Tooth Update

Thanks for all your kind words in the comments of the previous post. It meant a lot to me. I really appreciated hearing from all of you, even those of you who are complete strangers to me. Thank you for taking some time out of your busy life to comfort someone you don't even know.

I took Vinny to the dentist yesterday morning. He's a pediatric dentist who was recommended by my (extremely young, fresh out of school) dental hygienist, who went to him when she was growing up.

Anyhow, the dentist took a look at his tooth and told me that it was unlikely that the baby tooth damaged the bud of the permanent tooth growing above it, and if it did, it probably didn't do much damage. The baby tooth could take up to a year to descend back to where it belongs. At this point, we don't know if it's going to die or not, but I'm to bring Vinny back in a month and we'll reevaluate then.

I felt relieved by the visit, but it was a bit traumatic for Vinny. The nurse and I held him down while the dentist checked his tooth and squeezed out some gooey white substance from under his gum. "Is that pus?!?!" I gasped in horror. But he said he didn't think it was, because not enough time had passed for so much pus to form. But just in case, he wrote a prescription for amoxicillin. And Jeff and I are to wipe Vinny's front teeth with a piece of gauze with peroxyl mouthwash solution on it, twice a day. It feels so cruel to do because it makes Vinny weep and beg for us to stop. But it has to be done.

Anyhow, I'm feeling better about the whole thing and Vinny is getting better. He's able to sleep through the night now (unlike the first two nights) and his appetite has returned. The mouth is still tender, though; tonight, he was goofing off and waving his arms around and accidentally bonked himself in the mouth, which clearly hurt a lot. But in time, it will be better.

The Worst Mother Ever

I am the Worst Mother Ever.

We were having a pleasant morning. Vinny and I were horsing around on the bed. Jeff was in the kitchen making his own breakfast.

Vinny loves to tumble. I try to be very careful and make sure he doesn't tumble off the bed. But he was closer to the edge than I had thought, and I didn't catch him in time. He tumbled another full rotation and landed on his face.

He was screaming. I grabbed him and held him to my chest, trying to calm him down. Then I noticed that there was blood coming from his mouth. I thought he had cut his lip or bit his tongue, but he screamed bloody murder when I tried to take a look and I saw that his front left tooth was pushed up into his gums. I called to Jeff and said we'd better take Vinny to the doctor. He called the doctor's office and they said we should take him to the emergency room. We decided to take him to the Children's Hospital because they seem better equipped to handle children than does our local emergency room.

I tried to hold some ice to his mouth while we were going there, but any kind of pressure on it made poor Vinny scream bloody murder. After what seemed like an eternity, we arrived at the hospital. I took him in while Jeff parked the car.

His tooth got pushed up into his gum, and it should descend on its own, according to the doctor. We need to take him to a dentist sometime within the next week or two, to determine the extent of the damage and whether the baby tooth has damaged the permanent tooth bud. He said to give him tylenol and ibuprofen, and a soft diet for seven days.

I looked to see what Dr. Google says the prognosis is for our sweet boy's smile. According to this site, if it's pushed up more than 6 mm (which I think it is), it's pretty much hopeless. The tooth will die and will probably need to be extracted.

I feel like the worst mother in the world. I should have stopped him from tumbling so close to the edge of the bed. Thanks to me, he's probably going to have a huge gap where his front tooth belongs for the next 5 years. I will take him to the dentist next week and find out for sure.

A Landmark Decision

I was ecstatic when I heard the news that California's State Supreme Court has struck down the state's ban on same-sex marriages!

This is a very important issue to me, because half of the members of my first family (excluding myself) are gay. I have always been a proponent of marriage equality, but the fact that it directly impacts my loved ones raises its importance to me.

I am pleased by their decision because it underscores the fact that marriage, in the eyes of the government, is independent of religion. The only objections to same-sex marriage are religious in nature, and therefore irrelevant when making the decision on whose commitments can be granted legal recognition.

I got goosebumps when I read the wonderful news in an email from the ACLU. Here are some particularly poignant excerpts:

Simply having the California Supreme Court say that constitutional principles demand that marriage be open to same-sex couples is an enormous win. This Court has a remarkable history of leadership on civil rights and civil liberties. It made landmark decisions on race and sex discrimination, on freedom of speech and privacy, and on treatment of the disabled and poor people long before the U.S. Supreme Court. No court in America has more authority to say that marriage for same-sex couples is an issue of basic freedom than this one.

...

Marriage in California will transform the discussion of marriage nationwide. California has one of the largest economies in the world. Given the state’s economic clout, the fact that California is marrying same-sex couples will put considerable pressure on the rest of the country to recognize those marriages.

Even more important, the rest of the country recognizes that California is America’s cultural trendsetter, that cultural change in California is usually a preview of what is to come in the rest of the United States. Most Americans already believe that marriage for same-sex couples is bound to happen sooner or later. I think marriage in California will help persuade many of them that this is an issue of basic fairness, and that the time for it is now.

...

Thank you, California, for giving those of us who value equality a little hope for the future! With a lot of elbow grease and a little luck, the committed relationships of same-sex couples such as my own family members could be legally recognized within a decade, which pleases me to no end.

Vinny's Latest Obsession

• In the house, pointing at the front door: "Opeeeeeen! Opeeeeen! Moon!"
• In the car, looking through the sunroof: "Moon!"
• Upon seeing a streetlight: "Moon!"
• Upon seeing a cookie, a coaster, a croissant, a Boppy pillow, nearly anything round or crescent-shaped*: "Moon!"
• The only thing that can cheer the inconsolable little lad, who was sobbing late at night before he spiked a fever: Google image search for moon
• The top three entities in his life: "Mama, Daddy, Moon!"

*You might think that a pizza would also elicit this response, but pizza (or, as he calls it, either "piZAAAAAA" or "miZAAAAA") is fourth on the list of important things (beating out prunes, shoes, and refried beans), and is therefore generally not conflated with the moon.

Sickness Strikes the Family

It's virus central here at Casa Rebecca. Vinny had a temperature of 103.5ºF and spent most of the day moaning yesterday. We took him to the doctor and while he did not have strep, he did have a nasty sore throat. But today he bounced back and is doing fine, just in time for me to feel miserable. I'll probably be fine by tomorrow, just in time for Jeff to get it.

There are some foods that I crave when I'm sick. A big one for me is canned fruit with cottage cheese. The most comforting is canned plums, but I have to admit, I have a weakness for canned peaches and canned pears too. In fact, I just snarfed down a can of peaches that was rightfully Vinny's. Jeff always needs a grilled cheese sandwich when he's sick.

What's your comfort food?

May Scientiae Carnival

The May edition of the Scientiae Carnival is up at my blogfriend Flicka Mawa's place, A Cat Nap. The topic of the carnival is "career paths, perspective, and changing self-image." Flicka Mawa was kind enough to include my graphical interpretation of my ever-evolving self-image.

Go forth and read it! It's really great! Thanks, Flicka Mawa, for putting together such a fascinating carnival!

In Which I Solicit Advice from My Vast Readership

O Vast Readership, maybe some of you will be able to help me figure out what I should do with the latest opportunity I have been afforded!

Later this month, I will be speaking at my local middle school's career day. My topic is "Math Careers," and I have a half hour allotted to me during each math period. My target audience is 8th graders (13-year-olds), I believe, but the school goes from 5th-8th grades.

I'd like to engage them and keep them interested in what I'm talking about. I could drone on and on about things that they will probably find boring, but what I was thinking was that quite honestly, every job they will ever have will involve math to a greater or lesser degree. I thought I might inform them of that fact, give some examples (e.g., cooking, which requires knowledge of fractions, multiplication, division, addition, and subtraction) before proceeding on to more math-based careers (such as scientist, engineer, computer scientist, and mathematician). But I'm not really sure what specifically to say and I'm not sure how I'm going to fill a half-hour, or maybe let's say 25 minutes with five minutes for questions. ;)

This is where the expertise of my vast readership comes in. I was an atypical 13-year-old in that I would have actually been fascinated by anything an actual mathematician had to say, so I am lacking a certain perspective. Any suggestions?

Lilly Ledbetter Fair Pay Act

Sometimes I think I live in the land of dinosaurs! Neither of my senators voted for the Lilly Ledbetter Fair Pay Act. Oh sure, there are misinformed people who believe that by allowing someone to sue for back pay after 180 days, there would be a catastrophic ten percent decrease in everyone's pay, but quite honestly, this does not follow. (If you believe that one, I've got some waterfront property to sell you -- it has a view of the water from all directions! Very lovely!) What it would do instead is make employers that much more careful about systematically discriminating against anyone.

Anyhow, here's what I wrote to my senators:

Dear Senator:

I was disappointed to see that you voted against the Lilly Ledbetter Fair Pay Act recently. I cannot understand why you would vote against a bill that would remedy unfairness in pay.

Senator, I am a woman and the breadwinner of my family. It is a well-known fact that due to bias (both conscious and subconscious), women do not get the recognition that they deserve for their hard work, and this includes promotion and raises. I am a scientist, and there have been peer-reviewed, double-blind studies that have shown that a woman scientist has to be more than twice as productive as a man scientist in order to be considered equally competent. If a man must publish ten papers a year, then a woman must publish twenty in order to be considered his peer.

Therefore a woman who works as hard as her male counterpart will not receive the same reward, and fall behind in pay. It takes time to ascertain the pattern of iniquity, and this bill seeks to allow women and others subject to this type of discriminatory practice the time necessary to do so.

Senator, I encourage you to change your vote the next time this bill comes up. My family deserves your support in making sure that I am remunerated as much as I deserve.

Sincerely,

[me], Ph.D.
Computational Scientist

My Awesome Husband

My husband is the coolest guy in the world. Here are a few of the myriad of reasons why:

• He is very brave. Like me, he defies gender stereotypes in his chosen vocation. It takes a lot of courage to be a stay-at-home dad and face gender discrimination every day. I admire him for doing what he wants to do despite the pressure to do otherwise.
• He is very loving. I love to watch him play and interact with Vinny. It brings so much joy to my heart to see the way they interact! Most of what I know about parenting, I've learned from him.
• He is a really good cook. When we first got married, he didn't know much about cooking. I taught him everything I knew, but he took that knowledge and ran with it. He has a knack for combining flavors in just the right way. Last week he made this catfish with lime and some combination of herbs and spices, and it was just superb. He didn't get the recipe from a cookbook; he just made it up! He has a blank book in which he writes all his special recipes, and I told him he definitely needs to add that one to the list.
• He is amazingly artistic. He has more artistic talent in his big toe than most people have in their entire body. He's recently taken up drawing. Most of the time, people start with simple compositions, such as a still life. But Jeff started by drawing portraits and figures. He is getting really good. The drawings are becoming more realistic every time. I am amazed by how well and how quickly he has picked up this ability!
• He is incredibly creative. In addition to his artistic talents, he has a knack for storytelling, which he uses for role-playing but also to develop plots for books. He has several books of fiction all planned out in his head. I keep telling him to write one down already so that we can live off all the money it will bring in! ;)
  
• He is empathetic. There's nobody in this world who understands me as well as he does. He really gets everything I say. I think he's also the perfect feminist boyfriend. He sees the damage that our patriarchal society does to women and to men. There aren't too many men who really get it like that.
• He loves me. I am continually baffled by the fact that despite my numerous character flaws, he really cares about me and wants to spend his life with me. I try not to let on about my bewilderment, but I always keep in perspective how fortunate I am to have him as my partner in this journey through life!
  

Game Theory and Human Behavior, Part II

Remember the post Game Theory and Human Behavior, Part I? What, you don't? Okay, so I posted it back in February. Obviously it was meant as the first of a series of posts. You might want to read that one again before reading this post, because this one uses some of the concepts I explored in the first post.

The practice of torture is another signaling game. This one is a bit more complicated than the good employee signaling game or signaling in contract bridge, but the premise is the same. Let's call the person doing the torturing the investigator, and the person being tortured the detainee. (I refer to them both as male, since they always are on television and in the movies.) The idea is that the investigator seeks truthful information from the detainee. The investigator doesn't have the same knowledge space as the detainee, although the two may overlap. (If the investigator knew what the detainee knew, he wouldn't have to talk to the detainee at all!)

The detainee, on the other hand, seeks to provide the investigator with only enough information to satisfy him and stop the torture. So the detainee wants to give the investigator the very minimum amount of information that would satisfy his curiosity.

Economist Roger Koppl performed a thorough evaluation of the efficacy of torture in a 2005 paper entitled "Epistemic Systems." In this paper, he determines that torture works only if the following two conditions are met:

1. The investigator must be able to recognize the truth when he hears it.
2. The investigator must be able to credibly commit to stopping the torture upon hearing the truth.

If the investigator cannot recognize the truth when he hears it, then he won't find the truth regardless of the method of questioning. If the detainee doesn't believe that the torture will stop upon telling the truth, then he has no incentive to tell the truth.

Here's the problem: These two conditions are both necessary for torture to work, but they almost never occur together in real-life situations.

Consider the first condition. The more the investigator knows about the knowledge space of the detainee, the better he is able to recognize the truth. But how can he recognize the truth when he hears it, if he isn't privy to everything within the knowledge space of the detainee? This is how the detainee can get away with providing the least useful information out of the set of relevant information that he knows, or even worse, information that sounds truthful to the investigator but turns out to be false upon verification.

The second condition is similarly difficult to satisfy. Consider the case of a Chilean rebel, who broke down and confessed the names of the nuns and priests who had sheltered her. Her devout interrogators didn't believe that these holy men and women were involved, and continued to torture her.

Even if torture were effective on actual "bad guys," the consequences of using it on people who are actually innocent bystanders are the things nightmares are made of. In Discover Your Inner Economist, Tyler Cowen discusses the nightmare situation of being mistaken for a spy when you are nothing but a clueless American on vacation. How can you signal to your captors that you are not a spy? Anything that you do, from protesting your innocence, to providing them with information that sounds promising but turns out to be false, is indistinguishable from what a real spy would do in that situation.

There's always the situation in spy movies and political thriller TV shows such as 24, in which the entire population of New York City is in imminent danger from a ticking time bomb, which was set by a fanatical terrorist of some sort. The hero has to beat the information about the location of the bomb out of the bad guy. Usually it's a renegade cop played by somebody like Bruce Willis -- a guy who goes against the rules and saves the day, much to the chagrin and grudging gratitude of his superiors.

While these sorts of movies and shows are entertaining, they are not reality-based. Dr. Jean Maria Arrigo, a social psychologist with a mathematics background, refutes the ticking time bomb scenario in her papers "A Utilitarian Argument Against Torture Interrogation of Terrorists" (Science and Engineering Ethics (2004) 10, pp. 543-572) and "Torture, Terrorism and the State: a Refutation of the Ticking-Bomb Argument" (Journal of Applied Philosophy (2006) 23(3), pp. 355-373), co-authored with Vittorio Bufacchi. I focus upon this second paper.

Bufacchi and Arrigo first formalize the scenario. The dominant elements in the story are that first, a great number of lives are at stake; second, there is a time at which the catastrophic event will occur if it is not prevented; and third, a person privy to knowledge that will allow authorities to intervene and prevent the catastrophic event has been captured. Under these circumstances, some philosophers and legal experts believe that torture may be justified. A naive utilitarian argument might go something like, "we're causing one person some pain so to prevent that person from causing millions of others even more pain."

But this is an argument based on fiction. First, it assumes that torture is necessary for the captive to give up his information, when the opposite is more likely to be true. Consider the case described in Bufacchi and Arrigo's paper (p. 359):

Five foreign terrorists were captured by the local [counterterrorist police team]. All were found under arms with explosives and maps of targets....The question of how many [terrorist] cells were to be sent to the country to other targets was of interest. The first three terrorists were not even questioned, only shot. The next two were asked the question separately. One shot was heard. The officer said to the last terrorist, ‘Do you also want to remain silent?’ The guy began to lay out the entire operation, the training the cells had received, where they were to meet, where the weapons depots were located, and the route that the terrorists were to take to exfiltrate the country....The other cells were picked up along with in-country support personnel.

How was the fifth man chosen, you may wonder?

The group was searched and then fed and given tea as per the Shariat law of the Koran. Those who refused to eat or drink and made intense hostile eye contact were selected as the first three. The body posture decided who went fourth. The youngest, who ate the bread, drank tea, and thanked his captors was determined to be the least experienced. His AK rifle was not even clean...and he did not appear committed to the jihad.

So, they simply profiled the men they captured, picked the one who was most likely to confess first, and just asked him. No torture was necessary to obtain the information they needed. Once he was free from the influence of the more senior terrorists, and felt well-respected by his captors' adherence to local custom, he readily gave the police all the information they needed. Like the majority of "ticking-bomb success stories, the efficacy of torture interrogation is demonstrated only if the case is framed on that premise."

There are several premises upon which the argument for forward-looking interrogational torture is based. I will use Bufacchi and Arrigo's outline of the argument to explain this (Notation: P = premise, C = conclusion).

• P1: The terrorist has been captured.
• P2: Torturing the terrorist will make him reveal crucial information about the catastrophic event.
• C1: The terrorist must be tortured.
• C2: The crucial information will be revealed.
• C3: The catastrophe will be averted, saving the lives of many innocent people.

Unfortunately, the conclusions C1, C2, and C3 do not follow from premises P1 and P2. There are some hidden premises that must hold true before C1-3 can follow, and none of these premises are legitimate "from an empirical point of view" (p. 360).

P1 has an invisible, companion premise:

• P1A: This terrorist possesses crucial information about the catastrophic event.

Remember the overlapping knowledge spaces of the torturer and detainee? Well, they don't completely overlap; if they did, we wouldn't need to extract information from the detainee. So this premise is an assumption, and we could end up torturing some innocent bystander who was indistinguishable (based on our knowledge space) from a real terrorist.

P2 is problematic, too, as I have discussed above. But even if we assume that Koppl's conditions are both met, there are other problems. Many American POW's in Vietnam (e.g., John McCain) withstood a great deal of torture without confessing anything. Similarly, most victims of the Inquisition confessed to no crimes. Countless torture victims in the Algerian Civil War went to their graves with their secrets undiscovered. Terrorists are more than likely trained in the art of withstanding torture, making it even harder to draw any information out of them with this means. Finally, terrorists could be provided with false information to "confess" in order to deliberately throw their captors off the trail -- this tactic was recommended by Sun Tzu more than 2500 years ago!

And even if torture did yield information from the terrorist, what is to say that it will happen in a timely fashion? After all, if the time bomb is going to go off in 24 hours, anyone sufficiently devoted to the cause would use every ounce of their fortitude to resist for that length of time. Furthermore, torture interrogation is not a "quick coercion" -- it is a lengthy "degradation of the subject's resistance" over the course of months, not minutes (p. 361)!

According to Koppl, "It is a fair, but approximate, summary to say that, as a means of extracting information, torture works best when it is needed the least."

I couldn't have said it better myself.

My Ever-Evolving Self-Image: A Mathematical Perspective

The topic of this month's scientiae carnival is "our changing views of ourselves and our careers as we pass through time."

Since I am more mathematician than wordsmith, I thought I might express these changes in graphical form. The x-axis represents time, and the colored, dashed lines represent major life events that have exerted influence upon my life.

First up, the amount of change for the better I must bring to this world (or else face certain doom):
A few data points from the previous graph: {(4, President of the United States + Nobel Peace Prize), (8, Nobel Peace Prize), (12, Nobel Prize in Chemistry/Physics/Medicine), (18, Professor at top-ranked Research I University), (32, Best second-rate mathematician in the world)}.

Next, my level of self-confidence:
A few data points from this graph: {(4, I can never do anything right), (12, Junior High -- need I say more?), (23, Grad School is Hard!), (27, My family -- putting the fun in dysfunctional), (29, I made it out of grad school! I made it!), (32, I got a real job!)}.

Third, how convinced I am that I'm doing the right thing with my life:
A few data points for this graph: {(4, I'll never be able to stop being a total screw-up), (12, I have to be a violinist but I'll never make it), (16, I want to be a scientist even though my mother thinks my talents are best spent elsewhere), (21, Physics is hard!), (27, Why did I ever take my mother's opinion of what I should do in life as my own? WTF was I thinking?), (32, I am definitely in the right place.)}.

scientiae-carnival

Power of Two Cake

I know you've been long awaiting a post on my powers-of-two themed cake.  Now that I'm less swamped by work, I can oblige.

But first, a little math.  (I know you all come here for the math!)  Astute commenter Rico mentioned his fascination with the sum of the series 1/2ⁿ (where n = 0, 1, 2, ...).  If you add 1 + 1/2 + 1/4 + 1/8 +..., the sum approaches but never reaches 2.  For a given n, the sum of the series equals 2-1/2ⁿ.  So for n=4, the sum equals 2 - 1/16 = 1+15/16.

I decided to represent this sum in cake.  In order to do so, I needed to bake two square cakes.  I had one 8" square cake pan, but I needed another, so I did as The Joy of Cooking suggested and squared off a rectangular cake with aluminum foil, putting dried beans on the other side:

The cake was my favorite cake recipe: cocoa devil's food cake from The Joy of Cooking.  You've seen me make this cake twice already, so I won't duplicate it here.

Here's how it turned out: the foil edge is a little uneven, but not bad.  I decided to use it as the base of the cake, to hide the unevenness.

My dad and Marvis were visiting, and Dad helped cut the cake into the proper pieces.  You can see him measuring the fractions with a measuring tape.
Here are the pieces after he was done.  So, if the other square represents one, then we have a 1/2, a 1/4, 1/8, and two 1/16s (one of which will not be used), concretely (or maybe, more aptly, chocolately) demonstrating the sum I showed above.
The frosting is a simple buttercream: butter + milk + powdered sugar + vanilla.  Here's the 1 + 1/2 + 1/4.
And here is the completed cake.
Next I decorated it with chocolate chips, to illustrate positive powers of two.  I put four chocolate chips on the smallest piece, four more on the part of the next sized layer that's showing, eight on the part of the next sized layer that's showing, etc.
So that ideally, if you look straight down on the cake, it just looks like it has 64 evenly-spaced chocolate chips on it.
Here's the cake from its most photogenic angle, complete with tissue paper flowers in the background:
I ate the leftover 1/16th piece while I was decorating the cake.  Then, when I served the cake, I had the 1/16th piece, Jeff had the 1/8th piece, and Dad and Marvis split the 1/4 piece.  We all had equal portions of cake and equal numbers of chocolate chips.  How cool is that?!?

I took the leftover cake to work with me the following Monday, so that Jeff and I wouldn't eat it.  I told everyone it was made in celebration of powers of two.  One of my colleagues emailed me with a message of "Great cake -- terrible reason!"  I had to laugh.

I hope you all enjoyed this mathematically-inspired cake as much as I did!

Busy Busy Busy

I have been so busy for the past two weeks that it's just not funny. I have been burning the candle from both ends, and probably the middle too.

I'm the PI (principal investigator) for a proposal that's due the Monday after this coming Monday (or, as the British call it, Monday-week). This means I am in charge of writing the proposal, figuring out the budget, and filling out all the dumb paperwork. And because we are soliciting money from the gubmint, then there are all kinds of crazy hoops to jump through too. Adding to the confusion is the fact that this is the first time I have ever had these responsibilities, so I don't know what I'm doing at all.

I worked on the proposal all day on Sunday, and it looks like I'm going to need to devote some more time to it this weekend. I hate doing work on the weekend but it doesn't seem like I really have a choice in this case.

Last week, I taught a one-day course on supercomputing to students and professors from colleges and universities in the region who were attending a conference put on by my division. I had to add some new content to the course, unfortunately, so I couldn't just dust of the old slides and reuse them. It was kind of a last-minute thing and I had to make all the slides the day before. But it was a success and I got a lot of positive ratings. My colleague, who organized the conference, declared that I was amazing for daring to do a live demo of writing a parallel program on the fly. He was especially impressed because it compiled and ran correctly on the first try. I attribute that to the excellent audience members, who found my syntax errors so that the compiler didn't have to.

This week, we had our users' meeting. I gave a presentation on how to use our supercomputer. My boss, his boss, and a very important person from the gubmint thought that my presentation was really good. Personally, I thought it was so-so. But hey, if they were pleased, I was pleased!

I'm really glad that the worst of the busy times are over. I still have to finish up the proposal and jump through all the proper hoops, of course, but it's manageable. Now I can get back to my other projects (work-related and other).

Power Winners!

Sorry, my vast blogging audience, for my involuntary hiatus. I had intended to announce the winners of the contest on the 8th (=2³) but then my life and my job got in the way. So I decided that the 16th, which is two squared, squared, was a good alternative date. I'm future-posting this at 2:32 am, just to get a few extra powers of two in there. I may be awake at that time, but doing all the work I'm supposed to be doing rather than writing this post.

Anyhow, there were some terrific entries in the contest, and it was hard to decide who should win. In keeping with the theme, I will announce them in terms of powers of two.

2² place: A tie between Lost Clown and Rico, who describe their geektastic obsessions with cyclic groups and sums of 1/2^N, respectively. You are my kind of people! When I get around to it, I will post pictures of my powers of two cake, which illustrates concretely (actually, chocolately) the fact that the sum of 1/2^N is finite.

2²-1 place: Rachel, with the bimetalic Toonie, her favorite coin. If you bring those coins south of the border, they'll be worth more than $2. Heck, at the rate the dollar's plummeting, they might be worth $2² American sometime soon!

2¹st place: Laura, who shared a favorite song featuring the powers of two. See, math and music share a special relationship!

and...

2⁰th place: Pete, whose father, mother, and he are each twins. 'Cause that's just cool.

All those of you who are not my family members or colleagues have won a very exciting prize: your choice of a Tennessee ball cap or t-shirt.

Why? Because a) I live in Tennessee, and you don't, and b) Tennessee is a power-of-two themed state name. It has four (=2²) unique letters, each of which appears in it a power-of-two times (T -- once; e -- four times; n and s -- twice each). And, depending on where in the state you live, it is pronounced with either two or four syllables (TEN-see or TEN-uh-say-ee).

If you want to collect your prize, simply leave a comment with your choice, the size (if applicable), and a mailing address to which I can send your prize. The comments are moderated, and I promise not to publish your contact information. If you don't feel comfortable claiming your prize, don't worry about it; that's less work for me.

The Power of Twos

Today is a great day to celebrate the number two! Let me tell you why:

• The date: it's 4/4/08, or, as powers of two, it's 2²/2²/2³.
• The timestamp of this post: 4:00 (=2²) p.m. EDT, or in the 24-hour clock, 1600 (=2⁴) hours. (Attention my workplace: I created this post in the morning before leaving for work, and used the cool new scheduled posts feature in Blogger in draft.)
• Ages: Vinny is one (=2⁰), my nephew Byron is 4 (=2²), I am 32 (=2⁵), and my dad is 64 (=2⁶). Also, as of tomorrow (i.e., in =2⁰ days) Vinny will be one and a half, which is 2⁰+2^-1.
• Events: I've had my new job for half a year now (=2^-1). We've lived here for two (=2¹) years. My dad and Marvis have been married for four (=2²) years (although next weekend it will be five). I took my prelim exam almost four years ago. (Maybe those two almost average out.)
• Other interesting facts: Four (=2²) of the digits 0-9 are powers of two (1, 2, 4, and 8). Also, if we wrote out all these interesting numbers in base two, they would all consist of a single 1 followed by a number of zeros equal to the number of the exponent (e.g., 2⁵ = 100000₂).
  

In honor of this glorious day, I will (of course) be baking a powers-of-two-themed cake. (Stay tuned for pictures.) But also, I wanted to have a little contest for my vast readership.

Within the next 64 hours (that's 2 days and 16 hours, for more powers of two!), leave a comment with your favorite/most interesting connection to powers of two. I will select the "best" 2^N of them, where N could be 0 but might be more, and the winner(s) will not only gain fame but also an appropriately themed prize!

Deadline: 8:00 a.m. EDT, Monday, April 7, 2008
Your favorite/most interesting connection to powers of two.

A Toddler Taxonomy

Kingdom Animalia
     Phylum Mnao: cats, squirrels
     Phylum Ooh ooh aah aah: monkeys, apes, gorillas, Cookie Monster
     Phylum Oof: any other animal (dogs, geese, etc.)

Kingdom Plantae
     Phylum Owish: fruits of the citrus variety
     Phylum Nganga: banana
     Phylum Ppp: grapes
     Phylum Apple: any radially symmetric fruit not of Phylum Owish or Ppp (e.g., apples, pears, kiwi)
     Phylum Twee: trees, bushes

A Tale of Three Cakes

It was the best of cakes; it was the worst of cakes. I was unsure about what I could make to celebrate my sister Laura's successful dissertation defense. I looked around the internet for ideas for graduation cakes, but I couldn't really find anything that was exactly right for a Ph.D. dissertation defense. But it did help me come up with the idea of creating a gigantic tome out of cake and icing. Read on to find out how I managed to bake this beautiful cake:

I was nervous because I'd never done anything decorative with icing, but this idea required me to write her dissertation title in frosting. I decided to reduce my nervousness by rehearsing what I wanted to do. So a few weeks beforehand, I made a practice cake, using the same recipe and techniques I planned to use for the final cake. I learned a lot from my "concept cake" -- first and foremost, that the overall concept was good, but also that pastry bags and my orthopedic problems don't get along. It was good to learn this ahead of time, so that I could buy a mechanical pastry bag before I did the real thing.

But like I said, the concept was there, and it worked out pretty well:

I had one of those frosting comb things that I used to shape the white parts into what looked kind of like leaves of paper between the covers of the gigantic tome.
I had quickly run out of steam when I was using the pastry bag to make a texture on the cover of the book, but the below picture shows the part I did get done and what I had envisioned to simulate the grainy leather for the tome:

None of these pictures show it but one of the other sides of the comb thingy had a pattern to it that looked like the leather binding of a book, so on the fourth side, I used that to smooth the chocolate frosting.


The night before we left for Virginia, I began baking the real thing. The underlying cake is the same cocoa devil's food cake recipe from the Joy of Cooking that I used for the turkey-shaped cake in November. Here is a picture of some of the more interesting ingredients.

For this recipe, you mix the cake flour, baking soda, and salt in one bowl, and then the buttermilk, sugar, cocoa powder, and vanilla in another.
You cream the butter, add the sugar, and beat in two eggs in your main mixing bowl, and then add the flour and liquid mixtures, alternating, in parts.
I had to make two batches because the cake pan is 11" x 15". I didn't try it, but I suspect that my mixer bowl is too small to hold both batches, so instead I did one, emptied it into the cake pan, then hurried and mixed up another. Here's the cake just before I put it in the oven:
And here's how it looked after I took it out:
As you can see, it has a huge crater in the center, and also a crack that goes about 2/3 of the way through it. I was pretty upset, really, because it was after midnight and I was going to have to start all over again. I was also completely out of eggs. What to do?

I just went to bed and resolved to get up early and bake another one before we left.

I tried to assess what had gone wrong with that cake. I had a couple of suspicions so I tried to fix all of them. First, I had baked it at a lower temperature (325 rather than 350 F) because my practice cake was baked at the higher temperature and had been kind of dry. I decided to go back up to 350 because a dry cake is better than a crumbly cake. Second, I thought my baking soda could have been a little bit old so when I went to get the eggs I also picked up some more baking soda. And third, I realized that the center was where the crater was, and it's also where pretty much the entire first batch of batter had ended up, with the second batch filling out the pan to the corners. Its leavening may have all bubbled away while I was making the second batch of batter. So I spread that first batch across the entire bottom of the pan, and spread the second batch on top of it.

Whatever the problem was, I did manage to fix it, as the second cake turned out quite well:
Next, I packed up the non-perishable ingredients
and the recipe (which I copied from a website that has all kinds of icing recipes)
for the icing, packed the crumbly cake pieces to take along for a snack, and we headed for Virginia.

In Laura's kitchen, I made the icing.
I'd never actually used a handheld mixer before so it was a new experience for me.

In her dining room, I began decorating the cake:


I was going to write more than that on the cake (we had figured out a good play-on-words for the subtitle of her dissertation) but I actually ran out of vanilla frosting. This didn't seem to matter, though; the cake was a big hit:
And that, friends, is how I made the dissertation-shaped cake for Laura's party!

Adventures in Histrionics

The manner in which the toddler utters the word "no" conveys the nuances of his intricate thought processes. To wit:

• "No no" [while smiling]: yes.
• "No no no" [while shaking head and reaching for something off-limits]: I know you don't want me to touch this, but I have no impulse control.
• "No!" [shaking head violently while pushing the spoon away]: I'll eat it only if I get to feed myself.
  
• "Nooooo!" [in agony]: I'm so tired.
  

A New Doctor in the House

My sister Laura successfully defended her dissertation on Friday! Congratulations, Laura!!!

She invited her sisters and their families to converge upon Charlottesville for the occasion. Jeff, Vinny, and I came for the weekend, as did Rachel. It was an exciting time for all.

Although we were not able to attend the defense itself, in the afternoon she held a special, informal colloquium about her research. So we got a taste of what she has devoted the majority of her working hours to over the past couple of years.

On Friday night we had a sisters' night out, with just the three of us sisters. We went out to dinner and had a great time together.

On Saturday she held an afternoon soirée celebrating her successful defense, for which I made a delicious, dissertation-shaped cake. (Photos of the cake-baking and decorating process will be posted sometime later, after I get them downloaded from the camera.)

In the evening we left Vinny with Anne's mom and attended a wonderful play called The Elvis Project. A blurb from the venue's website:

The Elvis Project is a collage-like meditation on the multiple meanings of Elvis Presley. A look at how one man, incidentally the King of rock’n'roll, has affected the lives of many. Two short plays: Graceland and ‘Asleep on the Wind’ by Ellen Byron. Featuring works by Charlottesville playwrights Robert Wray and Doug Grissom. Performances are March 13-15 and 20-22 at 8:00pm. Tickets are pay-what-you-can.

During the intermission they served fried peanut butter and banana sandwiches. The guy who played Elvis was a familiar face to Laura and Anne: he works at a restaurant they frequent. Rachel got an autographed copy of a picture of Elvis (the actor's picture, signed as Elvis).

Rachel left early Sunday morning, but the rest of us enjoyed a delicious Easter dinner at Anne's mother's house. We left in the early afternoon and made it back home by about 11 p.m. I took a day of vacation today before I get up extra early tomorrow morning to head to Austin, Texas for a meeting.

Speaking of Children and Phones...

One time when I was a child, probably six or seven years old, I answered the phone. The person on the other end said something unintelligible, and I said I didn't understand and asked him to repeat himself. He said it again, this time interspersing his words with laughter, which made it even harder for me to understand what he was saying!

So I gave up and handed the phone to my older sister (who was nine or ten), who understood that the boy was reciting a long list of obscenities, and hung up on him.