I am going to forget about it.

I used chicken broth out of a box. The recipe called for scallions but I think I didn't have any handy that time. The recipe called for anchovy paste but I left them out because Lorrie doesn't like the way they taste. But I put did in some thin slices of zucchini. We do have a nice Korean-style glazed earthenware pot which I cooked in and then transported directly to the table.

Everyone in my family likes soondubu and it made me really happy that they considered my first one successful.


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Mon, 11 Nov 2019

MJD's law

All programming languages are equally crappy, but some are more equally crappy than others.


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Wed, 06 Nov 2019

Help me ask why you didn't just…

Regarding the phrase “why didn't you just…”, Mike Hoye has something to say that I've heard expressed similarly by several other people:

Whenever you look at a problem somebody’s been working on for a week or a month or maybe years and propose a simple, obvious solution that just happens to be the first thing that comes into your head, then you’re also making it crystal clear to people what you think of them and their work.

(Specifically, that you think they must be a blockhead for not thinking of this solution immediately.)

I think this was first pointed out to me by Andy Lester.

I think the problem here may be different than it seems. When someone says “Why don't you just (whatever)” there are at least two things they might intend:

  1. Why didn't you just use sshd? I suppose it's because you're an incompetent nitwit.

  2. Why didn't you just use sshd? I suppose it's because there's some good reason I'm not seeing. Can you please point it out?

Certainly the tech world is full of response 1. But I wonder how many people were trying to communicate response 2 and had it received as response 1 anyway? And I wonder how many times I was trying to communicate response 2 and had it received as response 1?

Mike Hoye doesn't provide any alternative phrasings, which suggests to me that he assumes that all uses of “why didn't you just” are response 1, and are meant to imply contempt. I assure you, Gentle Reader, that that is not the case.

Pondering this over the years, I have realized I honestly don't know how to express my question to make clear that I mean #2, without including a ridiculously long and pleading disclaimer before what should be a short question. Someone insecure enough to read contempt into my question will have no trouble reading it into a great many different phrasings of the question, or perhaps into any question at all. (Or so I imagine; maybe this is my own insecurities speaking.)

Can we agree that the problem is not simply with the word “just”, and that merely leaving it out does not solve the basic problem? I am not asking a rhetorical question here; can we agree? To me,

Why didn't you use sshd?

seems to suffer from all the same objections as the “just”ful version and to be subject to all the same angry responses. Is it possible the whole issue is only over a difference in the connotations of “just” in different regional variations of English? I don't think it is and I'll continue with the article assuming that it isn't and that the solution isn't as simple as removing “just”.

Let me try to ask the question in a better better way:

There must be a good reason why you didn't use sshd

I don't see why you didn't use sshd

I don't understand why you didn't use sshd

I'd like to know why you didn't use sshd

I'm not clever enough to understand why you didn't use sshd

I think the sort of person who is going to be insulted by the original version of my question will have no trouble being insulted by any of those versions, maybe interpreting them as:

There must be a good reason why you didn't use sshd. Surely it's because you're an incompetent nitwit.

I don't see why you didn't use sshd. Maybe the team you're working with is incompetent?

I don't understand why you didn't use sshd. Probably it's because you're not that smart.

I'd like to know why you didn't use sshd. Is it because there's something wrong with your brain?

I'm not clever enough to understand why you didn't use sshd. It would take a fucking genius to figure that out.

The more self-effacing I make it, the more I try to put in that I think the trouble is only in my own understanding, the more mocking and sarcastic it seems to me and the more likely I think it is to be misinterpreted. Our inner voices can be cruel. Mockery and contempt we receive once can echo again and again in our minds. It is very sad.

So folks, please help me out here. This is a real problem in my life. Every week it happens that someone is telling me what they are working on. I think of what seems like a straightforward way to proceed. I assume there must be some aspect I do not appreciate, because the person I am talking to has thought about it a lot more than I have. Aha, I have an opportunity! Sometimes it's hard to identify what it is that I don't understand, but here the gap in my understanding is clear and glaring, ready to be filled.

I want to ask them about it and gain the benefit of their expertise, just because I am interested and curious, and perhaps even because the knowledge might come in useful. But then I run into trouble. I want to ask “Why didn't you just use sshd?” with the understanding that we both agree that that would be an obvious thing to try, and that I am looking forward to hearing their well-considered reason why not.

I want to ask the question in a way that will make them smile, hold up their index finger, and say “Aha! You might think that sshd would be a good choice, but…”. And I want them to understand that I will not infer from that reply that they think I am an incompetent nitwit.

What if I were to say

I suppose sshd wasn't going to work?

Would that be safer? How about:

Naïvely, I would think that sshd would work for that

but again I think that suggests sarcasm. A colleague suggests:

So, I probably would've tried using sshd here. Would that not work out?

What to do? I'm in despair. Andy, any thoughts?


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Tue, 05 Nov 2019

Octopuses opening jars

A while back a YouTube video was going around titled Octopus Intelligence Experiment Takes an Unexpected Turn. Someone put food in a baby bottle with a screw cap and a rubber nipple. There was a hole drilled in the bottle so that the octopus could reach in to taste the food, but it was not large enough for the food to come out or for the octopus to go in. The idea, I suppose, was that the octopus would figure out how to unscrew the cap.

The “unpexected turn” was that instead of unscrewing the cap, the octopus just ripped the entire nipple out of the bottle.

A still from the video.
The bottle is transparent plastic and the octopus, about the same
size, is behind it, with its arms wrapped around various parts of the
bottle, looking at us.  The orange screw cap that secures the nipple
in place is still screwed to the top of the bottle, but the white
rubber nipple itself is floating away. One of the octopus’s arms is
inside the bottle, entering through the large hole previously blocked
by the nipple.

I have mentioned this before but it bears repeating: this outcome should not have been an unexpected turn:

Crabs can be wrested from containers (Pieron, 1911; Schiller, 1948; Cousteau and Diole, 1973) but the results appear to be achieved by chance and there is little indication that the octopus can learn to deal with the situation more efficiently with practice. The animal approaches and struggles with the apparatus until something happens; if it learns anything as a result of its experience it is only to be more persistent and vigorous.

(Martin Wells, Octopus: Physiology and Behaviour of an Advanced Invertebrate (Springer, 1978), page 241.)


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Mon, 04 Nov 2019

Wonders of the Internet, II

We have had two of these plastic thingies cluttering up our dish drainer for years. We didn't know what they were but we didn't want to throw them away because what if they are important?

A photograph of my hand, holding rectangular piece of white
plastic with a handle-shaped indentation on the front and two short
feet sticking out below

I got tired of them this weekend, and examined one more closely. There wasn't any indication of what it had been part of, or what it was for, but it did have the marking PART 2198768 on the back, so I handed that to The Goog.

And the result was instantaneous and unequivocal: it belongs to my refrigerator. Specifically, it goes in the back of the freezer compartment to keep food from falling down into the back and blocking the drainage path.

Partial screengrab of Google Search results.  At the top is a
link titled “Whirlpool 2198768 Read Fence”.  Then there follow some
images of the part, looking just as it does in my picture.  Then there
is a hit titled “What is the 'rear fence' and what does it do?”, and
one from Amazon offering a replacement part for sale.

Truly, we live in an age of marvels.


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Wonders of the Internet, I

I suddenly wondered: was Andy Warhol ever a TV show guest star? So I asked the Goog, and the answer was better than I could have imagined: in October 1985, Andy Warhol was a guest star on The Love Boat.

According to Wikipedia, one of the subpots of the episode was “a woman avoids Warhol, wanting to forget the time she was in one of his movies.” That is a lot more plausible than many Love Boat plots!

Also starring in that episode were Milton Berle, Tom Bosley and Marion Ross, Andy Griffith, and Cloris Leachman. TV Guide named it one of the hundred greatest episodes of any TV show ever.

Wow.


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Wed, 30 Oct 2019

Russian names in English and English names in Russian

One day I was surprised to find that Michael Jordan's name in Russian is “Майкл” (‘mai-kl’), and not “Михаи́л” (‘Mikhail’, the Russian translation of Michael.) Which is just what I should have expected; we don't refer to Mikhail Gorbachev or Baryshnikov as “Michael”, and it would be just as odd, in the other direction, if the Russians referred to the famous basketball player “Mikhail” Jordan.

When I was taking high school Russian we were assigned Russian versions of our names and I was disappointed to receive “Марк” (“Mark”) rather than anything more interesting. My friend Jeremy was stiffed in a different way. Apparently there is no direct Russian analog of “Jeremy” so the teacher opted for “Юрий” (Yuri). Yuri is not in any way a correct translation of Jeremy; it is the Russian version of “George”. Looking into it now, I wish she had thought to use “Иереми́я” (Jeremiah), or perhaps “Иерони́м” (Jerome).

1688 engraved portrait of Kimg
James I of England, cpationed “IACOBUS I · Rex. Angliæ”. James is
shown from the chest up, wearing a stiff lace ruff and a dark,
close-fitting jacket of some sort. His beard (chin only) is cut in a
square and the ends of hus mustache turn up.  His hairline has receded
and left a little tuft in the middle of the top of his head. His eyes
are wide and bright, and his lips slightly upturned, giving him an
appearance of private amusement.
King James I of England

It's funny how sometimes these names can be so easy to translate and sometimes so difficult. Mark is Mark, Aleksandr is Alexander, Viktor is Victor, Ivan is John, Yuri is George, Yakov is Jacob (or maybe James), Fyedor is Theodore, nothing is William, and Igor is nothing.

Italian Maria is obviously English “Mary” but how do you translate Mario? English has no male version of “Mary”.

(Side note: it is so bizarre that James and Jacob are somehow the same name, that when you turn Iacobus / Jacques / Iago (Latin / French / Spanish) into English it somehow turns into James. Another: What knucklehead decided to translate Frère Jacques as Brother John?)

[ Addendum: My previous article discussed the Korean translation of 邓小平, the name of Chinese leader Deng Xiaoping. Brian Lee points out that the usual Korean translation of Chinese小 (“small”) is 소 (pronounced, roughly, as /shoo/), but, just as in my Michael-Jordan examples above, the Koreans have chosen to translate the name so as to preserve the foreign pronunciation, 샤오 (/shya-oh/). Thanks! ]

[ Addendum: Dmitry Ivanov points out that there is a second Russian version of George, less common but closer to the English version: Георгий (“Georgy”). He also drew my attention to another Russian version of Jeremy, Ерёма (“Yerema”). This led me to discover that Russian Wikipedia has an entire page about Jeremy-related names, and mentions at least the following:

  • Еремей
  • Ереми́й
  • Ерене́й
  • Ерёма
  • Иереме́й
  • Иереми́й
  • Иереми́я
  • Ириме́й
  • Ярёма

Clearly, my high school Russian teacher blew it. ]


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Tue, 29 Oct 2019

Vowels in Korean and Mandarin

Something I've been wondering about for a while: there's this vowel in Mandarin which is usually written as ‘e’, for example in Deng (Xiaoping, 邓小平) or in feng shui (風水). But it's not pronounced like the ‘e’ in English “bed” or “pen”. It seems to my untrained ear to be more like the Korean vowel ‘ㅓ’, which is sort of between English “bought” and “but”. So I had wanted for a while to look up how Deng's name was spelled in Korean to see if they used ‘ㅓ’ or some other vowel. Partial success. Sure enough, Deng is spelled with ‘ㅓ’ in Korean: 덩(샤오핑).

“Feng shui” is spelled differently in Korean, with a different vowel: 풍수. But that's not too surprising, since the term “feng shui” presumably entered the Korean language centuries ago, and not only was the Chinese pronunciation probably different then, the Korean pronunciation would have changed over time after the adoption. In contrast, Deng's name presumably wasn't translated into Korean until sometime in the 20th century.

I was surprised that “Xiaoping” turns into three syllables in Korean. But Korean doesn't have that /aʊ/ dipthong, so that's the best it can do. This reminds me now of how amused I was by Corn Flakes boxes in Korea: in Korean, “Flake” is a four-syllable word. (플레이크).


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Mon, 28 Oct 2019

A solution in search of a problem

I don't remember right now what inspired this, but I got to thinking last week, what if I were to start writing the English letter ‘C’ in two forms, to distinguish its two pronunçiations? Speçifically, when ‘C’ gets the soft /s/ sound, we'll write it with a çedilla, and when it gets the hard /k/ sound we'll write it as usual.

Many improvements have been proposed to English spelling, and why not? Almost any change would be an improvement. But most orthographic innovations produçe barbaric or bizarre spellings. For example, “enuff” is still just wrong and may remain so for a long time. “Thru” and “donut” have been in common use long enough that not everyone thinks they look entirely bizarre, and I think only the Brits still object to “catalog” in plaçe of “catalogue”. But my ‘ç’ suggestion seems to me to be less violent. All the words are still spelled the same way. Nobody would have to deal with the shock of new spellings like “sirkular” or “klearanse”. I think the difficulty of adjusting to “çircular” and “clearançe” seems quite low.

On the other hand, the benefit also seems quite low. There aren't that many C’s to begin with. And who does this help, exactly? Foreigners who might otherwise have trouble deçiding how to pronounçe a particular ‘C’? Are there any people who actually have trouble reading “circle” and would be helped if it were spelled çircle”? And if there are, isn't c-vs-ç the least of their problems?

(Also, as Katara points out, ‘C’ is nearly superfluous in English as it is. You can almost always replaçe it with ‘S’ or ‘K’, accordingly. Although she did point out a counterexample: spelling “mace” as “mase” could be misleading. My proposal of “maçe”, though, is quite clear.)

I wonder, though, if this doesn't point the way toward a more general intervention that might be more generally helpful. The “ough” cluster gets a bad rap, but the real problems in English orthography are mostly in the totally inconsistent vowel spellings. Some diacritical marks might be a big help. For example, consider “bread” and “bead”. What if the close vowel in “bead” were indicated by spelling it “bēad”? Then it becomes easy to distinguish between “rēad” (/ɹid/, present tense) and “read” (/ɹɛd/, past tense), similarly “lēad” and “lead”. Native Anglophones will quickly learn to ignore the diacritical marks. A similar tactic might even help with the notorious “ough”. I don't really know what to do about words like “precious” or “ocean”, though. We can't leave them as they were, because that would unambiguously indicate the wrong pronunçiation “prekious”, “okean”. But to spell them “preçious” or “oçean” would be misleading. “Prećious”, maybe?

(I suppose someone wants to suggest “preşious” and “oşean”, but this is exactly what I'm trying to avoid. If you're going to do that you might as well go whole hog and use “preshus” and “oshun”.)

If you follow this path too far (and in the wrong direction) you end up with Unifon. I think this is a better direction and could end in a better plaçe. Maybe not better enough to be worth doing, though.

Peaçe out.


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Sat, 26 Oct 2019

Gringos and gringas

Wikipedia's article on the etymology of gringo is quite good, well-cited, and I did not detect any fishy smells. I had previously tried to look up gringo in the Big Dictionary, but it only informed me that it was from Mexican Spanish, which is not really helpful. (I know that's because their jurisdiction stops at the English border, and they aren't responsible for anything outside, but really, OED folks? Nothing else?)

Anyway Wikipedia helped me out. I had gotten onto this gringos thing because yesterday I learned about gringas, which are white flour tortillas. I immediately wondered: are they called gringas because (like gringos) they're made of white paste? Or is it because they're eaten by gringos, who don't care for corn tortillas? The answer seems to be: both explanations are current, but nobody knows if either is correct.

On the way to gringo I spent a while reading about yanqui, which Latin Americans use to refer to northerners.

So do people in the USA for that matter. Southerners will angrily deny being “yanqui”. They reserve that term to mean anyone from the north, such as myself. But folks like me from the Mid-Atlantic states also deny being Yankees and will tell you that it only means people from New England. Many New Englanders will disclaim being truly Yankee and say that to meet true Yankees you need to go to Maine or maybe New Hampshire. And I suppose people in Maine use it to mean one particular old Yankee farmer who lives up near the Canadaian border.

Anyway, I wonder: in Latin America, does “yanqui” always mean specifically USA-ians, or would it also include Canadians? Would a typical Mexican or Guatemalan person refer casually to Canadians as yanquis? Or, if they were drinking beer with a Canadian, and the Canadian refered to themselves as yanqui, would they correct them? (“You're not a yanqui, you're Canadian! Not the same thing at all!”)

If Mexicans do consider Canadians to be a species of yanqui, what do they make of the Québécois? Also yanqui? Or do Francophones get a pass? (What about the Cajuns for that matter?)


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Fri, 11 Oct 2019

More fair cake-cutting

In a recent article about fair cake division, I said:

Grandma can use the same method … to divide a regular 17-gonal cake into 23 equally-iced equal pieces.

I got to wondering what that would look like, and the answer is, not very interesting. A regular 17-gon is pretty close to a circle, and the 23 pieces, which are quite narrow, look very much like equal wedges of a circle:

A 17-gon divided into 23 equal pieces, as described in the
previous paragraph

This is generally true, and it becomes more nearly so both as the number of sides of the polygon increases (it becomes more nearly circular) and as the number of pieces increases (the very small amount of perimeter included in each piece is not very different from a short circular arc).

Recall my observation from last time that even in the nearly extreme case of a square divided into three slices, the central angles deviate from equality by only a few percent.

Of particular interest to me is this series of demonstrations of how to cut four pieces from a cake with an odd number of sides:

I think this shows that the whole question is a little bit silly: if you just cut the cake into equiangular wedges, the resulting slices are very close in volume and in frosting. If the nearly-horizontal cuts in the pentagon above had been perfectly straight and along the !!y!!-axis, they would have intersected the pentagon only 3% of a radius-length lower than they should have.

Some of the simpler divisions of simpler cakes are interesting. A solution to the original problem (of dividing a square cake into nine pieces) is highlighted.

The method as given works regardless of where you make the first cut. But the results do not look very different in any case:

The original SVG files are also available, as is the program that wrote them.


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Thu, 10 Oct 2019

Incenters of chocolate-iced cakes

A MathOverflow post asks:

Puzzle 1: Grandma made a cake whose base was a square of size 30 by 30 cm and the height was 10 cm. She wanted to divide the cake fairly among her 9 grandchildren. How should she cut the cake?

Okay, this is obvious.

Puzzle 2: Grandma made a cake whose base was a square of size 30 by 30 cm and the height was 10 cm. She put chocolate icing on top of the cake and on the sides, but not on the bottom. She wanted to divide the cake fairly among her 9 grandchildren so that each child would get an equal amount of the cake and the icing. How should she cut the cake?

This one stumped me; the best I could do was to cut the cake into 27 slabs, each !!\frac{10}3×10×10!! cm, and each with between 1 and 5 units of icing. Then we can give three slabs to each grandkid, taking care that each kid's slabs have a total of 7 units of icing. This seems like it might work for an actual cake, but I suspected that it wasn't the solution that was wanted, because the problem seems like a geometry problem and my solution is essentially combinatorial.

Indeed, there is a geometric solution, which is more interesting, and which cuts the cake into only 9 pieces.

I eventually gave up and looked at the answer, which I will discuss below. Sometimes when I give up I feel that if I had had thought a little harder or given up a little later, I would have gotten the answer, but not this time. It depends on an elementary property of squares that I had been completely unaware of.

This is your last chance to avoid spoilers.


The solution given is this: Divide the perimeter of the square cake into 9 equal-length segments, each of length !!\frac{120}{9}!! cm. Some of these will be straight and others may have right angles; it does not matter. Cut from the center of the cake to the endpoints of these segments; the resulting pieces will satisfy the requirements.

“Wat.” I said. “If the perimeter lengths are equal, then the areas are equal? How can that be?”

This is obviously true for two pieces; if you cut the square from the center into two pieces into two parts that divide the perimeter equally, then of course they are the same size and shape. But surely that is not the case for three pieces?

I could not believe it until I got home and drew some pictures on graph paper. Here Grandma has cut her cake into three pieces in the prescribed way:

A square with vertices
at ‹±3, ±3› and center at ‹0,0›. Three regions are marked on it: a
blue kite
with vertices
‹0,0›, ‹1,3›, ‹-3,3›, ‹-3,-1›; a pink irregular quadrilateral
with vertices ‹0,0›, ‹3,-3›, ‹3,3›, ‹1,3›; a green irregular
quadrilateral congruent to the pink one with vertices ‹0,0›, ‹-3,-1›,
‹-3,-3›, ‹3,-3›.  The three regions completely partition the square
with no overlap and nothing left over.

The three pieces are not the same shape! But each one contains one-third of the square's outer perimeter, and each has an area of 12 square units. (Note, by the way, that although the central angles may appear equal, they are not; the blue one is around 126.9° and the pink and green ones are only 116.6°.)

And indeed, any piece cut by Grandma from the center that includes one-third of the square's perimeter will have an area of one-third of the whole square:

A square with vertices
    at ‹±3, ±3› and center at ‹0,0›. Marked on it is an orange trapezoid
    with vertices ‹0,0›, ‹-3,-2›, ‹-3,3›, ‹0,3›. Also a pink pentagon with
    vertices ‹0,0›, ‹2,-3›, ‹3,-3›, ‹3,3›, ‹2,3›.  Both polygons
    include 8 units of the square’s 24 perimeter units, and both
    have an area of 12 square units.

The proof that this works is actually quite easy. Consider a triangle !!OAB!! where !!O!! is the center of the square and !!A!! and !!B!! are points on one of the square's edges.

The triangle's area is half its height times its base. The base is of course the length of the segment !!AB!!, and the height is the length of the perpendicular from !!O!! to the edge of the square. So for any such triangle, its area is proportional to the length of !!AB!!.

No two of the five triangles below are congruent, but each has the same base and height, and so each has the same area.

Five wedges
radiate downward in different directions from the center of the
square, each arriving at a different part of the edge but each with a
base of 1 unit.

Since the center of the square is the same distance from each of the four edges, the same is true for any two triangles, regardless of which edge they arise from: the area of each triangle is proportional to the length of the square's perimeter at its base. Any piece Grandma cuts in this way, from the center of the cake to the edge, is a disjoint union of triangular pieces of this type, so the total area of any such piece is also proportional to the length of the square's perimeter that it includes.

That's the crucial property of the square that I had not known before: if you make cuts from the center of a square, the area of the piece you get is proportional to the length of the perimeter that it contains. Awesome!

Here Grandma has used the same method to cut a pair of square cakes into ten equal-sized pieces that all the have same amount of icing.

A 10×10 square divided
    into five pieces from the center.  The pieces are three
    different shapes, but each piece contains 8 units
    of the square's perimeter and has an area of 20 square units.

The crucial property here was that the square’s center is the same distance from each of its four edges. This is really obvious, but not every polygon has an analogous point. The center of a regular polygon always has this property, and every triangle has a unique point, called its incenter, which enjoys this property. So Grandma can use the same method to divide a triangular cake into 7 equally-iced equal pieces, if she can find its incenter, or to divide a regular 17-gonal cake into 23 equally-iced equal pieces.

Not every polygon does have an incenter, though. Rhombuses and kites always do, but rectangles do not, except when they are square. If Grandma tries this method with a rectangular sheet cake, someone will get shortchanged. I learned today that polygons that have incenters are known as tangential polygons. They are the only polygons in which can one inscribe a single circle that is tangent to every side of the polygon. This property is easy to detect: these are exactly the polygons in which all the angle bisectors meet at a single point. Grandma should be able to fairly divide the cake and icing for any tangential polygon.

I have probably thought about this before, perhaps in high-school geometry but perhaps not since. Suppose you have two lines, !!m!! and !!n!!, that cross at an acute angle at !!P!!, and you consider the set of points that are equidistant from both !!m!! and !!n!!. Let !!\ell!! be a line through !!P!! which bisects the angle between !!m!! and !!n!!; clearly any point on !!\ell!! will be equidistant from !!m!! and !!n!! by a straightforward argument involving congruent triangles.

Now consider a triangle !!△ABC!!. Let !!P!! be the intersection of the angle bisectors of angles !!∠ A!! and !!∠ B!!.

The
triangle, as described, with the two angle bisectors drawn and their
intersection at P.  Centered at P is a circle that is tangent to all
of AB, BC, and CA at the same time.

!!P!! is the same distance from both !!AB!! and !!AC!! because it is on the angle bisector of !!∠ A!!, and similarly it is the same distance from both !!AB!! and !!BC!! because it is on the angle bisector of !!∠ B!!.

So therefore !!P!! is the same distance from both !!AC!! and !!BC!! and it must be on the angle bisector of angle !!∠ C!! also. We have just shown that a triangle's three angle bisectors are concurrent! I knew this before, but I don't think I knew a proof.

[ Addendum 20191011: Many illustrated examples. ]


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