## Homemade flavored salts, or quick wedding favors

13 Apr

Many apologies for delay since last post.  My Yale talk went over great, and I meant to make a Q&A: Grad School post with all the questions the undergraduates asked and my answers, but my amanuensis didn’t take any notes!  Since then I’ve been reading a lot of cool stuff, including this awesome paper by the awesome Matt Clay.  So hopefully I can get that paper into a blog post, as well as some cool math from the SECOND annual Midwest Women In Mathematics Symposium (the first one was last year at UIC, mentioned in that blog post).  But this is not a math post.

I cannot imagine doing this for a wedding of more than 25 people (and really I did favors per household so I only made 14), but if you had helpful friends then knock yourself out.  These are also “fill in the blank” favors or small gifts (housewarming?).  I made four types of flavored salts, which are SUPER EASY and fairly quick if you just let them dry out for a day or two instead of baking them.  They mostly follow the same idea:

Take X amount of salt, and mix thoroughly with Y amount of flavoring using your fingers.  Spread on a baking sheet and let dry in a not-humid room for a day or two.  Package cutely.  Note the lack of equipment, expertise, or active time!

Most flavorful: Sriracha salt

This one is a darling of the internet and is straight from the Sriracha cookbook, which I do not own nor think I ever will (we own two or three cookbooks right now: my mom’s 1970s copies of “Joy of Cooking,” a Three-Ingredient Cookbook from my sister-in-law, and possibly his copy of Rachel Ray’s 30-Minute Meals, which he claims is a misnomer).  I did 1 1/2 c salt + 1/4 c + 2 TB Sriracha.

It’d be great to make a robot that could play the music of British progressive rock bands. It’d be called “Bot-Tull” (Jethro in case you didn’t get the reference)

Mix that up with chopsticks until there are just little clumps left, and lay it out for a day or two.  This ends up being pretty chunky, so people recommend beating it (wrap in a towel and hit with a rolling pin) or pulsing it in a food processor, but I am lazy so my guests got chunky Sriracha salt.

Prettiest: Rosemary salt

This one is the opposite of the last, in that it’s not all over the internet and the Martha Stewart website I got it off of kept being down.  Other recipes call for a food processor and just pulsing the rosemary to teeny bits, but I actively hate having pieces of rosemary in my food so I went with rosemary scented salt.

You should go lie down, salt, you’re looking a little green around the edges

You heat up rosemary with salt in a pot (I did 1 1/4 c salt + 4 stick/twigs of rosemary) and stir every minute or two for 10-ish minutes until little pieces of rosemary start falling off and it smells good, then put it in a tupperware and forget about it for a few hours.

Some parts fell off, but I’m at piece with that

Or you’re Martha Stewart and you remember about it after five minutes and cover it.  In any case, pop a lid on that and leave it for a day or three.  It’ll smell SO GOOD when you take off that lid.  For packaging, I popped a teeny piece of fresh rosemary in to each little jar and topped it with salt, discarding the big pieces and leaving in the little cooked parts.

Most unusual: Vanilla salt

I still don’t really know what to do with this one.  I bought a vanilla bean from the store, which I’ve never seen before, and mixed it into a cup of salt using my hands.  You use your fingernails and scrape out the inside from the bean-holder-thing (pod?), and then use your fingers and massage it into the salt.  It smells good!  Then spread it out and let dry like the others.

Best smelling/most versatile: Citrus salt

This is a fun one: take 1 1/4 c of salt, and zest a bunch of citrus over it.  I used two oranges, two lemons, and two limes.

We’re naked… is that appeeling to you? Actually ‘fruit porn’ is a thing (sometimes it’s just food porn, and sometimes it’s I don’t know what I don’t want to click on the links)

Use your fingertips to crush the zest into the salt, releasing the yummy smelling oils, and then spread out a baking sheet.

A far better than usual interpretation of “yellow snow”

I tried to dry this in the oven and hated the results and did it over again- I cooked it too hot and it sucked all the citrus flavor out. Plus you have to pay attention to the oven and who likes to do that?  (Possibly readers of baking blogs, but let’s forget about that…)

Sometimes I just want to yell, “Ogre in da house!” when I’m feeling the opposite of orange.

I packaged these in small jars I found on Amazon, but my maid of honor sent me cuter ones which I thought were too small: about two ounces of flavored salt is pretty good.

We buy 4 oz jars of flavored salts when we do so (it’s always truffle salt), so half that for a favor sounds perfect to me.  Plus the colored lids precluded the need to label each jar, which would’ve been a pain for ~50 tiny jars.  I threw them in clear favor bags I picked up a Target, along with a printout of a Bible verse/Byrds song, some suggestions on how to use the salts, and a thank-you tag.

Total cost:

20 count Favor bags (Target): $2 Ribbon (Target):$3

Printing (Fedex/Kinkos): $5 Salt (Grocery store):$4

Vanilla Bean (Grocery store): $3 (!) =$17 if you don’t count stuff you’re going to use anyway

Lemons, Limes, Oranges (store): $5? (and then we ate them later, so basically free!) Bottle of Sriracha (store):$6? (we’ve had this for awhile)

Rosemary (store): $3 (and then we ate some lamb with it!) =$31 if you count things you will eat

I’ll leave this post with something I thought was ridiculous when I went to Target.  Also, shout out to the amazing Alliance Bakery for the BEAUTIFUL job they did with our cake!  I’ll post pictures of that cake up when I get some.  It was lemon with a mango mousse filling.  Yum!

## Hummingbird muffins (pineapple loaf)

30 Mar

Last Christmas was the first time I met a lot of my fiance’s extended family, and I wanted to make a good impression, so I brought one of those delicious fruitcakes from my very first blog post.  Turns out people don’t like fruitcake much (why?!  It’s SO GOOD!!!).  This year I brought a loaf of hummingbird cake, which is essentially banana-pineapple cake with some nuts.  Turns out you should refrigerate it, so keep that in mind if you make this.  It is still quite tasty and moist, though not as good as the fruitcake (but not nearly as labor and time-intensive).

I don’t have my Ph.D. yet, but you could still call me doctor for doctoring this cake mix!

The bananas look a little beat up in the picture above, but they aren’t actually that ripe.  So I cheated and threw them in the oven to roast to get a riper flavor.

We’re GOING (roasted) BANANAS IN HERE!

Another reason this was not time-intensive: it’s semi-homemade!  I used a box of “supermoist” yellow cake mix instead of measuring out flour etc.  I didn’t have instant pudding on hand (the only time I can recall buying instant pudding was in the future, for my barely-semi-homemade banana pudding), so I subbed in some powdered sugar and vanilla.  Mixed in some cinnamon, then made a little yolk-y picture.

If John Travolta and Nicolas Cage were in my kitchen, I bet they’d race to pull the eggs out of this bowl. It’d be a Face/Off.

This all happened while the bananas were roasting.  I also drained a can of crushed pineapple, so I could put the juice into the egg-cake mix-sugar batter.

Our dog was standing in the kitchen and looking at this and kept saying “fruit roof!” But since he’s a dog, we thought he was saying “Fruit Loops!” and gave him a handful of cereal.
That story is ridiculous. You know we have two cats, we certainly couldn’t also fit a dog in the apartment. Much less a talking one.

While the iced teas are popular, sometimes Snapple likes to make wacky promotional flavors, like pie. Pie Snapple, however, was a dud, and Mr. S. decided to pull out his support. Pi-napple was a huge hit!

Once those are mixed with the yogurt, you can mash the bananas.  I love doing this with super ripe bananas because they yield so easily under your fork!  Also, I appear to use banana a LOT when baking (I love bananas!)  Mix in the pecans and pineapple with the banana, and then mix those with the cake mix.

This guy used to mix up letters and numbers. So when his girlfriend asked him to pick up some home decor items at this one store, he was very confused to find himself at a historic pier. Though he messed up in that direction, he’d never do the converse: for instance, look at this bowl and say it’s a pur-one.

Batter batter batter SWING! Actually don’t, please, you’ll get cake all over the floor.

I initially meant to make this time and labor-intensive cake (just how labor-intensive?  You crush the pineapple by hand instead of buying the can), but went with these little muffins and a mini-loaf instead.

No pun: I suddenly wonder why they don’t sell cake batter the way they sell cookie dough: just pour and bake. It seems like something people might use… maybe not.

Bake til done!  While it’s cooling, make a quick icing drizzle with cream cheese, milk, and powdered sugar:

Even if something REALLY SAD just happened, you probably shouldn’t cry over this. You’d get saltwater in your icing

If you’re talented/pinterest-y, you can make a pretty drizzle.  If you’re me, just chuck it on.

It’s been a little while since we’ve had a picture of me eating, so here’s one!

Hummingbird loaf: adapted from this recipe, who took it from Dole (the company)

1 box yellow cake mix

1 1/4 c powdered sugar, divided

4 eggs

1/4 c full-fat yogurt (I use greek)

2 tsp cinnamon

1 20 oz can crushed pineapple

2 bananas

1/2 c chopped pecans

1 tsp vanilla

1 TB milk

2 oz cream cheese

First, roast the bananas: put on a cookie sheet, in their peels, and throw in the oven.  Set oven to 350.

Drain pineapple, reserving 1 c juice.

Mix cake mix, 3/4 c powdered sugar, cinnamon.  Mix in eggs, pineapple juice, yogurt until smooth.

Do something else for a few minutes (I emptied the dishwasher)

Remove bananas from oven.  Peels will hopefully be black by now (about 10 minutes).  Carefully peel, then mash.  Add pecans and pineapple to bananas, then fold the mixture into the cake mix.

Pour into greased mini loaf pans or muffin tins (I made 12 muffins + 1 loaf out of this).  Bake in that 350 oven for 30 minutes, check the muffins.  If toothpick comes out clean, they’re done.  Loaf should be done around 40 minutes.

While they cool, smash cream cheese with remaining 1/2 c powdered sugar and milk until smooth.  Drizzle icing over loaves/muffins.  Store in fridge if they last more than two days.

## Spring break! Actually, banana pudding

22 Mar

I’m in a hotel room in New York right now, enjoying my first day of spring break (it’s a Saturday so we’ll see how good I am at not working during the week).  What a great day so far: breakfast at Frontera in the airport (the ORD location is much cheaper than the real one), lunch at Katsu-Hama (I try to always go when I’m in this city), and dinner at the very young (4.5 month old) Wallflower for fancy-pants date night.  Thought I’d take advantage of the time difference and blog.  Actually now I’m hungry… I’m going to go get some food and come back to this post later.

Sated!  Grabbed a late-night falafel.  I love travel-eating.  Eat-cation.  Speaking of non sequiturs, each week in my advisor’s secret seminar, the speaker from the week before brings treats, preferably homemade baked goods (you can guess who started this tradition…)  Two weeks ago, our new and fantastic postdoc brought in a big bowl of his mom’s banana pudding.  So of course I emailed him to ask for the recipe and figured I’d make it for this blog, especially since I still haven’t bought lots of baking supplies for the new place (like a mixer… we did get bowls for Christmas though!)  Despite how much I LOVED my last banana pudding post, and I recommend that one if you want to make banana pudding from scratch, Mama Hull’s recipe is ridiculously easy.  It’s hard to even call this semi-homemade… as Michael said, “slicing the bananas is what’ll take the most time.”

Va-ped-atin sounds like “something missing substance,” per fiance. Which is perfect, because those are all the letters missing from these to make real words (va-nilla, whip-ped cream, jell-atin). Sorry if that didn’t make sense, I’m feeling a little daze-y.

The directions for this recipe are basically: mix everything, slice bananas, layer with Nilla wafers.

I have actually never made instant pudding, so this was AMAZING to me!  You just whisk milk with the powder and it turns into pudding in like three minutes!  So crazy!!

Are you sure this is food?

I think I have had cool whip before in my home, as a kid.  I used to be REALLY into making elaborate jello parfaits for my family, layering different colors mixed with ice cream, fruit, or whipped cream.  When I was seven or so, someone in my family got me a set of parfait glasses for Christmas (I have a tendency of receiving kitchen gear for Christmas), and that spurred this phase where everyone had to pretend to like my creations.  Thank goodness that’s over!  (…I hope.)

Mix ALL the different and confusing textures!

Once you mix all the stuff, you really do have to take a long time to slice all the bananas!

The three phases of a dead banana’s life: whole, peeled, sliced. Not so appeeling, is it?

At first I tried to slice the bananas in the banana peel and not use a cutting board, but it’s so much faster to cut six bananas on a cutting board.

Then layering time!  So pretty!  I, unsurprisingly, went a little dorky on it and made a smiley face out of Nilla wafers on top.  Michael’s looked much prettier with a circle of bananas on top and some crumbled wafers.

This recipe makes enough to feed about 10 people who like banana pudding a lot, based on our dinner party and the leftovers the next day.  Just realized I failed at reading the recipe, which was literally two lines in an email.  Sigh.  Well, do whatever you like.  This is delicious!  Make sure you let it sit for a long time (I did overnight) so that the flavors meld and the Nilla wafers sort of melt into cake-like texture.

Yum!

I have at least one friend!

Mama Hull’s Banana Pudding, emailed from Michael Hull

1 large box (~5oz) vanilla instant pudding (I used two boxes of the 3.5 oz size)
2 2/3 cups milk
1 cup sour cream
1 (8oz) container of cool whip
1 box nilla wafers
6 bananas
Mix pudding with milk; whisk for 2 min.
Add sour cream and cool whip and mix together.
Layer, in a large dish, nilla wafers, 1/2 sliced bananas, 1/2 pudding; repeat. (I made three layers)
Crumble wafers on top. (I didn’t read this and it still turned out okay!)
REFRIGERATE.  Best if left overnight, but a few hours is OK too.

## Happy pi day!

14 Mar

No radio interview this year, though I did give two talks within three hours yesterday (on two different subjects!)  I actually forgot about pi day until this morning, which also means that I didn’t make a pie =(  I did, however, run the “Pi Day Pi K,” which is a misnomer as it’s actually 3.14 miles, not 3.14 kilometers.  Just a little bit over a 5K.  It was extremely fun and extremely cold.

I’m not a “natural” runner (in fact, I once spent six weeks on crutches with no explanation, but definitely from a running-related injury), and I’d estimate about 100 people passed me during the first mile of the run (I went in the second group, for those taking it a bit slower, while fiance was in the super fast first group).  While running I almost always feel myself thinking “when will this be over?” and trying to estimate how far I’ve run (“maybe I’m one-sixth of the way through now…”).  On the way back I desperately seek out landmarks and try to remember how far I had run when I first saw them (“oh, that rock!  That’s when I was thinking about stopping to walk but then I didn’t because that old lady blasted by me.  That was around 1/3 of the way, maybe?”)  Also, I’ve never felt the mythical “runner’s high.”

I’m the person on the bottom with my tongue out dripping sweat. Other runners are that guy up there with the sweatband

Afterwards I’m incredibly achy and sore all over my body.  BUT I feel really accomplished- even if I’m the slowest, least coordinated, wheeziest person at the finish line, I still finished the race, and no one can ever take that from me.  I’m incredibly proud of my accomplishment, probably because of all that misery I described above.  Same thing with math: I’m never going to win a Fields Medal or anything like that, just like I’ll never win a running race, but I sure as hell am going to get my damn participant certificate that says I finished (a.k.a. a Ph.D.)  Maybe I’ll take a really long time and be super wheezy or nervous or stuttery at my defense, but I will finish this race and no one will ever be able to take that away from me.

I tell my students about those six weeks last semester that I did CrossFit (I got a groupon for it), and how I was objectively the slowest, weakest, least coordinated person in the room (there was a pretty pregnant woman in there too).  But I still tried really hard and knew at the end of every day that I did my best in there, and that’s all anyone can ever ask of me.  I don’t expect all of my students to get A’s in every subject, but I do expect them to try really hard, to try hard to ask questions (I know it’s hard), and to get their damn participant certificate.  It helps that I teach a “satisfactory/unsatisfactory” class, so I just want my students to deserve their “S”s.

Wow I didn’t mean to write all that.  Just wanted to say happy pi day, and suggest reading Evelyn Lamb’s post on the other kind of pi (not the circle constant we know and love).

## Assorted stuff I’ve been reading

4 Mar
• On the insecurity of manliness: Actual title is “Is there anything good about men?”  Interesting speech from 2007 with a couple of good points in it.  It’s a bit long, and parts of it have become outdated, but I still enjoyed the read (Thanks Chris!).  In fact, in 2007 I took an Intro Psych class and first learned about evolutionary psychology, which used evolution to explain differences in men/women and their sex drives.  Since then, I’ve read a few places about how evo psych depends too much on social constructs, and if you measure just physical outputs (e.g. blood flow to genital areas) and ignore what people say (which is constrained by what they’ve been taught growing up), sex drive becomes much more equal.  At one point in this 2007 article the author says

It’s official: men are hornier than women.

I just googled “are men hornier than women?” and came up with this 2013 book saying the opposite.  (Now I want to read this book!)

• On challenging the status quo with lots of vocabulary words I don’t know: Actual title is “Feminism and Programming Languages.”  I’m not a usual Hacker News reader (believe it or not I don’t like spending a lot of time on the computer in general), but Jeremy Kun pointed me to this a few months ago and asked for my thoughts- the article itself has lots of vocab words, but the comments are interesting.  One summed up the article well:

This article raises the question: ‘where do our ideas about what programming languages should be like come from?

I’ve done some, but minimal programming in my life.  This isn’t really my field, but I find the above question intriguing because you can replace ‘programming language’ with ‘mathematics’ or really anything.  Or as Jeremy asked me:

Do you feel like the direction of mathematics, what questions are asked or believed to be important, what’s relevant and irrelevant, is shaped by male dominance of the field?

In other words, are there other paths in mathematical inquiry that you feel ought to be taken but aren’t, and that this could be linked to the fact that all the leading researchers are male?
Short answer to question one: yes, absolutely, completely.  Mathematics doesn’t care about us humans, but we certainly care about it.  The directions of research, grant money, what gets published in top journals: everything follows trends that we as a mathematical community create and enforce.  And we are dominated by males, so yes, the male perspective does shape where we’re going and what we’re doing.
Question two is more complicated and not the same as question one.  What “should” we be doing as mathematicians?  What is the goal of mathematics?  I can’t pretend to know the answers to these questions, or even if satisfactory answers exist.  We do what we do, we pursue what we find interesting and are either rewarded by our peers who also think it’s interesting, or not, and we have to find something else to keep the money and publications coming in.  If we were living in the world of Y: The Last Man, would mathematics be different?  Probably.  Better, worse?  Who can say?
That said, stereotypes aren’t so much about people totally projecting things that completely aren’t there but about people having a framework with which they interpret things that actually are there. It’s not that racism causes people to see (for example) belligerent teenage boys where there are none, but that a white belligerent teenage boy is just seen as himself while a black belligerent teenage boy is part of a pattern, a script, and when people blindly follow the scripts in their head that leads to discrimination and prejudice.

Look, we all know that there’s a trope in the movies where someone of a minority race is flattened out into just being “good at X” and that the white protagonist is the one we root for because unlike the guy who’s just “good at X” the protagonist has human depth, human relationships, a human point of view—and this somehow makes him more worthy of success than the antagonist who seems to exist just to be good at X.

So we root for Rocky against black guys who, by all appearances, really are better boxers than he is, because unlike them Rocky isn’t JUST a boxer, he has a girlfriend, he has hopes, he has dreams, etc. This comes up over and over again in movies where the athletic black competitor is set up as the “heel”—look at the black chick in Million Dollar Baby and how much we’re pushed to hate her. Look at all this “Great White Hope” stuff, historically, with Joe Louis.

So is it any surprise that this trope comes into play with Asians? That the Asian character in the movie is the robotic, heartless, genius mastermind who is only pure intellect and whom we’re crying out to be defeated by some white guy who may not be as brainy but has more pluck, more heart, more humanity? It’s not just Flash Gordon vs. Ming the Merciless, it’s stuff like how in the pilot episode of Girls Hannah gets fired in favor of an overachieving Asian girl who’s genuinely better at her job than she is (the Asian girl knows Photoshop and she doesn’t) and we’re supposed to sympathize with Hannah.

On the gendering of toys, or the free market.  Also videos here, which I highly recommend.  IF YOU CLICK A LINK IN THIS POST CLICK ON THIS ONE.  My fiance suggested there’s no malice in the Lego Corp., just a desire for more sales- gendering the legos may have caused an uptick in sales.  This is the hard part about the free market: of course Monsanto is going to trademark its crap, it wants more money.  Obviously Lego is going to divide the market and embrace stereotypes: they see what Barbie is doing and they want a piece of that money too.  At some level, corporations have a responsibility to society, but that seems totally unenforceable without governmental regulation (part of the point of government).  We just watched the Lego movie and loved it, but we’re also keenly aware of the Bechdel test and female characters in everything =(

I just want my kids to not feel like there’s a monster in them for being female, or half-Asian, or whatever.  Quoting myself

This particular little monster is the one that says boys save the day and overcome obstacles and girls get rescued, even when they try to save the day.  Or the one that sees the handwriting on the exam and braces itself for a bad proof.  The one that thinks you’re more like Amy and not like Penny at all (from Big Bang Theory, a show I actively hate for reasons I’ll go into later if ever), but that wants to be “normal.”  It’s the monster that says you don’t know what you’re talking about and you don’t know what’s going on so why even try.

Wow really long post!  In honor of my officemate, here are a bunch of red panda pics.

Click on photos for links to original sites

## An open problem in group theory

25 Feb

My last post was about Hee Oh‘s talk at CIRM from that conference I went to last month-it actually covered the first third or so of the first of four lectures she gave.  Étienne Ghys gave seven short talks on his favorite groups, which was a huge blast, so I thought I’d try to share some highlights.  This post is a surprisingly simple open problem in group theory, which talks about functions on a circle.  A circle!  Who would’ve thought we still don’t understand everything there is to know about circles?

Who knew? This guy! This is me attempting to draw a smug circle.

If you don’t know what a group is, check out my quick intro post for some examples.  Wikipedia also has a significantly more exhaustive page.

You may remember that I once did a series of posts 1, 2, 4, on the homeomorphisms of the torus.  You don’t need to read all the posts to get this post, I just wanted to point out that at one point I used the notation $Homeo_0(T)$ to indicate the homeomorphisms (continuous functions with continuous inverses) of the torus which are isotopic (wiggle-able) to the identity.  In fact, $Homeo_0(T)$ is a group, very related to the group we’re discussing today.

Instead of homeomorphisms, we can also talk about diffeomorphisms: these are homeomorphisms which are differentiable, whose inverses are also differentiable. Rather than dive into a definition of differentiable here, I’m just going to give you an intuitive definition: differentiable functions are “smooth” instead of chunky.

Top is smooth and a differentiable. Bottom isn’t; there are weird kinks in its frown

Some functions are differentiable, and some aren’t (see illustration above).  You can also take second and third and n-th derivatives, and we say functions are n-differentiable if it’s possible to take derivatives.  So in the above example, the red rectangle function is at least 1-differentiable (maybe 2 or 3 or more), but the blue function isn’t differentiable at all.

Notation time: we call a circle $S^1$, the sphere in one dimension.  So a hollow ball would be $S^2$, and so on.  In this post, we’ll be talking about twice-differentiable diffeomorphisms of the circle that preserve the orientation of the circle: so if a point is clockwise from y is clockwise from z, then f(x), f(y), and f(z) are also in clockwise order.  This group is written $Diff^2_+(S^1)$.

Great, now we know the group we’re talking about.  Now let’s get into the nitty-gritty of the problem.  First, a subgroup is a subset of a group which is itself a group.  For instance, a subgroup of the integers, $\mathbb{Z}$ under the operation of addition, is the even integers, $2\mathbb{Z}$.  This is because adding two even numbers gives you an even number (2+2=4).  In contrast, the odd integers are not a subgroup of $\mathbb{Z}$, since adding two odd numbers gives you an even number (3+5=8), which doesn’t lie in the set of odd integers.

Next, we need the concept of a normal subgroup.  FYI, mathematicians really care about normal subgroups: they give us lots of insights about the structure of groups, and they help us cut up groups into smaller, more manageable chunks- lots of times we’ll prove things about normal subgroups in order to say something about the larger group.  We start with a subgroup, call it N.  Then N is normal if for every group elements and in N, $xyx^{-1}$ is also in N.  The $x^{-1}$ means the inverse of with respect to the group operation.  So in the integers under addition, the inverse of 2 is -2, because 2 + (-2) =0.  In the real numbers under multiplication, the inverse of 2 is $\frac{1}{2}$, since $2 \cdot \frac{1}{2} = 1$.

In our example, the even integers is a normal subgroup of the integers (you can convince yourself of this).  It’s pretty easy to find subgroups of most groups, but finding normal subgroups (which aren’t just the identity element or the whole group) can be a little harder.  We say a group is simple if it has no non-trivial normal subgroups.

So here’s the open problem I promised at the beginning: Is $Diff_+^2(S^1)$ simple?

And if you want, here’s another one: is $Homeo (D^2, \partial D^2, area)$ simple?  Those are homeomorphisms of the disk that fix the boundary circle and respect area.

I just find it crazy that we don’t understand everything there is to know about functions of a circle or of a disk!  It’s amazing!

In terms of personal blog time, I did in fact bake last weekend, a lot (we were at a vacation rental house full of bakeware), but I didn’t take photos.  Mostly I baked the cookies, plopped them down in front of our awake and lively friends, and went to bed every night- turns out I’m not great at adjusting to living at 9000 feet.  Day one was those awesome salty shortbread cookies, day two were 3-ingredient peanut butter cookies with added peanuts and chocolate chips, and day three were double chocolate bacon cookies which I totally screwed up on but were still delicious.  Ten people made it through a pound and a half of butter in three days, which is glorious.

## Introduction to Apollonian circle packings (tangent)

17 Feb

This is not my area of research at all, but I think it’s super cool.  The first time I heard of Apollonian circle packings was at that conference I went to in Marseille last month, during the first lecture of Hee Oh’s minicourse.  So here’s a quick write up  background of the first third of that lecture.

These packings and all this theory come from one dude, Apollonius of Perga, who wrote a bunch of math books back around 200 BC.  Literally this math has been around for 2200 years.  Here’s a paper submitted a month ago which is a generalization of Apollonius’s problem from circles to spheres.  Math is so amazing!  We live in history!

To understand the problem, we’ll have to do a quick geometry brush up.  We say that two shapes or curves or lines in the plane are tangent if they touch at exactly one point.

Left: tangent; line hits circle at exactly one point.
Center: not tangent, line hits circle at two points
Right: not tangent, line doesn’t hit circle

So, here’s a theorem of Apollonius: Given three mutually tangent circles (so each one is tangent to the other two) in the plane, there exist exactly two circles tangent to all three of them.  Remember, this theorem is from 200 BC or so.  Here’s a picture representing the problem:

We’re so sad! We want more mutually tangent circles!

Woohoo! Found two! (there’s a little hot pink one in the middle)

A few other ideas that immediately come to mind if you see this theorem: what if the circles aren’t mutually tangent, but just lying around the plane? (per Wikipedia this is the actual Problem of Apollonius)  What if you use spheres in the three dimensions instead?    What if instead of circles you use other shapes?  Can we tell, given the radii of our first three circles, how big the other two will be?  These are all big problems in math that many people have thought about (but I have not).

Here’s a cool thing that can happen: if you take three mutually tangent circles, per the above theorem we can draw two more that are mutually tangent to them.  Now if you take one of these new circles, and two of the other ones, per the theorem again we can draw two more mutually tangent to those three.  See picture below.

Adding the pink circle, which is mutually tangent to green, orange, and purple

Then you can do this over and over again: for instance, we’ve still got lots of circles to build which are mutually tangent to three of the circles in this picture (how many did you count?  I count eight on first glance.  Remember that the dark blue and light pink circles, for instance, aren’t tangent, so you wouldn’t count a triplet with those two in them).  And then once you build those new circles, you get more circles.  Do this forever and you have an Apollonian gasket, a type of fractal.

I’m not going to build an Apollonian gasket to show you, but the internet has lots of pictures.

Wikipedia: click on image for link

Fractal Science Kit: click for link

I just wanted to put in the pretty pictures.  Let’s prove Apollonius’ theorem!

Review: the complex plane looks like the real plane, with coordinates (x,y), but the y-axis represents multiples of the imaginary number i, where $i = \sqrt{-1}$.  So if you see a coordinate like (2,3) in the complex plane, it represents the complex number 2+3i.  If you aren’t that familiar with i, you can think of it like a variable.  So in algebra, if you wanted to simplify the expression 3(x+4), you would have 3x+12, not 15x nor 15, because the x lives on its own.  Similarly, we add complex numbers like this: (2+3i) + (5-7i) = (2+5) + (3-7)= 7-4i.

If we add a point at $\infty$ to the complex plane, we’ll get the extended complex plane, $\mathbb{C} \cup \{\infty\}$.  One way to think of the extended complex plane is as a big ball, with the point at infinity at the north pole, and the origin (0,0) at the south pole, the unit circle (the circle with radius 1) lying on the equator, and the rest of the complex plane wrapping around to get closer and closer to infinity.  This way of thinking about the extended complex plane is called the Riemann sphere.  You can drop it back down to the complex plane by drawing a line from a point on the sphere to the point at infinity, and figuring out where the line hits the sphere.  In more concrete terms, imagine putting a tennis ball on a piece of paper.  Use a marker to draw a point on the tennis ball, then use a super high power laser and shine a line from the north pole of the tennis ball to hit that point.  You’ll burn a hole in the paper, which is exactly where the corresponding point on the complex plane lies.

Wikipedia is great.

Circles in the plane might look a little different on the sphere.  Lines can go through the point at infinity, just like any other point.  Here’s a picture of two circles that are tangent at the point at infinity:

The blue and red circles both wrap around the sphere.

If we project these circles down onto the plane as we were saying before, you get parallel lines: the point at infinity heads toward “infinity” in all directions.

So now we have a few new configurations for tangent circles.

Here’s a fact: if you multiply all the points in two tangent circles in the extended complex plane by a matrix $A=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ so that $A(x+iy) = \frac{a(x+iy) +b}{c(x+iy)+d}$, the resulting circles will still be tangent to each other.  A condition you need is that ad-bc=1 for this to be true, which can be a bit tricky since a,b,c, and d are all complex numbers.  More facts: if you’ve got any two circles, you can always find a matrix that will send one circle to another one.  You can actually do this for any three circles to any other three.

Okay, now we’re going to prove the theorem.  Take your three tangent circles, and choose a matrix A to map them to two parallel lines with a circle in between them.

Where do we put mutually tangent circles?

This makes it pretty clear that there are only two choices for mutually tangent circles.

Adding the blue and green circles!

Now multiply everything by $A^{-1}$ to send the pink circle and parallel lines back to your original three circles.  From our first fact, when we multiply the blue and green circles by $A^{-1}$, they’ll still be parallel to the pink circle and parallel lines.  So that’s all the ways you can find circles mutually tangent to three given, mutually tangent circles.

That’s your introduction to Apollonian circle packings!  I will probably never blog about them again, unless I randomly see another talk on them.  This was the first quarter or so of the first lecture by Hee Oh.  Other references: wikipedia.

P.S. Sorry that I’ve been doing such a bad job over the past few weeks of blogging every week (I think I’m averaging every 9 days).  I moved in with my SO three weeks ago (yay!) and am still trying to figure out how to live with someone with whom I want to hang out all the time.  Also we don’t own lots of baking ware (I just bought a glass pyrex casserole dish, which we didn’t have before.  We don’t have a cookie sheet.  We had to buy some measuring spoons.)  So I haven’t been baking as much.  And I can’t seem to find my camera battery charger!  It’s a little hectic.  Happy winter!