Perfect cinnamon twists

This is a great recipe for beginning bakers. They come out light and fluffy. If I can make them, so can you.

Bring to boil in large saucepan:

• 1 cup sour cream

Remove from heat. Stir in until well blended:

• 3 tbsp shortening
• 1/4 cup sugar
• 1/8 tsp baking soda
• 1 tsp salt

Cool to lukewarm. Add:

• 1 large unbeaten egg
• 1 package of dry yeast

Stir until yeast is dissolved. (At this point you could also throw in a dash of vanilla, almond, or orange extract, but I didn't and they were great. The sour cream alone gives them plenty of flavor.) Mix in with spoon:

• 3 cups sifted plain flour

Turn out onto a lightly floured board. Knead slightly a few seconds to form a smooth ball. Cover with a damp cloth and let stand five minutes to firm up. Meanwhile, in a small bowl mix:

• 1/3 cup brown sugar
• 1 tsp cinnamon
• 1/2 cup finely chopped nuts (pecans work fine)

Roll dough 1/4 inch thick in a rectangle 6"×24". Spread the entire surface with

• 2 tbsp soft butter or margarine

Sprinkle half of the dough (the long way) with the sugar-cinnamon mixture. Bring the unsugared half of the dough over the sugared half, pressing the top surface lightly to seal in the sugar mixture. Cut in strips 1/2 to 1 inch.

Grease 2 cookie sheets. Take each strip of dough, give each end a half twist (in opposite directions of course), and place on the cookie sheets about 2" apart, pressing both ends firmly and flatly to the baking sheet. Cover with a damp cloth and let rise at 85° or so until very light, about 1 hour 15 minutes.

Bake 12-15 minutes at 375°. While baking, mix in another bowl:

• 1-2 tbsp milk
• 1/2 cup powdered sugar

to make a fairly thick icing. Drizzle over the twists.

Makes about 2 dozen.

Mathematicians in training

First off, if you haven't played Set, you really should, because you're just the sort of person who would love it. It's a clever idea elegantly executed, and it just happens to be great competitive fun. I taught the kids to play, and the four-year-old has a definite edge on the six-year-old. I couldn't be more pleased.

We had a bit of a drive yesterday, and along the way we gave the kids some analogies to puzzle over. You know the sort of thing: “Ice is to water as rock is to what?” This turns out to be engrossing and surprisingly fun. (I might try it with adults sometime. I sense opportunities for nerd humor.) Here the kids were not evenly matched at all. The four-year-old would get the easy ones (cow : moo :: pig : x) but would guess random related words on the harder ones, apparently with equal confidence. The six-year-old saw more clearly what the game was about, so he was able to bring his greater general knowledge to bear.

Why is this post titled “Mathematicians in training”? Well, Set is a transparently mathematical game. There are 81 cards because 81 is 3⁴. The deck is the Cartesian product of four three-element sets. They form some kind of algebraic structure with extraordinary symmetry (of a kind I don't really know anything about—it's not a group—such that I'm tempted to get completely sidetracked here). But the kicker here is, the gameplay itself is mathematical. As far as I can tell, the only good strategy is to try to prove there are no sets.

Analogies are just little homomorphisms, which is to say, structure-preserving transformations. The idea that deep sameness is more interesting than superficial differences is more important to mathematics than numbers.

On the surface it seems like analogies are less mathematical than Set. Appearances can be deceiving.

(P.S. Figured it out. The sets in Set are the cosets of cyclic subgroups of (Z/3Z)⁴. The symmetry I was referring to above was that after you erase the underlying group operation, there's no privileged element. There are isomorphisms on the deck of cards, preserving the sets, mapping any given card to any other given card.)

Milestone

I probably learned about variables from playing around with a Commodore 64 when I was about the age you are now. But I didn't see them used in mathematics for many years, until they were finally introduced, in about 7th grade, as a tool for solving problems. Take a problem, write down the equation, putting variables for the unknown quantities, and then you have something you can solve.

A little while ago I realized that this isn't the only way, or even the most important way, that variables are used in math. Variables are used to write laws.

The other day you were getting a shower, and I told you that letters could stand for numbers, that you can use letters to write rules about numbers. The letter could stand for any number, and the rule would always be true. I wrote in the condensation on the glass:

A + 0 =

Then I stopped and said, well, A plus zero equals what? You said zero. I said I didn't think that was right, because what if A was seven? Seven plus zero equals zero? So then you said A. And I couldn't be sure but I thought you really got it. That's right, I said. I'm sure you could tell I was very pleased. Actually I was surprised and excited.

Later I wrote some math pages for you to solve. The first one said,

Here is a rule:

0 < A

Do all the numbers follow this rule?

If there is a number that doesn't, that's called a counterexample. It means the rule is false.

In math, a true rule is always true, for all numbers.

The other one had some mathematical statements on it and asked you which ones were true.

You surprised me.

(Click for a larger image.)

You got them all right. I asked you about the rule on the first page, A < 0, and you had to look up what the < symbol meant, but then you told me right away that it was false, because zero isn't less than zero.

I was amazed. I asked your mother, “Did you see those pages J. did today? What does this mean?” She wasn't surprised. “It means first-graders can learn pre-algebra,” I said, insistent.

“Some can,” she said.

She is half right: you are special; you are bright; and you are interested. But I know there are millions of special, bright, curious kids like you in this country, and I think by and large their schools are selling them short. You sure are lucky you've got me, kid. But not as lucky as I am to have you.

Prehistory, continued

I guess I always thought of technology as strictly cumulative, but reading a little about the Stone Age dispelled that notion. Archaeologists have had to reverse-engineer, from the artifacts they left behind, the toolmaking skills and technology of extinct cultures. How did Cro-Magnon mammoth hunters store meat? Wouldn't it spoil? Well, we really don't know that sort of thing anymore, but there's a way to find out:

…[U. Michigan researcher Dan] Fisher butchered a draft horse using stone tools he'd knapped himself, then cached the meat in a stock pond. Naturally preserved by microbes called lactobacilli in the water, the flesh emitted a faintly sour, pickled odor that put off scavengers even when it floated to the surface. To test its palatability, Fisher cut and ate steaks from the meat every two weeks from February until high summer, demonstrating that mammoth hunters might have stored their kills in the same way.

–Tom Mueller, “Ice Baby”. National Geographic, May 2009.

People showed up on Australia forty thousand years ago. Even though the sea level was lower then, they would have had to cross a channel 55 miles wide to get there. It's a little mysterious to me, as the oldest boats ever excavated anywhere are dugout canoes at most nine thousand years old.

So the first real agriculture (planted fields) happened about ten thousand years ago, dogs were domesticated about fourteen thousand years ago, but apparently people were exploring the South Pacific in logboats, miles from shore, forty thousand years ago. (The oldest stone tools predate H. sapiens and appeared 2.5 million years ago.)

This week I learned...

It's prehistory week at the jorendorff household. This week I learned:

• The fossilized skeletons of a 14-foot Xiphactinus (a mean-looking Late Cretaceous fish) and its last meal, a merely 6-foot bony fish which it swallowed whole, are on display at the Sternberg Museum of Natural History in Hays, Kansas.

• Woolly mammoths had a three-inch-thick layer of fat underneath thick skin, fur, and long shaggy hair. (Yet their environment was such that occasionally one would be flash-frozen, without spoiling the meat, to be eaten by modern dogs ten thousand years later.)

Last week:

• The original Nintendo GameBoy had an 8-bit processor with a HALT instruction which games were supposed to use to wait for interrupts. At least one game would sometimes busy-wait instead (yuck!).

• There's a tool, dwarfdump, that dumps DWARF debug info from an executable or object file.

• On Mac there's a lazy debug-info-linking scheme that causes dwarfdump not to see any DWARF in compiled executables. The Mac tools that come with Xcode are aware of this magic, but dwarfdump isn't.

Earlier still:

• The Spiral of Theodorus shows that the square roots of integers can be constructed with straight edge and compass.

• According to Tim Sweeney, quoted in this DDJ article:

  Any loop written in a traditional programming language can be vectorized, to execute 16 iterations of the loop in parallel on Larrabee vector units, provided the loop body meets the following criteria:

  • Its call graph is statically known.
  • There are no data dependencies between iterations.

  So compilers will be able to do a lot more vectorization.

Steampunk 4 life

“But surely (said I) that would make us Slaves to these Automata, if indeed they did not destroy us altogether.”

“Nay (quoth Doctor Albertus), it is not the Extinction or Enslavement of our Race that I see when I gaze into the Future of Mankind. On the contrary, I see naught but Liberty. There are some among us destined to be Monarchs, but how many are they? Each Country admits of but one Monarch, for that is the very Meaning of the Word. The Rest of us, and thou and I, Sir George, are in that Number;–the Rest of us, I say, are destined not to rule, but to be ruled; and in such Circumstances, our Happiness depends upon the Virtue of the Ruler. Now, who would not chuse rather to be guided by Reason, than to be subject to arbitrary Tyranny? Therefore I proclaim the Manumission of the Race of Man: For now we are Slaves to the Whims of Tyrants; but soon, when the Automata take their Place as Heirs of the whole Earth, we shall be guided only by Reason, and live under Rulers which cannot hate, or persecute, or lie, or sin in any Way.

“But if we shall be ruled by Automata, why should we not also be served by Automata? Machines have always served Men, tho’ in a limited and primitive Capacity; but what great Accomplishments lie within our Grasp, when we shall have Machines of greater Capability to serve us!–Machines that shall build, or dig, or plough the Earth; Machines that shall row our Ships faster than the Wind, or push our Carriages; Machines that shall fly through the Air like Birds, and carry us away with ’em on Wings like those Daedalus once dreamed of. Famine shall be unknown; the most impossible and artistic Constructions shall be put up in a week; the most distant Climes shall be brought near, and the most distant Peoples made our proximate Neighbors. In short, the Want, Misery, Ugliness, and Hatred of our current Existence shall give way to an Age of Plenty, Happiness, Beauty, and Peace.”

—The Wonderfull Automaton, a gothic novel in letters from (as literary conceit would have it) the eighteenth century

Dr. Boli's online magazine is, for lack of a better word, curious.

Lockhart's Lament

Lockhart's Lament (PDF, 25 pages) starts out like this:

Everyone knows that something is wrong. The politicians say, “we need higher standards.” The schools say, “we need more money and equipment.“ Educators say one thing, and teachers say another. They are all wrong. The only people who understand what is going on are the ones most often blamed and least often heard: the students. They say, “math class is stupid and boring,” and they are right.

and ends up like this:

How sad that fifth-graders are taught to say “quadrilateral” instead of “four-sided shape”, but are never given a reason to use words like “conjecture”, and “counterexample”. ...

Mathematics is about problems, and problems must be made the focus of a student's mathematical life. Painful and creatively frustrating as it may be, students and their teachers should at all times be engaged in the process—having ideas, not having ideas, discovering patterns, making conjectures, constructing examples and counterexamples, devising arguments, and critiquing each other's work.

The author is a bit crazed, but that just makes it more fun to read. In the unlikely case that you somehow got here while thinking math is stupid and boring, or if you've ever found yourself teaching a stupid, boring math class, take a look.

P.S. As the previous post maybe suggests, I've only recently discovered how to learn math by making conjectures and trying stuff, which is what Lockhart recommends.

In unrelated news, J. is pretty sharp at finding lines of symmetry. I need to give him a circle to play with and see what he says. (evil chuckle)

P.P.S. I got this link from humph, who is also a one-of-a-kind teacher (but not crazed).

The ring Z[i]

This is a little self-portrait, "The Artist Trying to Learn Abstract Algebra", probably of no interest to anyone else.

I read an introduction to rings (in Gallian, fifth edition, which I enthusiastically recommend). Now I'm trying to come up with some conjectures and prove or disprove them before I start on the exercises. (This book has great exercises, and doesn't bother teaching anything in the text that it can teach in an exercise.)

I figured maybe every ideal of the ring Z[i] is a principal ideal generated by some element of the ring. This morning I think I have the proof. It's a consequence of Z[i] being enough like the integers to support Euclid's algorithm.

That in turn is a consequence of Z[i] having something like integer division. You can define a well-ordered metric M on Z[i] such that M(0) < M(a) where a is any other element; and for any a and nonzero b, there exist a quotient q and remainder r such that a = bq + r and M(r) < M(b). That the domain of M is well-ordered implies that Euclid's algorithm terminates.

Z[i] also has something like prime and composite elements. For example, 5+i can be factored into (1-i)(2+3i). I wonder if these two properties are actually the same thing.

I think the ideals of Z[i] generated by "prime" elements are prime ideals.

A storytelling game

I've played this game a few times now and have really enjoyed it. I've only tried it with two players. It might work with more.

Rules

Each player starts by jotting down a very brief story outline: just five lines. Each line should be seven words or less. It's OK to steal the outline of a familiar story, as in the example below.

1. a girl in red
2. a wolf with a plan
3. the wolf eats grandma
4. the wolf's disguise is unconvincing
5. someone saves the day with an axe

Or of course you can make up your own. You do not have to stick to fairy tales.

Each player passes their story outline face-down to another player. (No peeking!) Then the game begins.

From here on out, it's very simple. All the players are going to cooperate to tell one big story including all the elements in all the story outlines. On your turn, look at the first line of the story outline that someone placed in front of you. Suppose it says ‘a lonely house on a hill’. Tell the first little bit of the story. Keep it brief, and be sure to work in a lonely house on a hill.

Then it will be someone else's turn. Perhaps they will read: ‘a girl in red’. They will pick up the story where you left off, adding a girl in red.

Take turns for five rounds.

The end.

A few game design notes

I haven't played many storytelling games. This game mostly tries to avoid the mistakes of Once Upon a Time, which I greatly enjoyed back in the day, but which has some flaws. It features actual competitive gameplay and a winner. But you don't win by telling the best story. You win by getting all the cards out of your hand.

The fixed deck of character, setting, and item cards in Once Upon a Time got boring after just a few games. At some point, the cards stopped helping. Instead I would find myself trying to scrape together yet another storyline involving a shepherdess, a ring, and a disguise. (I have not played Nanofictionary, but it seems like it might have the same problem.) It was usually fun anyway. But I think having the players instead supply fresh material for each game might help. And story outlines seem better than cards in other ways. They naturally provide characters and exposition early in the game, plot developments in the midgame, and endings at the end. They can revisit previously introduced elements instead of constantly adding more folderol to the story. And they're fun to make.

The endgame in Once Upon a Time was especially unsatisfying: as soon as a player could play all the cards from his hand, he would turn over his Happily Ever After card and crash-land the story into the overly specific, predetermined ending printed on it. This would be something like, ‘...and for all I know, they may be dancing still.’ Constraining the last words of the story just seems like a mistake. In this game, everyone gets to tell an ending, and when it's your last turn of the game, you know it. So every player gets their crowning moment, and loose ends tend to get tied off.

Stoicism, Christianity, and Mother Goose

I read the Handbook of Epictetus. It's very brief, just a few pages really. I'll quote a few paragraphs that should make it clear what Stoicism is about. (I'm quoting a recent translation by Nicholas P. White which I really like. The translations I found on the Web seem stilted, or florid, by comparison; though Higginson isn't bad. Of course you can try the original Greek.)

Some things are up to us and some are not up to us. Our opinions are up to us, and our impulses, desires, aversions—in short, whatever is our own doing. Our bodies are not up to us, nor are our possessions, our reputations, or our public offices, or, that is, whatever is not our own doing. (1.)

You are foolish if you want your children and your wife and your friends to live forever, since you are wanting things to be up to you that are not up to you, and things to be yours that are not yours. (14.)

If you are fond of a jug, say “I am fond of a jug!” For then when it is broken you will not be upset. If you kiss your child or your wife, say that you are kissing a human being; for when it dies you will not be upset. (3.)

Do not seek to have events happen as you want them to, but instead want them to happen as they do happen, and your life will go well. (8.)

The best of it is extremely well said, but the content is troublesome. If the sample above does not convince you that you should prepare yourself to be unperturbed when your wife and children die, then the rest won't either. Still, given that you're reading this, something of Stoicism is very likely in you. It's in Western culture.

I guess none of the quotes I picked addresses the evident problem of whether a Stoic may act, or whether he must be distant and docile at all times. The Handbook doesn't seem to offer a head-on answer. Modern Christianity's interpretation of Stoicism does, though.

God, grant me the serenity

to accept the things I cannot change;

the courage to change the things I can;

and the wisdom to know the difference.

—Reinhold Niebuhr, around 1944.

The wisdom, that is, to know what is up to us and what is not up to us. It is exactly this wisdom that is on offer in Epictetus: nothing is up to us except how we see things and how we comport ourselves. I think I like the modern Christian philosophy, demanding as it is of courage and wisdom, better. But then, I tend to like messy, perilous things in principle even when they are not so enjoyable in practice.

The Wikipedia article on the serenity prayer offers this lovely postscript. I have no idea why this rhyme is not a widely-known classic.

The philosopher W.W. Bartley juxtaposes Niebuhr's prayer with a Mother Goose rhyme (1695) expressing a similar sentiment, but without comment:

For every ailment under the sun

There is a remedy, or there is none;

If there be one, try to find it;

If there be none, never mind it.

Sonnets

Once I wrote two lines of what would have been an awesome sonnet.

Shall I compare you to my friend Matt Jones?

You are more lovely and not half so drunk.

In San Francisco, while everyone else was napping, I managed to sneak out to City Lights Books. There I stumbled on Sonnets by Giuseppe Gioacchino Belli, translated by Mike Stocks. Translated. Imagine translating sonnets. I opened it up and read one, and it was pretty good, so I bought the book.

I emerged unconvinced that translating sonnets is possible or sane. I don't know what I expected. The translations are real sonnets that rhyme and scan, an amazing technical achievement already, and they have a nice spontaneous feel. Not everything works. Many lines, inevitably, are awkward. The English slang Stocks uses is so different from what I grew up with as to sound inauthentic (a real shame).

Oh but what a pleasure to be introduced to Belli. He wrote these sonnets in Rome in the 1830s. If half of them are as true to life as they feel, it was a city alive with outrageous characters, illicit sex, and poor anger management. The best ones are little candid character sketches, slightly or heavily satirical, of harried mothers, old men, annoyed girlfriends, corrupt priests, and so on. There's one from a furious stutterer. Reading them is like eating cupcakes while falling in love.

I'll post one of the translations that I like, with the warning that this one is atypically clean.

A Very Roman Pastime

The treat that us lot liked the most when small,

the biggest thrill, the real McCoy, the biz,

was finding new-built homes and palaces

and using lumps of coal to trash the walls.

So here we'd doodle numbers, little sums,

and Gordon knots and those of Sollymom,

and there some Lotto stuff; and then move on

to filthy words and pricks and twats and bums.

Or else we'd take a stone or nail or stick

to gouge the plaster out, and draw a pic

so deep we'd hit the bricks and stuff below.

Those were the days all right, my God. Although,

that said, I like to dabble still, it's true...

and when I see a nice white wall, I do.

At the moment, a few more translations are posted on the book's web site.

Recently I learned...

• Your browser uses the public suffix list to determine whether two web sites may share cookies. This is not very robust but better than the previous strategy.

• If you take a long strip of paper, fold it in half as many times as you can, and unfold it, it'll make an approximation of the fractal shape called the Heighway dragon.

• If you take two fractions, say 1/2 and 1/3, and add the numerators and denominators, you get the mediant, in this case 2/5. I don't know much about the mediant, but it is linked to Ford circles in a way I don't really understand, and I'm told mediants give a startling way of approximating the value of continued fractions.

• glibc's qsort only does an actual quicksort as a last resort. If there's enough memory, it does a merge sort.

  This is old news, and I kind of figured it was the case, but I never looked at the source code before.

And I was reminded that complex numbers are key to quantum mechanics, something I missed when I was writing about complex numbers a year or two ago.

We've been unpacking books that have been in boxes for a year and a half. It's like meeting old friends. I read the epic of Gilgamesh, the book of Joshua, and Six Easy Pieces. I love small books.

Learning in games vs. applications

I was talking to one usability expert and he was describing how they measure task completion. Did the user press the buttons in the right order? Their ideal app resulted in new users completing tasks 100% of the time. This isn’t exploratory learning. You need to be able to fail and explore the possibility space of a particular tool. Through repeated failure and success, users build up robust skills that can be applied successfully in a wide variety of situations.

—Danc, Mixing Games and Applications (PDF)

Huh!

Well, maybe that's just what you'd expect a game designer to say about application design. Amusing slides anyway. The blog is Lost Garden.

Handlebars

Hey, listen to this song:

Listening to the first 15 seconds, I didn't think I was going to like it. The reasons I like this song are interesting enough to write about at length, but... oh, just hit the button.

In which something is beeping

Several years ago I predicted that in the future there will often be a mysterious beeping which no one can identify or locate. This has happened to me several times since then, but tonight was special. Something kept going BEE-doo loudly, about once every two minutes. I suspected the smoke detector and my laptop before tracking the noise to a cell phone.

It was 3 AM. My house is full of guests trying to sleep.

J. listens to a story

I started to read J. the story of Moses last night. As a kid I probably scorned story bibles, but someone gave us this one when J. was a baby, and flipping through it I was pleasantly surprised. Now I think you can probably do a whole lot of deleting and clarifying without hurting much. You do lose the texture of the Bible. But we've already read the kids D'Aulaire's Greek Myths (miraculously told without any hint of debauchery) and Julius Lester's version of the Br'er Rabbit stories (retaining occasional cruelty but excising the stories' roots in a culture enslaved). So much for texture.

Still, just setting the scene for the story of the little baby in the bulrushes requires a whole new vocabulary of evil and suffering. J. did not know the words slave, taskmaster, or anguish. Also I forgot that the first thing of note Moses does is murder someone.

When J. is surprised, he doesn't look surprised. Just very thoughtful.

I think we'll proceed. J. likes being challenged. And, though I am not religious, there's a message here. The Israelites are not a particularly well-behaved people. Their relationship with God is anything but smooth. Sometimes they are stubborn and refuse to do as God tells them. Sometimes God abandons them to the consequences of their actions. Sometimes He punishes them rather arbitrarily (it seems to me). At one point they get a “time out” that lasts forty years. He never stops loving them.

Did I just write a whole paragraph comparing myself to the Almighty? I'm just saying I can relate. I expect J. can relate too.

J. writes a song

The handwriting is our babysitter's.