Flattered or Flattened

Last night I got an interesting question from my nephew Phineas: Do 2D objects exist in our world? How do they work? In my exhausted state, I managed to cobble something together about electrons in a conductor, but I felt I owed him a more complete answer once I thought a bit more.

First let's discuss a bit what it means for something to be 2D (or 3D, or even 0D). The number of dimensions refers to the extent of an object, or the space it takes up. However, we can talk about embedding an object of lower dimension in a higher-dimension space. For example, we can put a plane, which is 2D, in a 3D space:

The equation for this particular plane is z = 1, with x & y between -1 and 1. That means we can pick a point on the plane, like (0.5, -0.5, 1), but if we go even a tiny bit off of z, like (0.5, -0.5, 0.99999), we'll no longer be on the plane.

Even if it's mathematically possible, it's still hard to believe such a thing could exist in the real world. That's where my example from above comes in: Imagine a metal ball that we charge up with some free electrons. Metals conduct electricity, so the electrons can move around freely, but they also are trying to stay as far from each other as possible since like charges repel. If you imagine trying to find the largest separation between a set of points in a sphere, you'll see that the solution requires all the points to be on the surface. This surface is 2D, even if it's curved and embedded in 3D, like the plane above. You can tell it's 2D because you only need 2 coordinates to position yourself on the surface of a sphere, like latitude and longitude.

Now that answer may not be satisfying, since if we imagine ourselves like the electrons, moving around the surface of the Earth, we're not 2D, just the surface we're walking on. However, based on the measurements that have been made in particle colliders, electrons act as single points, or 0D objects. In response to this, Phineas asked, If electrons are 0D how do they exist? They must exist to become 1D, or 2D, or even 3D! I still don’t understand how if something is 2 or 1D how it exists. This is fair point – If something has fewer than 3 dimensions, it will have a volume of zero, so how can it have mass? If they do have mass, then they should have infinite density, which would make them a black hole. My feeling is that the answer lies in the fact that electrons, and similar sub-atomic particles, are quantum objects, and quantum mechanics works very differently from anything we're used to our own lives.

These questions are really at the limits of what we know right now – While looking up info to write this post, I found an article from 2020 about the discovery of 2D particles. If you're curious about this stuff, Phineas, you can become a physicist and maybe explain it to your UncOrion some day!

Chilling Reports

This weekend, Mt. Washington in New Hampshire set a record for the lowest measured wind chill. I've always been a bit bothered by the idea of wind chill, since it doesn't represent what temperature it is, but instead what temperature it feels like, which seems a bit subjective. I thought I'd take a closer look at how wind chill is calculated.

There are several different models for calculating wind chill, but the principle is the same: In calm conditions, a boundary of warmer air will build up between your body and the colder air, which will slow the heat loss, but strong winds will replace that boundary with the cold air, cooling you faster. In spite of this faster cooling, you'll never get colder than the actual temperature of the air, so if you sit on top of Mt. Washington you can rest assured you won't go all the way to -108°F, but instead stay at a balmy -47°F.

Thinking about it a bit more, I suppose my real problem with wind chill is that it's using the wrong units for what it's measuring: Wind chill is meant to convey the rate of heat loss, so that's how it should be measured. That's what the original model, called the wind chill index, measured with units of kilocalories/hour/meter^2. What's nice about this is that you can clearly see that your total energy loss will depend on how long you're outside, and how exposed you are.

If you look at the Wikipedia page above, you can see the different equations for the wind chill index and the wind chill temperature. On the surface they look similar, but the wind chill temperature has a very unusual 0.16 power, which only makes me more uncomfortable with the measure. We can look at how the two compare for a range of temperature and wind speed values:

The shapes are similar, but the wind chill index has more of a bulge, indicating that it is more sensitive to the wind speed than the wind chill temperature.

The Wikipedia article states "Many formulas exist for wind chill because, unlike temperature, wind chill has no universally agreed upon standard definition or measurement. All the formulas attempt to qualitatively predict the effect of wind on the temperature humans perceive." It seems a bit silly to tout a record based on an arbitrary perception of temperature – Ever since chemo, I've found myself more easily chilled than I used to be, so maybe I'll define my own wind chill measure: The usual wind chill temperature, minus 5 degrees. New record set!

Banditopod

Recently as part of my research, I've been trying to measure a probability distribution – specifically, the chances that we've seen a certain signal in LISA. The trouble is, there are many random noise factors that go into the calculation of whether we see the signal or not, so it's not a straight equation I can plug things into. Instead we need to sample it many times to estimate the distribution, and this can be expensive. My colleague Henri suggested I could use a technique called Markov Chain Monte Carlo (MCMC). I thought to get a better feel for the method, I'd try out a simple example here.

There's a traditional problem in probability theory called the "two-armed bandit." Imagine a slot machine with two levers – You insert a coin and choose which lever to pull. Each arm has a certain probability of paying out, but the only way to find out is by playing, and looking at how often you win or lose. What then is the best strategy for choosing a lever? You may have gotten lucky your first few pulls of one lever and overestimated its chances of winning.

We can make this more like my research by extending to a multi-armed bandit – Each arm represents a set of parameters we're searching for, and we want to pick the arm with the biggest payout/best fit to the data. Still to be answered though is how we pick which arm to play: Imagine a set of players, who can choose an arm at each step based on the wins/loses they've seen. Each one is more likely to pick an arm with lots of wins, but might try another arm just in case. Now, if we look at the estimated probabilities for each arm as time goes on, we might think we'd get a good idea of the true values:

The blue line is the true probability for each arm, and the orange dots are the estimates based on the average number of wins. The dots are jumping around so much though that it's hard to see how well we're doing. Instead of animating in time, we can try looking at how frequently we play each arm:

Pretty quickly, each arm gets a consistent rate of pulls, but it looks like we're undersampling the highest-probability arms. I think this may be due to the top-probability arms having fairly similar values – As I pointed out above, we can't tell whether we have the best lever, or just a streak of luck, so we hedge our bets. A common technique with MCMC is run a "burn in" for a while to let the players move around the parameter space, then reset the probability estimates and continue running.

As a final view of the data, we look at how the players distribute themselves among the arms through time [NB: The x-values are off by 1 compared to the earlier plot due to the way I gave the distribution to the MCMC tool I used]:

It starts off fairly flat – the parameter exploration I was talking about – but after a certain point, the distribution establishes itself, and from there the shape simply scales upward. However, even if you could afford to play tens of thousands of times, I think you'll be hard-pressed to find a slot machine paying out as frequently as these!