## Stop what you’re doing and make avocado coconut chocolate chip ice cream

22 Aug

I was in the middle of writing a blog post about the apple-blueberry-yogurt pie I made for the fourth of July, but then I tasted this ice cream that I just finished and changed my mind.  It’s SO GOOD.  The light avocado flavor plus the coconut milk and a teensy bit of lime and salt just make the creamiest indulgence.  This still has cream in it, but could be done vegan (just use more coconut milk).  I’m in love with this ice cream.  And also, I’m in love with ice cream (I didn’t bring the ice cream maker with me to Boston, but since coming back I’ve made strawberry ice cream and strawberry-lime-mint sorbet which were both also awesome and I’ll put the recipes at the bottom of this post).

If I used maple syrup or agave or something instead of sugar, this would be paleo, right?  It’s my husband’s birthday on Sunday and I want to make him something as good as the cake I made two years ago.  I’ve been going through a big carb phase for the past seven months, so this would be a thank-you to him for putting up with all the pasta/bread/not-paleo-at-all stuff we’ve been eating.  And what would be better than an avocado coconut ice cream cake with an almond-chocolate base?  I’ll update if I do that!

Incidentally, let me brag about him for a second: he’s currently in the woods in Oregon, running in one of the world’s largest relay races from Mount Hood to the coast.  I’m currently sitting at home with a bowl of ice cream and a laptop.  This is exactly what I imagine we look like right now:

I’ve never been in a competitive eating contest. I’m more of an amateur-portion person- you can tell from this picture (both because of the size of the bowl, and because it’s not in PRO-portion)

Anyways.  I picked up five small avocados for a dollar at the amazing produce shop down the street earlier this week, and I set them in a basket with some bananas waiting for them to ripen.  The problem with buying five avocados at once when it’s just you at home is you have to use them all at once, or you’ll have overripe ferment-y ones (remember the raw avocado pie from last summer?).  So I thought of ice cream or massive smoothies- avocado is definitely for sweets in Vietnam, with avocado smoothies being a huge thing (blend an avocado with ice and sweetened condensed milk.  Drink.)  I also didn’t have any milk at home, but I did have two limes and a can of coconut milk.  Also some friends brought me chocolate with coconut in it.  So that’s what happened!

Lately I’ve been very clumsy and stubbing my toe a lot. This makes me mix up things- am I an avocado for this ice cream, or an AdVOCAte? Stubbing my toe made me lose the extra ‘o’ (and also made me anagram)

Non-custard ice creams are so easy- just put everything in a blender.  Plop your avocado pieces in (I cut avocados in half, then use a spoon to scoop out the flesh), some sugar, some cream, the juice of two limes, and a can of coconut milk.  Blend.

She’s been in some complicated relationships-it’s still hard for the coconut milk to open up.

She’s just sick of all the li(m)es, but realizes that we all li(m)e sometimes to make life a little smoother (or perhaps more zesty- have i mentioned you look REALLY good in that?)

Blend that up til smooth, then toss it in the fridge while you go do other things.  Also, cut up a chocolate bar into little chunks.

We could try to be honest all the time, but everyone has a DARK side- and BARing all might not be a great idea

People will CHOC up outcomes to what yOu say- so there’s no need to explain you’re LATE because you want to spend less time with them. Just blame traffic- a little li(m)e won’t hurt anyone.

This was my first time leaving the mixture in the blender instead of transferring to a bowl.  It makes SO MUCH MORE SENSE because then you don’t splash all over when you pour the cream into the ice cream maker.

I’m not advocating lying all the time, but hey, any PO(u)Rt in a storm!

Note: DON’T put the chocolate chips, nuts, mix-ins, whatever in when you first pour in the mix.  Wait until it’s about the thickness of greek yogurt so that the chips get evenly distributed.

1 can coconut milk

3/4 c sugar (or agave syrup/other sweetener for paleo)

2 limes

1/2 c cream (or more coconut milk)

pinch of salt

chocolate (I used a 3 oz bar of dark chocolate with coconut in it)

Blend all ingredients except chocolate together.  Chill in a fridge for at least 20 minutes.  Start your ice cream maker and throw in the blender-ful of stuff.  Chop chocolate into small chips, and add in about 10 minutes into churning process.  Enjoy!

Bonus recipe: Strawberry-lime-mint sorbet, adapted from the Cuisinart recipe book that came with my ice cream maker

Note: While for ice cream I just throw the sugar in, by now I’ve decided that simple syrup is the way to go with sorbets.  Otherwise you’ll get grains of sugar ruining the smoothness of the sorbet

1 c water

1 c sugar

1 bunch mint leaves (I took them from my garden so I don’t know how many there were.  Maybe 15)

3 limes

2 c strawberries (one pint according to this helpful website)

pinch of salt

Bring the sugar and water to a boil to make a simple syrup.  Let it boil while stirring for a minute, then turn off the heat.  Throw in the mint.

Meanwhile, cut up and measure your strawberries.  Blend them with the juice and zest of the limes and the salt.  Discard the mint.  Add about 1/4 c of the simple syrup to your berry blend, and blend.  Taste.  This is probably too sour.  Add more syrup 1/4 c at a time until the mixture is a little bit sweeter than you’d like.  Chill in the fridge, clean up, and throw in the ice cream maker.  Yum!

I used my leftover mint simple syrup (I had about 1/4 c) in a strawberry iced green tea.  Delicious.

## How to cut a watermelon (for science!)

13 Aug

It’s summertime, which means watermelon!  There are actually two parts to this post: first, the smarter way to cut watermelon (so you don’t end up with watermelon juice all over your face/hands/clothes), and second, a goofy little experiment I did the other day to figure out the best way to buy watermelon.

For my entire life I assumed eating watermelon, like eating a ripe juicy peach, was just a naturally messy ordeal best done outside or over a sink.  Then this summer a friend showed me the right way to cut watermelon- watermelon fries!  The traditional wedge shape means every bite releases more juices to go everywhere else, while the fry fits entirely in your mouth and is a lot like eating a fry.  See schematic of happiness:

On the left I’m positively melon-choly. On the right I don’t understand w(h)at ‘er problem is.

You start the same, cutting a watermelon in half and laying it flat side down for safe cutting.  Instead of making on vertical slice and lots of horizontal slices (as you would for wedges), you make equal numbers of vertical and horizontal cuts.  This forms a lattice- it’s best if each square is about an inch to an inch and a half long.

PRO TIP: cutting board that fits inside a baking sheet = no mess to clean up after cutting

Wedges were very in fashion a few years ago, but now they’re passe

It’s all about the stilettos now (I’m not sure what stilettos are but they’re definitely skinnier than wedges)

Discard the corner/end pieces that have no watermelon, and voila!  Watermelon fries!

Now you don’t need to go to a generic American chain restaurant to say TGIF! (Because everyday can be Fry-day)

Here’s a real life version of the right side of the cartoon above:

Now for the second part of the post!  What’s the best way to buy watermelon?  That is, if you can’t carry a 15-20 pound watermelon yourself (for whatever reason).  Sometimes stores carry big slices of watermelon, so I’ve bought plenty of 5 pound slices this summer, which I highly recommend.  But yesterday the store only had a box of precut watermelon for \$3.90 or a personal sized watermelon for \$4.99.  So of course I bought both, for you blog readers (or because I wanted watermelon).

We wait & watch… which watermelon will win the war?

Any guesses?  Because of the labor involved we guessed that the mini watermelon would win, but then again, mini seedless watermelons seem like a luxury item, like a labradoodle or a pluot, so they might charge a premium for them.  Volume wise they look similar, but humans are pretty bad at estimating volumes- we’re good at linear estimation but bad once we get to higher dimensions (how many acres is the lot where you live?) [For that matter, what is an acre?]

I decided to go by weight, but also tried a rough volume calculation ahead of time.  Clearly there’s space in the box, but I didn’t want to measure each piece and calculate volume so I just did the volume of the box.  Similarly, there’s rind in the watermelon and it’s not a perfect sphere, but hey, this is why I’m a mathematician and not an experimental scientist.

Box: 7″ x 6″ x 3″ = about 126 cubic inches of watermelon

Mini watermelon: 6″ diameter = 4/3* pi * 3^3 = about 113 cubic inches of watermelon

Again this volume calculation is pretty bad: you can see how much space is in the box above, and once we cut open the watermelon, how much rind there actually is.  After measuring the box/melon, the first thing I did was weigh the precut melon using my husband’s fancy scale and a bowl.  Then I ate a lot of that melon, and started cutting the mini watermelon (not the smart way).  I wanted to get as much watermelon as possible out of the mini melon, but still cut off all the white parts since the precut didn’t have any white parts.

Cutting a melon this way is not very appeeling- I just did it for science

Mini-melon fries! I guess fries are already mini potato fries (holy crap regular-sized potato fries are often called ‘wedge fries’ all the puns don’t work!)

One thing that I thought was crazy was the next picture: I’ve unwound the measuring tape to 6″, the diameter of the mini melon, but you can see how much more melon there is.  Then again, the bowl isn’t so deep so this isn’t that crazy.

So that’s how long six inches is!

Also, I needed to do three weighings for the mini melon because the scale couldn’t hold that much.

Box: 542.0 grams

Mini melon: 1162.0 grams

The mini melon held twice as much melon as the box did!  It was really crammed in there vs. all the space in the box, I suppose.

So the final tally:

Box: 138.97 g/\$

Mini melon: 232.87 g/\$

So the mini melon is clearly the better deal, right?  You get almost twice as much melon per dollar spent, even though you spend an extra dollar.

HOWEVER.

The precut watermelon was perfect- sweet, juicy, with just the right amount of bite.  I ate almost the entire box while doing this “experiment,” which means I ate an entire pound of watermelon in half an hour.  It was like every piece was from that magical inner zone of watermelon which isn’t too mushy (the exact center) but is still sweet (not touching the rind).

Meanwhile, just about every piece in the personal watermelon was a rind-piece.  It’s just not big enough to get to that magic zone.  Even the store clerk thought I’d picked out a good one!  I’m going to make gazpacho with those two pounds of watermelon because I want to hide the lack of sweetness.

Next time I want watermelon (like tomorrow), I’m going to pay the premium and buy the precut.  I would encourage you, however, to buy a whole watermelon and cut it into fries as above.  That is, if you can carry a watermelon.

## The Apology and why it bugs me

3 Aug

I want to remark at the beginning of this post that I love math people.  We’re a little weird, very friendly, and generally quite open-minded and supportive (at least, this is true of the math people I know, a.k.a. geometric group theorists and friend fields).  There’s one thing that really, really bugs me that many (definitely not all) math people do when talking math with each other.

Also, I’m really into lists right now.

As I’ve mentioned earlier, I’m doing an exciting research program this summer involving four faculty, five graduate students, and three undergraduates doing at least five research projects.  With so many different experiences and different personalities interacting, there are lots of times when apologies are required:

• Interrupting someone in the middle of a productive thought (actually people don’t apologize for this enough.  Reminds me of this post from the What is it like to be a woman in philosophy? blog)
• Stealing someone’s notes/pen/paper/seat
• Talking over someone (similar to the first thing here)
• Probably more things I can’t think of right now

And I’m totally down with all of those.  They make complete sense- apologies are a nice lubricant for social and professional interactions.  But there’s one apology that really bothers me, which comes up in these situations:

• Not knowing something that you’ve never been exposed to/had a reason to explore
• Not being able to read the mind of someone who isn’t communicating clearly (related: this old post on teaching)
• Having a different background than someone else, mathematical or otherwise
• Being better at processing things in a visual rather than audial way, or vice-versa

These all come down to one thing: you’re a different mathematician than whoever you’re talking to.  And this is the thing that you might say in this situation:

Sorry, I’m slow.

I dislike this so much!  I’ve heard very many mathematicians say this over the past few weeks, whom no one would call “slow.”  One reason for my distaste ties in with the whole “women apologize more” bit, explored in a Pantene ad, dissected by Time, and perhaps most effectively explained in this spoken word video.

To be clear, this is not a women-only problem (while I’ve noticed more women do so than men, men also do this).  I dislike the phrase “sorry, I’m slow” because

1. I’m apologizing for an adjective that I’m applying to myself- ->I’m apologizing for who I am.  [I am not a person who likes doing this.  I certainly apologize when I make mistakes/do bad actions, but to judge myself on my character, and invite you to pass that same judgment?  Not fun.]
2. I’m devaluing my contributions to this conversation.  If I don’t take myself seriously, how can I expect you to?
3. By saying these words aloud, whether I believe them now or not, I convince myself and you that I am, in fact, slow.  Just like if I looked in a mirror everyday and said “I’m ugly” I would eventually believe it.
4. I’m perpetuating a system of these apologies- now whenever you’re in a conversation and struggling to understand what’s going on, you’ll be tempted to say “sorry, I’m slow” and cause 1-3 to happen to you.

Maybe the worst part of “sorry, I’m slow” is that there are good reasons to say it: when faculty/those further along say it, it encourages undergrads/younger folks that they aren’t the only ones who feel this way.  Similarly, if you say it in a group of peers, it builds camaraderie (in the way that teenage girls insult themselves in order to get compliments from each other).  When younger people say it to older people, mentorship instincts kick in and older people often share personal stories of some other time they felt slow.

Really what I’m saying is that “sorry I’m slow” is bad because it makes you believe that you’re slow, and it’s good because it tells everyone else that you also think you’re slow.  I just wish people didn’t pass these value judgments on themselves.  =(  I suppose this post is why I’m a mathematician, not a psychologist or sociologist.

From here: http://cheezburger.com/5218979584.  Also, I’m the puppy and the cat.

## Pineapple upside down cake

24 Jul

Last month I thought about going to a party for a friend’s birthday, but then sat inside and played board games instead.  To make up for it, I baked him a cake!  This actually happens fairly often, but without the cake part.

Anyways, pineapple upside down is my second favorite cake (it’s hard to beat carrot cake).  Also, I’ll take this inopportune time to give a shout out to the incredible Alliance Bakery down the street from my old apartment, who made our hilarious and wonderful wedding cake.  It was a delicious light lemon cake with a mango mousse filling and a simple buttercream frosting.  Also, because our initials spell out YAM, we asked for it to be made in the shape of a sweet potato, but to try very hard to make it not look like poop.  They did a fantastic job and I wish I had more pictures of the cake… unfortunately I only have one and it’s in B&W

You should try to get there early; there are often hordes of people at Alliance (WOW reference…)

Also that photo (and all my wedding photos) were taken by my best friend who happens to be a professional photographer, so if you’re in Singapore and need photos taken I recommend her.

ON TO CAKE I PROMISE IT’S NOT A LIE.

One, I already have a spouse. Two, why would you wed pants? Marry-chinos, yeah right.

All I had to buy for this cake was the pineapple and the cherries.  I’m curious what I’ll do with maraschino cherries (any ideas?  Shirley Temples, I guess?).  Also, this is a one-bowl cake, which is so great! (Two if you use a separate bowl to microwave butter)

First, melt half a stick of butter and pour it in the bottom of your cake pan, then sprinkle brown sugar over it.  Takes about 30 seconds in the microwave to melt half a stick of butter.

This is getting me so many brown ie points with my friend

Then cram as many pineapple rings as you can into the pan-I got in six.  If you’re using a square pan you might have better luck.  Also throw in as many cherries as will fit, because when else will you use them?  I realize that I did not need to put maraschino cherries in the cake but they’re literally the cherries on top so it’s hard to say no.

If you like it you should put at least six rings on it

Fun fact: pineapples don’t come from trees.  They’re actually the only edible species of the bromeliad family of plants- this is the only fact I remember from signs at the University of Washington greenhouse in 2011, when I visited to see a corpse flower.  Those are flowers that smell like rotting carcasses, hence the name.  I recommend checking it out if you’re ever near one (they don’t bloom often).

Back to cake.  Melt a stick of butter in the bowl (again I did 30 seconds and it gets almost all the way melted, and if you stir the blobby butter it’s basically melted), and mix in the sugars.

Everyone was bowled over by this cake

Does not melting the butter all the way really work? You butter believe it!

Then add the rest of your wet ingredients.  It really does help to mix butter and sugar together at the beginning of your recipes when baking, then adding wet, then dry.  Otherwise the sugar doesn’t incorporate as evenly/well.

Yes, I milk my jokes as far as I can- please laugh to protect my eg(g)o

I sort of gave away what happens next.  Here’s another chance to use more bowls, if you mix your dry ingredients first.  Or you could sort of sprinkle the other dry ingredients over the flour and whisk it all together (one less bowl to clean!)

Whenever someone says they like my blog, I’m floo(u)red

Also have you used rubber spatulas (not an Amazon affiliate link I’m not that fancy I just googled ‘rubber spatula’)?  They’re amazing for scraping!  Fun fact I was first introduced to these when I was about seven and baking a cake with my Swedish grandmother (long story not actually related to her but did go to her house every month and called her “Grandma”), and she showed it to me and called it something like a ‘fun ruiner’ or ‘child spoiler’ because it doesn’t let you lick the bowl.  Cool story Yen.

If you couldn’t tell the program I’m at right now is sort of intense and I’m a little unfocused because I’ve been mathing all day.  Yay math!  It’s SO MUCH FUN I’ll have to blog about this summer’s math sometime too.  Random groups!  Nilpotence!  I like that word because it’s like you’re so helpless you aren’t even impotent, you’re nilpotent (this has nothing to do with the definition of nilpotent).

BUT SERIOUSLY LOOK HOW AMAZING THIS SPATULA IS

Right, cake.  Pour the batter over the pineapple topping, pop it in the oven for 40 minutes, let it cool for a bit, and invert it onto a plate.

Yes I love this spatula.

Also new cake pans, thanks wedding registry!

Pineapple upside down cake- recipe lightly adapted from Sally’s Baking Addiction- honestly I suspect hers might be better than this one so you should check that out.  But if you want the cake I made see below

3/4 c butter (1 and a half sticks)

1 1/4 c brown sugar

1/4 c white sugar

can of sliced pineapple

maraschino cherries

1 egg

1 1/2 c flour

1/2 tsp baking powder

1/4 tsp baking soda

1/4 c oil

1/2 c milk

1 TB vanilla

First, melt half a stick of butter in the microwave, then spread out into a cake pan.  Sprinkle 1/2 c brown sugar over, then top with pineapple slices and cherries to your liking.  Don’t throw out the can of pineapple!  Measure out 1/4 c of the juice for later.  Turn on oven to 350.

Next, melt the other stick of butter, and mix in remaining brown sugar and the white sugar until smooth.  Add in the egg, pineapple juice, oil, milk, and vanilla, and mix until smooth again.  Then add flour and sprinkle the baking powder/baking soda over.  Mix well.

Spatula that batter into your pan, and bake for 40 minutes or so, until a toothpick in the middle comes out clean.  Let cool.

Run a knife around the edges of your cake, then put a plate on top of it.  Flip the whole thing upside down and lightly bang on the bottom of the cake pan to drop the cake onto the plate.  Yum!

## Chris’s super-easy and super-impressive French toast souffle

15 Jul

I’m a huge sucker for deals.  The other day I wanted ice cream and was going to buy a pint, but then saw that for just \$1 more I could get TWO half-gallons (a two for the price of one deal).  So I ended up getting those, only to realize when I got home that I’d accidentally bought LOW FAT ice cream (egads!)  So I feel like I got suckered into buying a product that I didn’t want.  That said, we’re still eating the ice cream; it’s just not as good.

That story was just an intro to say that this dish is a really good deal.  It’s very, very little work for a lot of return-literal “ooh”s and “aah”s.  My friend Chris (who, incidentally, made one of my favorite sites involving Rodents Of Unusual Size)  sent this to me via one of those chain-email recipe exchange things that never work.  Lucky for me I got this recipe, which I’ve made again and again since 2011.

You’ll maybe not want to whisk life and limb for this souffle, but definitely try it to your fullest CAPacity

I realize I took the above photo a little bit late- the first thing you want to do is put a stick of butter in your pan of choice (Chris likes deep and narrow; I like whatever I have on hand), and stick it in your oven, which you’ll heat to 425.

Not a pun: I think I’ll start calling grilled cheese sandwiches “butter melts”. Sounds so good right now.

While the butter is melting, put all your other ingredients in a bowl and whisk, whisk away.  I do the eggs first, then other stuff.

I hate it when people sugar-coat bad news

Just give it to me and maybe I”ll figure out if it’s a mixed blessing or not

By the time your oven is preheated, your butter should be melty and bubbly.  Pull out the pan, pour your bowl of delicious stuff right on top of the butter, and toss it back in the oven.

If someone with ombre blonde-brunette hair hangs from her knees on monkey bars, would you call her upside-Brown?

This POURcelain dish is perfect for what’s happening here.

Who needs root beer floats when you can have a butter float?

After thirty minutes, you’ll have a gorgeous, poofy souffle that took none of the work compared to usual souffle recipes.  I’ve had it with cinnamon sugar on top, maple syrup, or my favorite, toss on cut up berries (any kind) and a spoonful of sugar- the heat will slightly cook the berries and sugar.  Serve IMMEDIATELY (or the poof will go down) with a side of berries.  I love the almost crispy chewy crust in contrast with the creamy center pieces.  It’s all delicious.
Chris’s French Toast Souffle
1/4 cup (1 stick) butter
3 eggs
1.5 cups milk
6 TB sugar
3/4 cup flour
1/4 teaspoon salt
Optional toppings: more sugar, cinnamon, maple syrup, berries…
Put the stick of butter in a pan, place in oven, and preheat oven to 425.
Whisk eggs in a bowl.  Add milk, sugar, flour, and salt and mix until smooth.
If the butter isn’t bubbling yet, wait.  When it is, pour batter into the pan over butter.
Bake for 30 minutes.
Serve immediately with desired toppings.

## The fault in “The Fault in Our Stars” (Cantor’s diagonalization argument)

8 Jul

I love The Fault in Our Stars- I’ve read the John Green book three times and my husband and I have discussed at length how much we like it.  We just watched the movie last weekend and thoroughly enjoyed it (not as much as the book, but it’s a tough act to follow).

There’s just one little thing wrong in it-a bit of math!  Both my officemate and my fairy blogmother have posted on social media about a flaw in something that Hazel says, so I thought I’d take a post to explain both a) why Hazel is wrong and b) how the argument that she refers to works.  For the record, I believe that c) it’s fine for teenagers in books to not fully understand these sorts of mathematical arguments, and it’s unclear whether John Green believes that the infinity between [0,1] is the same “size” as the infinity between [0,2] or not.  This is true and is what we will show, but Hazel says (beautifully) in the book:

There are infinite numbers between 0 and 1. There’s .1 and .12 and .112 and an infinite collection of others. Of course, there is a bigger infinite set of numbers between 0 and 2, or between 0 and a million. Some infinities are bigger than other infinities… I cannot tell you how grateful I am for our little infinity. You gave me forever within the numbered days, and I’m grateful.

Makes me tear up a little every time.  And not because the math is wrong, but because the writing is so pretty.  But the math is, in fact, wrong.  All those infinite sets are the same size.  But she is right that some infinities are bigger than others.

Speaking of math, I’ll do just a little bit more reminiscing before we dive in.  During my sophomore year of college, I took Set Theory with the other two math and philosophy majors (some of my now-best friends) and we saw this argument for the first time.  Blew.  My.  Mind.  This is also one of my two cocktail party math tricks (I’ve already written about my other one).  I’m not that fun at cocktail parties.

So.  Let’s talk about sizes, shall we?

Let’s first talk about finite counting.  How do I know that I have three potatoes?  I look at the potatoes and I assign a number to each one.

These potatoes are hiding their sadness- not enough eyes compared to their peers.

Another way to say this is that I have written a bijection between my set of potatoes and the set {1,2,3}.  That is, a function from one set to the other so that each item in set A gets assigned to exactly one item in set B (1), and so that each item in set B has someone assigned to her (2).  Property 1 makes a function injective, and property 2 makes a function surjective.  If your function is both injective and surjective, we say that it’s bijective.  One way to imagine a function is lining up all the items of your sets next to each other, and drawing a back-and-forth arrow between the items of the set.

These po’ taters… these smiles are hiding so much pain

And we can do this for any finite set.

Click photo for credit. I see a bijection to the set {1,…,14}, so there are 14 bunnies.

So far so good, this is just normal counting.  But what about counting to infinity?  Is there more than one kind of infinity?  (Spoiler alert: yes).  How could we know?

To deal with infinity, we extend how we count from finite sets.  We say that two sets are the same size if there exists a bijection between them.  For instance, the set of bunnies and the set of potatoes are not the same size, because there is no bijection between them.  I can see that by trying to make a bijection between A={1,2,3} and B={1,2,3,4,5,6,…14}.  No matter how I assign a number to 1, 2, and 3, there are still 11 numbers in set B that aren’t assigned a number in set A.  So I can’t have a surjection, therefore I can’t have a bijection, therefore A and B are different sizes, ergo the size of the bunny set is not the same size as the potato set.

Technical aside: if there is a bijection between sets A and B, and another one between sets B and C, you can put them together and get a bijection between sets A and C.  This jives with our intuition that if A and B are the same size, and B and C are the same size, then so are A and C.  “Being the same size” is an example of an equivalence class, a math term that we may use in the future (but not right now, so no formal definition just yet).

Here’s a first example of playing with infinity.  I claim that the set of natural numbers {1,2,3…} is the same size as the set of even natural numbers {2,4,6…}.  Proof by (completely unnecessary) picture

Eye just like putting eyes on things

This is our first idea of infinity.  In fact, we use this infinity so much that we have a word for when a set is the same size as the natural numbers: we say that set is countable.  Here’s the next question to ask: are all infinite sets countable?  Is there an infinite set which isn’t countable?  Or as Hazel puts it, are there some infinities that are bigger than others?

Here’s where Georg Cantor’s ingenious argument (from the 1870s) comes in.  Let’s try to see if the numbers between 0 and 1 are countable.  We’ll do so the same way we tried to see if there were the same number of bunnies and potatoes: start building a bijection, and come up with a contradiction.

Fact: every number between 0 and 1 can be written uniquely as an infinite decimal expansion.   By infinite, here, we mean countable.  Examples: 1/100=0.010000000…. 1/3= 0.33333333….. 1/11=0.09090909090….

So let’s start building our bijection by putting the natural numbers on the left, and infinite decimal expansions on the right.  I started with these examples here:

Assume I have finished building my bijection and have a list of all the natural numbers on the left, and the numbers between 0 and 1 on the right.  We’re going to build a number between 0 and 1 that doesn’t show up on this list.  This proves that our assignment function isn’t surjective, and therefore isn’t a bijection.  And our argument will hold for any way that you try to build a bijection, and so it’ll show that there are in fact no bijections between the two sets.

How do I build the number?  Let’s look at the first coordinate in its decimal expansion.  I look at the first number in my list, 0.01000000, and look at its first coordinate: it’s a “0”.  So we’ll assign any digit besides 0 to the first coordinate in our special number.  Let’s say it’s 1.  So far, my number is 0.1_________.

Now let’s look at the second coordinate.  I look at the second number on my list, 0.3333333333, and its second coordinate: it’s a “3”.  So let’s make our second coordinate a 4.  Now my number is 0.14______.

Third: I get 0.141________________

I keep going.  To make my ith coordinate, I look at the ith number in my list, and the ith coordinate of that number.  Then I pick any digit which is not that one, and assign it to my ith coordinate.

Now let’s say we’ve finished building our special number coordinate by coordinate.  Is it somewhere in our list?

Think about it for a second…

….

Answer: no!  If our special number were on our list, it’d have to be assigned to some natural number n.  But by the very way we built our special number, its nth coordinate is not the same as the nth coordinate of the number assigned to n.  So it can’t be on the list.  So our special number, which by construction lies between 0 and 1, is not in the image of our function.  So our function isn’t a surjection, and hence it isn’t a bijection.

This means that we do have more than one infinity: the infinity in [0,1] is not the same size as the infinity in {1,2,3….}.  Some infinities are bigger than others.  (we finished goal b)

But some aren’t. (goal a).  In particular, the bijection we did earlier between the sets {1,2,3…} and {2,4,6…} works to show that the infinity between 0 and 1 is the same size as the infinity between 0 and 2.  That is, $x \mapsto 2x$ is both injective and surjective.  Also, $x\mapsto 1000000x$ is also a bijection, so the infinity in [0,1] is the same size as the infinity in [0,1000000].  Sorry Hazel, you’re wrong in your example (but the idea is correct!).

I hope you’re enjoying a little bit of infinity in your day!  (Actually, a lot of infinity.  Or at least, uncountably much.)

## The best burger ever

2 Jul

I went through a big burger phase a few weeks ago (we may have had four burgers in three days), and this recipe really is the best burger ever.  It was better than the \$14 burger at the fancy butcher shop across the street.  It’s better than any burger I’ve had.

Don’t get me wrong, I love a good barbecue and grill marks and all that (in fact I had my first grilled burger of the summer last night!).  But you just can’t get the same juiciness on a grill as you can in your cast-iron (because that juiciness will just drip down the grill and away from your burger).  Honestly I’m not sure how much the “smashing” step does for the crust of the burger (wouldn’t throwing a patty into a searing hot cast iron sear it just as well?), but I do it anyway because this recipe has worked out so well for me.  It’s also made me realize why people follow recipes closely- because they work, over and over again!

After we got that cast-iron for Christmas, I started cooking a lot more meat.  By now I’ve made these burgers three or four times and they’re delicious every time.  I also got a meat grinder attachment for my KitchenAid (thanks in-laws!), so the first few times I ground my own meat (half chuck, half sirloin).  But we live across the street from an excellent butcher and their fresh-ground meat is just as good.  However, if you don’t have access to fresh-ground meat and just see the stuff in the store, I highly recommend seeking it out.

Yeah, I work out. That’s how I got such nice-looking buns.

I love ketchup, but these burgers are so good that ketchup would just distract you from the flavor-all you need is that melty cheese and sweet grilled onion (pressed right into the patty), and maaaaybe a slice of tomato/lettuce.  Honestly I put on the tomato just to please my husband (because then there’s a bit of health on the plate).

Anyways, let’s start with meat grinding.  If you’re using the one I used, you’ll want to slice your meat into strips (maybe 1-2 inches wide), and then throw it in the freezer while you put together your machine/do something else for awhile.  Don’t forget when cutting chuck to AVOID the white ligament-y parts (they’ll get stuck in the grinder and be a hassle).  Another reason to use sirloin (which is delicious!)

I was really nervous to meat his parents

I thought it’d be a terrible grind

Apparently if you want it to be as finely ground as at the store, you should grind your meat twice.  I didn’t do that and it was still delicious.

Next, start heating up that cast-iron skillet on medium-high or high if you like to live dangerously.  The cast-iron skillet is key.  Incidentally, the original website I got this from uses a big green egg for cooking, and a friend of ours has a crazy amazing website all about the Big Green Egg if you’re into that.  I’m very impressed by it.

While that’s heating up, slice your onion up (not the smart way) laterally so you get some rings, and separate those out.  Take out your cheese of choice.  Defrost your buns.  Slice your tomato.  Then make some meatballs!  Each of mine was 1/4-1/3 lb.

I was afraid my jokes would be too cheesy, I’d turn red as a tomato, I’d make someone cry, or flip out, and/or all of the above. But my in-laws think I’m the greatest thing since sliced bread!

Now put two or three meatballs in your cast iron and let cook for 30 seconds (I used my microwave timer).  Reset your timer for 2 minutes.  Smash down the meatballs (I need a metal spatula!), and press some onion rings into each one.  Then generously toss on some salt and pepper (THIS IS ALL THE SEASONING YOU NEED- use good meat!).

I shouldn’t have worried- neither of us has ever really gone ball-istic in stressful situations.

Overall it was a smashing success!

I just had to remember to be myself, not un-Yen.

Now’s a good time to think about toasting those buns.  I thought the next step would be hard, but it was actually super easy- flip the burgers so that they land on the onions.  You can smash them down a little to hold the onions there.  Let that cook for another two minutes, then put on a slice of cheese (if you want).  If you don’t want the cheese, you should still do the next step: cover with a lid and let cook for one more minute.

Cheese Louise I’m done with the in-law stories, I promise