The Number Field

I have been amazed this week by the sheer number of people who read this blog. Moreover, it seems that from the initial flood of people who read my post about vaccination and mathematics, at least a few of you have decided to stick around and see what else I might write. I suppose that it's the goal of any blogger to find an audience, but at the same time I am astonished that there are real people out there, people I have never met in person, who would be interested in reading what I write.

I never thought we would be featured on a list like this. Screencap from January 16, 2010

For the past eighteen months, the blog has served two purposes. First, it's a way to force myself to keep a personal journal and family history. I even published a book version of the blog from its inception until Elena's birth * . Second, it's a convenient and unobtrusive way to let Elena's grandparents and extended family know what's going on in our lives. I plan to continue with those two purposes.

But there's also something appealing about having other readers, who may not be quite as interested in the minutiae of our personal lives. You probably liked my mathematical perspective on vaccination, and would like to see similar material in the future.

Please forgive an anecdotal interlude. On one family vacation, we joked around about doing a TV spoof called The Mundane Race , based on one of our favorite shows . In our version, we planned to have extremely boring, ordinary activities replace the extreme challenges in faraway places. One of us would do our best Phil Keoghan impression and say:

In this challenge, a team member must enter the McDonalds, order a hamburger and then eat it. They will receive their next clue once they finish the burger.

What does this have to do with the blog? I envision a similar kind of project, except with Charlie Eppes as the role model. Instead of a two minute explanation of how math can help solve that week's crime (the standard conceit in Numb3rs ), I plan to put up a weekly post on how math is all around us. I'll keep the posts short and readable, informed by the entrepreneur's elevator pitch . Sadly, I won't have the benefit of CharlieVision ; I'll have to rely on words, along with a few well-chosen illustrations.

For those who want more detail, I'll also provide a more detailed technical summary of the issue at hand. It will still be concise--about two pages, but will include more details, formulas, and references for further study. Because of the smaller audience, I'll make that available as a PDF at the end of the related blog post.

As I write all this down, it appears to be an ambitious project. I'm not sure I'll accomplish the goal of weekly updates, especially with everything else I already have on my plate, but I'm going to try. I want to show people the wonder and usefulness of math, so I won't be writing exclusively about controversial social topics. For instance, some of the things I plan include: a description of music compression, how GPS works, and maybe some thoughts on earthquakes. If you can think of a topic you would like me to address, please leave a comment or send me an email.

As before, you'll mostly find updates about my happy little baby and all of her adventures. In a sense, I'll be thinking of her future self as the audience for my math posts--I want to do my best to help her understand the complicated world in which we live.

* The book is certainly not worth the listed price; I set it up that way to deter buyers. If you are interested in obtaining a copy, please contact me directly.

I have several little incidents from the past few weeks which didn't merit their own posts, but I just realized that they all happened in the kitchen. So that's enough reason for me to group them together.

First, the successes. Last semester I wasn't in the kitchen as much as I (or Jenny) would like. So over the break I've tried to be more helpful and also experiment with new things. One day we had tacos on the menu, but were out of tortillas. Jenny has had limited success with making our own tortillas in the past, so I offered to give it a try. She did have some new information for me: a woman with years of experience in a family breakfast taco business told Jenny that it's essential to let the dough rest, even if the cookbook recipe lists it as an optional step.

I'm no chemist, but I did see this principle in practice when I was younger and my family had apple strudel parties. We used high gluten flour, kneaded it vigorously, let it rest in a warm spot, and were able to stretch the dough paper thin. So I looked up gluten, and found a good description of what happens . Armed with that knowledge, I made the dough at noon and let it rest until dinnertime. The tortillas rolled out thin and round, and made for great tacos, filled with leftover slow-cooker pork from the night before.

This week I took on another project that had frustrated Jenny. She found a recipe for homemade yogurt . It seemed simple, just a few steps. The process took so much time that it turned out to be much harder in practice. Jenny tried three batches, only one of which was really successful. Her last attempt was so discouraging that I offered to take a shot at the next one.

I didn't use the slow cooker, instead opting to heat the milk in a regular saucepan. After mixing in the starter yogurt, I left it overnight on a heating pad inside an insulated cooler to maintain the proper temperature of about 100 degrees. When I checked it the next morning, I was amazed to find a nice firm yogurt under the layer of whey. It reminded me of the first time I made pizza dough that actually rose, after several previous disappointing efforts made me think of myself as a yeast killer. Because of that, being successful on my first yogurt attempt was even more satisfying, as the underlying mathematical process, exponential growth, is the same in both cases.

Now for my admission of kitchen failure. We invited friends over for dinner, and I chose to make chicken parmesan. Jenny was taking a shower and Elena was playing in the front room while I prepared the chicken. Elena came in to see what I was doing, so I picked her up and showed her the breaded chicken sauteing on the stovetop, then took her back out to the front room because I still had more to do.

After Elena comes into the kitchen once, she'll almost certainly make a return visit. The next time my fingers had egg, flour, bread crumbs and raw chicken all over them. Elena must have sensed it, because she made a beeline for the trash can. I turned on the faucet and tried to wash them off as quickly as possible, but she was moving so fast and a vision of the baby in the middle of a pile of raw chicken bits and other assorted grossness flashed through my mind. Without even thinking, I yelled Elena, No!

I had never yelled at Elena before. I have spoken sternly, even raised my voice once or twice when she insisted on spitting her food out, but I always kept myself under control. She knew this was different: she stopped immediately, turned towards me with a scared face and started crying. Then she crawled over to me, and by this time my hands were clean so I was able to pick her up, comfort her, and apologize. Not long after, Jenny finished her shower and took Elena and so I could finish dinner.

There are two takeaways, both about distraction. First, I had so many things going on--chicken on the stove and in preparation, messy hands, baby on the move, that I didn't have the mental resources available to assess and resolve the situation quickly enough in a more thoughtful way. I know I'm not a great multitasker, but this just proves it. Second, I'm thankful for our subsequent discovery that Elena really likes refrigerator magnets. We don't yet have very many that are big enough for her to play with, but Jenny put some good ones within her reach. They have proven to be a good first line of defense that keeps her far away from the trash can.

The barrier for entry is so low, the prize is so large, and Elena is so cute. We're planning to enter her in the Beautiful Baby Search . This morning we spent some time taking pictures and got a few good ones. We can't decide which one to submit, so I'm soliciting your help. Please pick your favorite and vote in the poll below.

Picture 1

Picture 2

Picture 3

Picture 4

For those interested, all the pictures were taken with a Canon Digital Rebel XT and a 50mm/f1.8 lens.

Elena has been sick for more than a week now. She had simple cold symptoms, but last weekend she started having periodic coughing fits that went on for what seemed like a long time and were almost uncontrollable. I have a hard time describing the cough--sort of short and full of mucous, and while she's in a coughing fit she hardly has time to breathe between the coughs. We have two effective ways to stop the coughing: give her some juice, or steam up the bathroom with hot water and take her in there until it's over. The coughing fits only happen a few times a day, often after she's been doing something more strenuous while playing, but they have woken her up in the middle of the night.

On Tuesday we decided to get some medical help, so Jenny called the nurse line and they had her make a doctor's appointment. The doctor decided to treat her for whooping cough based on her symptoms and the fact that there has been a recent outbreak of whooping cough around here. The tests for the whooping cough take several days to run and are not entirely reliable, and in this case it's best to start a course of antibiotics without waiting for results. Another name for whooping cough is pertussis, which is the P in the DTaP vaccination. Since Elena has followed the recommended immunization schedule, she's already had three of these shots, so you might wonder why she still got sick. We discussed this briefly in my mathematical biology class so I can give an overview of the dynamics. I'll try not to get too mathematical or launch into too much of a rant, but be forewarned, because I can't make any guarantees.

Like almost all immunizations, the DTaP shot is only partially effective. Studies of effectiveness are always done at the population level, not on individuals, so it's unclear what "partially effective" means. It could mean that most people are completely immunized while a few are completely unprotected, or it could mean that everyone immunized has a smaller chance of falling ill, but is still slightly susceptible. Of course, it's probably some combination of the two: inoculated individuals are protected to varying degrees, depending on a wide set of complicated factors. This matters when combined with the idea of how contagious a disease is. Pertussis has a contagion rate of 70-100%, so if an non-immunized person is exposed, it's almost a certain thing that they will fall ill.

The most basic model used by epidemiologists to study the spread of disease is called SIR, an acronym for "Susceptible, Infected, Recovered," which is the progression that individuals go through. In the model, you look at the percentage of the population in each of those categories over the time of a disease outbreak. In order to do so, information about contagiousness, population susceptibility, and recovery rates is plugged into a matrix, which then can be multiplied against the population percentages. The resulting system can be analyzed to see whether the disease is likely to die out, maintain a steady level, or explode into an epidemic. All of that depends on some things called the eigenvalues of the matrix. You don't need to know exactly what that means, but you do need to know that matrix multiplication doesn't work like regular multiplication. Two matrices that look very similar could bring about very different behaviors, because they have qualitatively different eigenvalues. Specifically, any eigenvalues bigger than \( 1 \) mean the outbreak will grow, while eigenvalues smaller than \( 1 \) mean that it will die out. So even though there's a much bigger difference between \( 1.05 \) and \( 3 \) than there is between \( .95 \) and \( 1.05 \), the first pair would imply similar similar behavior and the second pair gives very different behavior.

If we want to reduce the spread infectious disease (at the demographic level of policy and public health), there are only a few things that we can try to do, such as improving the effectiveness of our vaccines, but that's likely to take a long time. There's really just one thing that can be done on a large scale in a short time: increase the level of vaccination in the population. When you model these kinds of changes, the non-linearity of the system becomes clear. If you guess that 50% of the population is vaccinated, and see what happens when you move that up to 65%, it might be that there are no visible changes--the disease goes on as before. But then if you move it up just a little bit more, say to 70%, then you cross over the threshold and change the eigenvalues, which makes the outbreak less likely to persist. Sometimes this is called "herd immunity" because it works only on a large scale, and means that it's actually okay for a certain number of people to remain unvaccinated, because they're effectively protected from exposure by the immunity of the rest of the group.

So what does this all mean? First, it's okay for vaccines to be only partially effective, because a second layer of protection kicks in when enough people get vaccinated. Second, it's unrealistic to vaccinate every person in a population. Some people just don't go to the doctor, some are still too young for the vaccine, some have legitimate allergies (eggs are the foundation of many vaccines, and some people are allergic). But that's okay, because those people get the protection of the entire group. It's even okay for a few crazy people to not want immunization for irrational reasons. What's not okay is when those people write about it on blogs or appear on Oprah --then this influences others and it becomes a movement, which changes the dynamics and threatens everyone's immunity.

Vaccine schedules are created by smart people for a reason, which is to optimize the health and safety for each individual. But they optimize in two dimensions--the miniscule risks presented by the vaccine itself, and the much greater risk of catching the disease if not enough people are immunized. The second dimension is the important one, but it's the one that people ignore, because they don't understand how it works. They don't understand that the risks are nonlinear. By not vaccinating, they take advantage of the group, but this only works up to a point, after which outbreaks and sustained epidemics become likely.

When a non-vaccinator makes their choice, they probably don't accurately consider the risk that their child will get sick, because serious, dangerous diseases are extremely rare in our modern world. We've eradicated many of the terrible diseases of yesteryear. The way we did it was--no surprises here--through vaccinations! If the vaccination campaigns hadn't worked so well, and terrible diseases like polio and smallpox were still prevalent, I can guarantee that nobody would worry about nonexistent links between autism and vaccination. But these things can come back when large numbers of people opt out of vaccination. There was a tragic story about measles on This American Life some time ago.

I'm not worried about Elena. She's happy and strong, the cough seems not to bother her too much and the coughing spells have been far shorter in the past few days. She may not even have whooping cough; we still haven't received the results. But if she does, it's a little bit frustrating to know that she might have a cough that lingers for months, when we did everything we could to avoid it. So my next step is to try to use this tiny platform of mine to sway the public discussion in the right direction, and help people understand why non-vaccinators are misguided. I hope that all of you vaccinate your children on the recommended schedule, and can encourage people to get the real facts when they are trying to understand how it works.

Updated: I changed a few small things at the beginning for clarity and accuracy, based on Jenny's first-hand account of Elena's doctor's visit. She also summarized things succinctly for me:

When a person decides to not vaccinate their child, or even to follow a delayed scheduled, they are not just deciding to take on the risks for their own child of not vaccinating. They are putting every person their child comes into contact with at increased risk. This could be compared to the unfairness of people affected by second hand smoke - but at least in those cases you can see and smell the smoke.

As Elena approaches her twelfth month, she acts more and more like a thinking, planning little person. She can't talk yet, but does she have a meaningful internal dialogue? There must be something because her actions sure seem to have motivation.

Yesterday I was sitting on the couch with my computer and she was playing on the floor. She deliberately picked up her ball and threw it towards me. That definitely got me to play with her, but how did she know that would achieve the desired effect? The other day, she found her toy car, held it up to me and started making vroom-vroom sounds, I think. But maybe I'm just attributing that meaning to the sound that she did make. Regardless, she does like to watch me drive the car all around the room, then up, down, and around her legs, arms, and head.

What about when she crawls into the kitchen, then waits for me to come in before she makes her final dash to the trash can? What motivates that? It's probably the same thing that made her crawl up the ramp to the podium at church, then stop and look back at me just before she scooted off towards the choir pews.

In the evenings just before bed she laughs and laughs at just about anything we do, but other times I can make the silliest face imaginable and not even get a smile. And if I leave her in her high chair for too long before bringing her food out, she'll cry like a newborn that has never eaten in her life. Is the crying just as purposeful as the laughter, or does she really revert back to a less developed state of mind when she's hungry?

I'd like to think that I'll understand these things when she can talk, so we can communicate. But I know that it's likely to be just the opposite--she'll be even more of a mystery.

Just for good measure, here are a few last holiday pictures.

Christmas is over, and now all that's left is to play with the toys, keep gorging on candy and cookies until they're all gone, and write about what happened. Our holiday started on Christmas Eve morning with a trip to visit John, Karina and the cousins for brunch. We also got to see some other friends that we don't see very often. Elena jumped right in to play with the big kids while we enjoyed a tasty variety of food.

Later in the afternoon, after an unplanned nap on my part, Jenny unilaterally announced the start of a Christmas Eve tradition. She decided that we should have a simple dinner of fruit, protein shakes, string cheese, and popcorn. I thought this was pretty strange, but Jenny had several convincing explanations for why this would be good not just for that one dinner, but as an ongoing holiday tradition * . I agreed, but only on the condition that we both do our planned workouts before dinner. Jenny headed to the gym first, but it was already closed by the time she got there, so she came home to stretch and do a Pilates video. This gave me time to go up to the boathouse and erg for the first time in several months. There must be something useful about carrying a giant baby around all the time, because my workout went better than I thought it would.

When I got home we had a short visit with my friend Ben, who dropped off some tasty frozen cheesecake that he made himself. Ben always amazes me, and this is a perfect example. How many senior Mechanical Engineering students would be so thoughtful or capable of making delicious desserts?

I do have to admit that after several very heavy meals in a row (including slow-cooked pulled pork sandwiches the night before), the fruit and shake dinner was just what we needed. Somehow the popcorn added just the right kind of contrast and variety. If we do this next year, I'd recommend that we upgrade our cheese from string cheese to something a little bit more classy. There was some popcorn left over after dinner, which was okay because we were watching a movie and didn't feel bad about continuing to munch.

Even though we didn't have too many presents to wrap, we somehow managed to stay up too late and had a hard time making our early wake up call on Christmas morning. We should have appreciated Elena's insensitivity to that particular Christmas tradition, but we had another brunch and wanted to prepare our contribution, open presents, feed Elena, and put her down for a nap all before we had to go, so we woke up at about 6:45 to get things going.

We thought that Elena would enjoy the wrapping paper and boxes more than her actual presents, but we were completely wrong--she loved her presents and wanted to play with all of them at once. I guess that's the advantage of having a very brief attention span. Instead of writing more, here's a photo montage:

Like I said, Elena loves all of her presents. She got lots of new toys. In my mind, the key additions are some dolls from her grandmothers, a singing book of nursery rhymes from another grandma, some stacking rings, a ball, and a matchbox car, since she likes her cousins' cars so much when we visit them. Jenny loves the cute new outfits that Elena received, and they all seem big enough for her to wear for a while before she grows out of them.

Our plans for Elena's nap didn't turn out so well. Maybe she was overstimulated, or maybe she just wanted to play some more, but her nap didn't last very long and she was still tired when we went to the brunch with our friends Martin and Christine. Elena fell asleep in the car and we drove around for ten extra minutes to let her sleep. She didn't wake up when we got to their apartment, so I stayed out in the car to let her sleep longer while Jenny went inside. After another ten minutes, we decided she had enough, woke her up, and headed inside. I was glad that I did--there was even more yummy goodness, including a syrup concoction for the pancakes that Christine just called 'liquid gold.'

After we finished eating, the adults were sitting around talking about Christmas past and other topics (whenever Martin's around, there's always a discussion of nutritional supplements). Elena started playing with a little snow globe, and was demonstrating the improbable feat of putting the whole glass dome in her mouth, leaving just the base sticking out. She tired of that and started tapping it on a table. We all assumed that it was made of unbreakable plastic, but Elena decided to disprove that hypothesis. She whacked it on the table and the glass shattered, leaking water and glitter everywhere. We whisked Elena away and cleaned it all up, so nobody was hurt, and it wasn't an heirloom or anything valuable, but I am sorry for the needless, preventable destruction.

In the afternoon we met up with Yasmin, a friend from church, and headed back to have Christmas dinner with John and Karina. At their house, we found two boys very happy with their Christmas presents, and baby Ainsley in an adorable Santa suit. Both Warren and Gavin wanted to show us a wind-up skateboard guy that did a complete flip and then kept rolling.

Karina made her signature ham for dinner, and we all ate way too much. Even Elena got in on the action--we gave her a few small bites after she finished the rest of her dinner, and she kept wanting more. I put her in the living room to play, but she crawled back to the table and begged for scraps like a little puppy. My meager contribution to dinner was jello with fruit cocktail mixed in, but Elena did enjoy sampling that too. For the record, Elena also had some of Jenny's really good mashed potatoes.

After dinner, dessert consisted of Jenny's cinnamon rolls and apple pie from Yasmin. Somehow we managed to find a little bit of extra room for it. While we enjoyed that, Elena was exploring again and found the staircase. A few weeks ago at another house, Elena discovered her first staircase and knew just what to do--she went right up, with me following along behind to keep her safe. She did the same again last night, except at the top of last night's staircase she could look out and see everything below. She was very excited to see everyone down there, and started to squeal, laugh, and wave at them.

Our Christmas evening did end on a sad note. Elena was getting tired, and I put her on the couch to play with Gavin. They were having a good time and I was standing guard, but there didn't seem to be too much of a need for me there; they had played for a while without being in danger of falling off. So when Jenny asked for help with her camera, I thought nothing of stepping away for just a moment to grab the camera. As soon as I took two steps away, I heard the awful thud of baby on tile, immediately followed by Elena's loud wail. She calmed down when we gave her a bottle and some baby Tylenol, but was more irritable than usual for the rest of the night. She did have a bump on her head, but it seems to be gone today and she's back to her usual happy self. So all's well that ends well, like this tale of Christmas.

I hope you had an equally memorable and enjoyable Christmas, except possibly with fewer head bumps and broken snowglobes! I've uploaded more pictures than I can include in this post. You can find them over on flickr .

* If Jenny would like to explain this tradition to more people, I would welcome it as her first blog post, but I don't want to misrepresent her by explaining it myself.

On Saturday evening we finally got a Christmas tree. We had been running some errands and Elena fell asleep in her car seat, so we didn't feel an obligation to wander through the tree wonderland at Home Depot. Instead, we noticed that Lowe's had just a few trees outside the garden section. I got out to take a look while Jenny stayed with Elena, and found a sign saying that they had a 50 percent discount, so we snapped one up. I managed to find one that wasn't too big--it mostly fit inside the trunk of our compact car, so we didn't have to try to put down the back seat to squeeze it in.

A deeply discounted tree is likely to be one that came from the farm a long time ago. Even after having its trunk trimmed, ours won't absorb the water we put in the tree stand, so it already looks droopy. That's okay because we don't need it to last very long, and it will definitely make it until Friday. Elena likes the tree an appropriate amount, by which I mean that she likes to look at it, but hasn't tried to pull it down yet. Jewel probably enjoys the tree the most. Sometimes she likes to hide behind it and pretend she's in the primeval forest. Other times she plays with the ornaments, including a low hanging bell that she can ring by hitting it with her paw.

In an after-Christmas sale two years ago we bought a red star for a tree topper, but none of our other ornaments match; they're mostly purple and silver, from Leisa's wedding four years ago * . We got the red topper because we couldn't find any silver ones, and we figured it would be easy to paint it. It was so easy that we completely forgot about it. Maybe I should go do something about it right now; if I'm lucky, Jenny won't read this until after I'm done.

Update: Done.

* Congratulations, Sam and Leisa. It's hard to believe it's already been four years! Is this the first year since your wedding that you had snow for your anniversary?

Last evening while Jenny was at a meeting, I turned on the television and started flipping through the channels. Usually Elena just plays on the floor and ignores the screen, unless she's pulling herself up and slapping it after she's done with poking the DVD player and adjusting the volume on the receiver.

This time was different. I happened across the season finale of So You Think You Can Dance , and whenever the music and dancing started, she stared at the screen and bounce-danced along in time. When the song stopped, she clapped along with the applause.

Jenny says that Elena dances to any music she hears, and claps whenever she hears clapping, even if it's just in the background. Many thanks to the Allens/Hikidas for loaning us a baby bouncy chair. Elena really enjoys jumping around in it, and we don't have much time before she's too big, so we have to make the most use of it now.

Last week Elena passed another milestone--she's now ten months old. Since she was born on February tenth, I'm declaring this to be her lucky month. She's been doing so many new things recently, it's hard to know what to write about.

On one front, Elena is finding lots of ways to communicate with us. It's probably overly optimistic, but we think she may be saying her first word: "amen." It isn't that she's very reverent, instead she knows that she gets to eat after we bless the food, and "amen" marks the end of the prayer. Along the same lines, she seems to have picked up the baby sign for "more." We're still working on the appropriate usage--sometimes she just makes the sign for fun.

Elena has been especially giggly lately, and is learning to laugh socially. For instance, when everyone else around is laughing, she'll join in. Also, she had a good time playing with cousin Gavin who visited us one day last week. His dad had shoulder surgery, so we got to take care of him. For a while, Elena laughed at every little thing that he did, whether he was trying to be funny (waving around a kitty toy) or not (blowing his nose). It was similar to the way she laughed at cousins Sammy and Aaron this summer. She really likes her cousins.

Elena's physical development definitely keeps us on our toes. Sometimes it seems like she's everywhere, and her range is extending every day, especially in the vertical direction. Her hands are fast, and if I don't notice that there are two things within her reach, she'll fake me out by reaching for one and then grab the other before I realize what's happening. Today I rearranged and cleaned up the mess of cables connecting the home entertainment equipment. I hope that now it's less noticeable and less accessible. The front side of the equipment remains a popular target for her inquisitive fingers. While I was away at Mutual this evening, she found a tiny button on the receiver that turned off the speakers, so Jenny couldn't watch any TV.

Elena spends a lot of time on her feet these days, holding onto couches and shelves and using them as guides while she cruises from one end to the other. She can even use just the wall to steady herself. I went outside to get something from the storage closet and closed the door behind me. Elena noticed and scooted right over, then stood up and watched me through the glass. I had to call Jenny for help, because I didn't want to slide the door open with Elena standing there. We think that she'll be free standing before long.

One final note: we've put a lot of time and effort into our Christmas packages over the past few weeks, but they're finally done and in the mail. If you're one of our immediate family members, be on the lookout. We hope you don't think they're too weird this year. More on that later.

After getting back to our hotel after midnight on Thanksgiving and a fitful night of sleep for Elena, hence also Jenny and me, we got off to a later start on Friday morning. We still managed to pack in an extremely full and tiring day. Here are my impressions.

Hotel hospitality: Because of our upgraded status, we had vouchers for free breakfast at the hotel. I stayed in our room with a sleeping Elena, while everyone else went down to breakfast. Nobody was around to take their vouchers, which they didn't think was unusual. As it turned out, they had inadvertently gone to a banquet area set up for a marching band group. Later I went down with Elena and found the right place. I hope no poor band kid went hungry for lack of food.

The financial district: I liked Federal Hall, but a fussy baby prevented me from reading about everything that happened there. I also didn't get to take a picture out front, mimicking the pose of the George Washington statue. I was the main person who wanted to go to Wall Street, but once we got there I realized that I really wanted to see what was going on inside the NYSE and the Federal Reserve bank, not just look at their imposing exteriors.

The UN building: Jenny's best information said the UN building would be closed on Thanksgiving. It said nothing about the day after, so she thought it would be open. While Elena and I took a break in my parents' nearby hotel room, Jenny took her family on the long walk to the UN. They were disappointed to find it completely closed--even the flagpoles were bare. It wasn't a total loss, though, because they had the experience of eating lunch at an authentic New York hot dog cart.

Jeffrey's Hook Lighthouse : On the map it looked like just a couple of short blocks from the subway stop to the base of the George Washington bridge, home of the famous Little Red Lighthouse . In real life, we found a winding pathway down a steep hill. There weren't any signs pointing the way, but at two key moments we found people to ask for directions. It was already cold and windy and got worse close to the water. Still, it was a beautiful little lighthouse, and very picturesque at the base of the gigantic bridge. Although it took a lot of time and energy and killed our desire to go ice skating, we all really enjoyed the Little Red Lighthouse. Now we just need to get a copy of the book for Elena.

Times Square: I used the last bit of my cell phone's power to arrange a meeting with my family at the entrance to Toys-R-Us. Apparently, everyone else in the entire New York Metro area decided to meet at exactly the same place and time--it was an unbelievable crowd. The overwhelming mass of people provided a perfect foil for the surreal, garish displays lighting up the area from the walls of all the skyscrapers.

Inside Toys-R-Us: It seemed that everyone who had just been outside the iconic toy store was now inside with us; we should have played Where's Waldo in there. At least I got to spend some time with my family. I felt sorry for the very pregnant woman, probably a local resident, who was trying to take advantage of a Black Friday deal on some needed baby gear. But she had to compete with tourists form all over the world for a spot in the register line.

Dinner in the city: Only half a block away from Times Square, it's dark and calm, and there are even homeless people sleeping under the fire escapes. We stopped at the first reasonable-looking diner, right next to the Port Authority. Elena had finally reached her limit and I had to carry her the last ten minutes before we found it. The food was good and wasn't very expensive. The highlight of dinner was a funny old Italian man working as waiter/busboy/maitre d' who smiled at Elena.

Macy's: Our last stop for the night, it was well worth the trip. The picture windows on the front side were bright and interesting, but we exited on the side of the building and chanced upon the windows with the story of Miracle on 34th Street . They were less crowded and more to our liking. On the inside, we were amazed that the store seemed to have no top floor--we kept going up and up and up. Up high there are old wooden escalators and squeaky wooden floors. We were so intrigued by the possibility of going higher and higher that we didn't explore enough to find Santa. We bought one souvenir: a "Baby's First Christmas 2009" ornament in the shape of a pair of shoes.

The long trip home: The people all around us on the crowded train back to New Jersey were very friendly, even when Elena finished her bottle and cried. We didn't get much sleep that night, but we made it out of the hotel and to our flight without any trouble. I have one message for the couple sitting in front of us on the plane: you can put your seats back or you can glare at the baby behind you, but you shouldn't do both. Jenny put in a heroic effort in keeping Elena from grabbing the lady's hair. The second flight, from Dallas to Austin, was even shorter than usual, which was great for us.

Back in our apartment we said hello to Jewel, then finally crashed and took a long nap. I'd like to thank our families for all their help with Elena and all the fun times we had together. Although they'll never read this blog, I also want to thank all the nice people who smiled at Elena. Watching the parade may have been a once in a lifetime experience, but who knows--maybe we'll go back someday!