Tag Archives: Fun

Too Close to Home

I haven’t been blogging much recently because I just moved from Chicago to Boston. Also, I don’t currently have access to internet in my new apartment. As always, there’s an XKCD comic to capture this scenario:


Hopefully, I’ll be back and posting more often soon!

Lunar Eclipse and the 22 Degree Halo

The beautiful thing about atmospheric optics is that (almost) everyone can look up at the sky and see stunning optical phenomena from the sun, moon or some other celestial object. In this post I’ll focus on two particularly striking phenomena where the physical essence can be captured with relatively simple explanations.

The 22 degree halo is a ring around the sun or moon, which is often observed on cold days. Here are a couple images of the 22 degree halo around the sun and moon respectively:


22 degree halo around the sun


22 degree halo around the moon

Note that the 22 degree halo is distinct from the coronae, which occur due to different reasons. While the coronae arise due to the presence of water droplets, the 22 degree halo arises specifically due to the presence of hexagonal ice crystals in the earth’s atmosphere. So why 22 degrees? Well, it turns out that one can answer the question using rather simple undergraduate-level physics. One of the most famous questions in undergraduate optics is that of light refraction through a prism, illustrated below:


Fig. 1: The Snell’s Law Prism Problem

But if there were hexagonal ice crystals in the atmosphere, the problem is exactly the same, as one can see below. This is so because a hexagon is just an equilateral triangle with its ends chopped off. So as long as the light enters and exits on two sides of the hexagon that are spaced one side apart, the analysis is the same as for the triangle.


Equilateral triangle with ends chopped off, making a hexagon

It turns out that \theta_4 in Fig. 1 can be solved as a function of \theta_1 with Snell’s law and some simple trigonometry to yield (under the assumption that n_1 =1):

\theta_4 = \textrm{sin}^{-1}(n_2 \times \textrm{sin}(60-\textrm{sin}^{-1}(\textrm{sin}(\theta_1)/n_2)))

It is then pretty straightforward to obtain \delta, the difference in angle between the incident and refracted beam as a function of \theta_1. I have plotted this below for the index of refraction of ice crystals for three different colors of light, red, green and blue (n_2 = 1.306, 1.311 and 1.317 respectively):


The important thing to note in the plot above is that there is a minimum angle below which there is no refracted beam, and this angle is precisely 21.54, 21.92 and 22.37 degrees for red, green and blue light respectively. Because there is no refracted beam below 22 degrees, this region appears darker, and then there is a sudden appearance of the refracted beam at the angles listed above. This is what gives rise to the 22 degree halo and also to the reddish hue on the inside rim of the halo.

Another rather spectacular celestial occurrence is the lunar eclipse, where the earth completely obscures the moon from direct sunlight. This is the geometry for the lunar eclipse:


Geometry of the lunar eclipse

The question I wanted to address is the reddish hue of the moon, despite it lying in the earth’s shadow. It would naively seem like the moon should not be observable at all. However, there is a similar effect occurring here as with the halo. In this case, the earth’s atmosphere is the refracting medium. So just as light incident on the prism was going upward and then exited going downward, the sun’s rays similarly enter the atmosphere on a trajectory that would miss the moon, but then are bent towards the moon after interacting with the earth’s atmosphere.

But why red? Well, this has the same origins as the reddish hue of the sunset. Because light scatters from atmospheric particles as 1/\lambda^4, blue light gets scattered away much more easily than red light. Hence, the only color of light left by the time the light reaches the moon is primarily of red color.

It is interesting to imagine what the earth looks like from the moon during a lunar eclipse — it likely looks completely dark apart from a spectacular red halo around the earth. Anyway, one should realize that Snell’s law was first formulated in 984 by Arab scientist Ibn Sahl, and so it was possible to come to these conclusions more than a thousand years ago. Nothing new here!

European League Underdogs

I’m taking a little detour away from physics blogging today to dip my toe into the world of football (soccer). With Leicester City’s success in the Premier League this season, I feel compelled to dedicate one post to this most unlikely of events. To those not familiar with English football, prior to the beginning of the season, Leicester City FC was famously a whopping 5000/1 bet to win the Premier League. There is an amusing BBC video demonstrating how unlikely this victory was by comparing to some other 5000/1 bets. Teaser: bookies are giving 5000/1 odds on Elvis being alive.

The last decade of world football has been dominated by a style of football colloquially referred to as “tiki-taka”. This type of football has been most often associated with Barcelona FC (and more recently with Bayern Munich) at the club level and by Spain on the international front. Tiki-taka is built on a heavily possession-based game. This high percentage of possession has two effects:

  1. It denies the opposition chances to score by denying them the ball
  2. Players of extremely high technical ability are required so that:
    • They can retain possession in tight spaces
    • They are able to find openings in the opposing team’s defense

Tiki-taka has probably been the most successful strategy in modern football, but it requires the best players in the world to execute the strategy. Therefore, this is a luxury that only teams with a great deal of money can employ.

The question then naturally arises: how is a team with fewer resources supposed to compete against the teams with better players?

To answer this question, I would like to concentrate on three teams, one from the Premier League (Leicester City), one from the German Bundesliga (Borussia Dortmund) and one from La Liga in Spain (Athletico Madrid). In the past seven years, these three teams, despite being at a huge financial disadvantage compared to other teams in their leagues, were still able to win their respective domestic leagues.

For those who are unfamiliar with football, in the Premier League, the teams with the most money are Manchester City, Manchester United, Chelsea and (despite what Arsene Wenger will tell you) Arsenal. In the German league, it is without a doubt Bayern Munich. In the Spanish league, Real Madrid and Barcelona are by far the richest teams in the league.

So how did the aforementioned underdogs overcome the odds and win their respective leagues on a shoestring budget? I contend that the answer is with a three-pronged strategy:

1. Tackle Hard

Take a look at the plot below. It ranks the season’s champions in terms of tackles per game. For example, in the Bundesliga season ending in 2013, the champions were Bayern Munich and they ranked 18th (dead last) in their league in terms of tackles/game. It is important to take into consideration when reading the plots that the English and Spanish leagues have 20 teams, while the German league has 18 teams.


This plot shows that on the way to their league titles, Athletico Madrid and Leicester ranked 1st in the league in the number of tackles per game. Borussia Dortmund ranked 2nd in tackles/game for both their 2011 and 2012 trophies.

Note that it is rare for a team to win the league with the highest number of tackles per game. This is especially true in Germany where from 2013-2016 Bayern Munich, who won the Bundesliga in those four seasons, ranked dead last in tackles/game.

2. Tackle Smart

Just because an underdog team tackles harder, however, doesn’t mean that they will miraculously start winning games. Take a look now at the plot below. It shows the end-of-season league position of the team with the highest number of tackles/game.


One can see that there is no pattern at all! In fact, if we take the average league position over the past seven years of the top-tackling teams we get:

  1. Premier League: 9.4
  2. Bundesliga: 8.4
  3. La Liga: 9.4

Keeping in mind that the Bundesliga only has 18 teams, this says that the tackles/game statistic is basically irrelevant! However, I showed in the previous section that Leicester, Athletico and Dortmund all had a high tackles/game trait in common. What gives?!

The data shows that Leicester, Athletico and Dortmund do something different. They don’t just tackle hard, they tackle hard and smart. This means three things:

  1. They seek to win the ball back in dangerous positions
  2. After recognizing a dangerous position, they win the ball back by pressing with more than one player
  3. They take advantage of the turnovers by producing goal opportunities, especially on the counter attack

This means that what these three teams do is defend well as a team. It’s no use having many great tacklers on the team if they do not collectively pressurize the ball when they see an opponent in a weak position. If one watches as much football as I do, one will realize the extent to which Athletico Madrid has turned this collective defending almost into an art form.

3. Don’t Worry about Possession and Take Advantage of Counter Attacks

The last important ingredient is to take advantage of this collective tackling and use it to benefit on the counter attack. Many teams that tackle hard are too slow to get the ball forward and therefore lose their tackling advantage.

Leicester (8 counter attack goals) and Dortmund (11 counter attack goals) ranked 1st in the number of counter attack goals scored in the 2016 and 2012 seasons, when they respectively won their domestic titles. Athletico Madrid (8 counter attack goals), on the other hand, ranked 3rd, behind (you guessed it!) Barcelona (10 counter attack goals) and Real Madrid (13 counter attack goals). It should be kept in mind, however, that Real Madrid and Barcelona outscored Athletico over the season by 27 and 23 goals respectively. Therefore, it can be said that these three underdogs put a huge emphasis on effective counter-attacking football.

Needless to say, this means that these three teams do not put an emphasis on possession-based football unlike their tiki-taka counterparts. In their championship winning seasons, the highest possession any of these teams had over the course of the season was Dortmund in 2012 with a measly 53.3%. In fact, in the 2016 season, Leicester had 44.3% possession over the course of the season, third last in the league! Athletico also routinely finishes outside the top 5 in La Liga in possession stats, including their title-winning season.

Putting All of it Together

Okay, so that was probably a lot to process, so let me summarize and paint a picture of what the data tells us.

It turns out that the underdog champions have quite a bit in common with each other. One significant revelation from these statistics is that it is not only important to tackle hard but to also tackle smart. This intelligence in tackling requires the entire team to be on the same wavelength. Teammates need to help each other to win the ball back and the team’s forwards need to make themselves available for a pass immediately after dispossessing the opposition. Take a look at this video (taken from here) to see how fast Leicester can turn defense into attack. Athletico and Dortmund also employ similar lightning-paced counterattacks.

There is an important lesson here for teams that do not have the wealth of the Barcelonas and Real Madrids of the world. There is a way by which one can beat these teams. This requires an extreme dedication to a collective defensive strategy coupled with an emphasis on direct counterattacking football on the offensive end. To play with this strategy requires the entire team to buy into this ethos. These teams cannot have “luxury players” that don’t graft.

Lastly, let me say that following Dortmund and Athletico’s domestic success, both teams went on to play in the UEFA Champions League (UCL) final. In Athletico’s case, they are in the European Cup final again this year (to be played May 28th). This begs the question as to how far Leicester can go next year in the UCL. Unfortunately, bookies have wised up and are only giving Leicester 100/1 odds to win the entire thing.

Most of the statistics were obtained at http://www.whoscored.com

Update: An article about Atheltico Madrid has appeared today discussing similar themes and can be found here: http://www.espnfc.com/club/atletico-madrid/1068/blog/post/2880642/diego-simeones-magic-touch-keeps-atletico-madrid-in-contention.

What’s NDT Been Up To?

Readers of this blog will know that I’m a big fan of what Neil DeGrasse Tyson does for science in the public eye. Recently, he sent out a couple tweets, which I thought were hilarious that I thought I’d share here as well. I hope you enjoy these as much as I did!



I’ve long had an interest in those magic-eye images that I used to look at (and could never solve!) when I was a kid. These are images that look like a regular tiling, but turn out to contain some sort of embedded image when viewed in a certain manner. Here is one of these for instance:


An autostereogram of a lovely butterfly

Magic-eye images are actually part of a much larger set of images known as stereograms. Stereograms were discovered in 1959 by Bela Julesz, a scientist at Bells Labs, who invented random dot stereograms to study depth perception. Humans are particularly adept at depth perception due to the fact that we have two eyes that are horizontally separated from one another. This allows us to triangulate, and our brain turns this information into depth. In fact, our depth perception is vastly superior to our ability to discern lateral displacements. Take a look at this image for instance:


If one looks at the image normally, it takes a while to figure out which tiles (i.e. black, pink or orange) are spaced further apart. However, if one looks at the image stereoscopically (i.e. how you would look at a magic-eye image), then one can immediately tell that from back to front, we have black, orange then pink, indicating the differences in spacing.

What one is doing when one views the tiling above stereoscopically, is that the left eye is seeing one element of the pattern while the right is seeing another. Since they are not focused on the same element, but the elements are physically the same, the brain is tricked into triangulating distance, resulting in the perception of depth.

Let’s now return to random dot stereograms. They are probably the easiest kinds of stereograms to generate. Here is an image of a random-dot stereogram that I made in MS Paint in about a minute:

Random Dot Stereogram


The idea behind making a random dot stereogram is outlined well at the bottom of this webpage. Again, a lateral movement results in the perception of depth because one is tricking the brain, which is using triangulation to calculate depth.

There are other types of stereograms as well. One is the single-image random dot stereogram (SIRDS). This type of stereogram is a little more sophisticated than the previous ones, and you can read about how to generate them in this paper (pdf!). Here is an example of a SIRDS:


Single-image random dot stereogram with an embedded annulus

If you are having trouble viewing the embedded images, I’m sure you’re not alone. However, let me prove to you that there really is something in there. The power spectrum (or power spectral density) of a random signal is known to be a constant. This is because there is no correlation between the random pixels in the image. Below is an image of random dots and its spectral density:


Random Dots                                                                      Power Spectrum

However, look what happens when I look at the power spectrum for the SIRDS with the embedded annulus from above:


Not so random dots                               Demonstration of non-randomness

While this is not a “proof” that there is an image embedded, it suggests that there is some sort of periodicity in the supposedly random dots. If you look hard enough, you may even be able to see this. Note that the periodicity is in the horizontal direction only, the vertical direction is indeed quite random.

Now that we know the image is non-random, is there a way to reveal the image embedded? Indeed, there is. We can exploit the repeating pattern. We can take two identical images of the not-so-random-dots, put them on top of each other, and subtract the pixel intensity of one image from the other. Of course, this would just yield a black figure. However, when one starts to slide one image over the other (translating while subtracting!), the image reveals itself. Below is solution for the annulus:


Annulus solution

I had actually posted a solution to the autostereogram on the Wolfram Demonstrations project website a little while ago, where you can download the very simple code if you want to play with it.

A Physicist’s Proof

As an undergrad, I took a couple “math for physicists” courses that I found to be quite helpful. One of the more humorous concepts a professor of mine conveyed to me was the idea of a “physicist’s proof”. These would be sort of intuitive proofs that a mathematician may scoff at, but physicists tend to appreciate. Below is an example of a physicist’s proof showing the following mathematical relation:

\sum_{i=0}^n (2i+1) = (n+1)^2

This relation can more easily be stated in words. It says that if you add up consecutive odd numbers starting at 1, you get a perfect square. For instance, 1+3+5=9=3^2 or 1+3+5+7=16=4^2.

Here is the idea:

For n=0, you get a 1×1 square as is seen in the following image:


For n=1, you add the 3 next squares and you get the following 2×2 square:


For n=2 and n=3, one adds 5 and 7 squares respectively to get the following 3×3 and 4×4 squares:




By now, you can probably see the pattern and why this mathematical relation holds (hopefully where I have put the dashes and dots helps you see this!). To me, these kinds of proofs, while lacking in mathematical rigor, do much for the intuition. In fact, they even suggest an algebraic manner by which to prove the sum rigorously.

Long live the physicist’s proof!

Does Frivolity Border on the Edge of Creativity?

Sometimes, for the sake of letting one’s imagination run around a bit, it may be advisable to indulge in a seemingly frivolous endeavor. In the sciences, these undertakings can sometimes result in the winning of an Ig Nobel Prize.

This year’s winner of the Ig Nobel in physics studied the “universal” urination time of mammals. The main conclusion of this paper is that mammals that weigh more than 3kg urinate for 21 \pm 13 seconds per session. I will not comment on the rather large error bars.

I reprint the abstract to the paper below:

Many urological studies rely on models of animals, such as rats and pigs, but their relation to the human urinary system is poorly understood. Here, we elucidate the hydrodynamics of urination across five orders of magnitude in body mass. Using high-speed videography and flow-rate measurement obtained at Zoo Atlanta, we discover that all mammals above 3 kg in weight empty their bladders over nearly constant duration of 21 ± 13 s. This feat is possible, because larger animals have longer urethras and thus, higher gravitational force and higher flow speed. Smaller mammals are challenged during urination by high viscous and capillary forces that limit their urine to single drops. Our findings reveal that the urethra is a flow-enhancing device, enabling the urinary system to be scaled up by a factor of 3,600 in volume without compromising its function. This study may help to diagnose urinary problems in animals as well as inspire the design of scalable hydrodynamic systems based on those in nature.

I present a translation of the abstract in my own language below:

We don’t know if humans and other mammals pee in the same way. Here, we study how both big and little mammals pee. We creepily filmed a lot of mammals (that weigh more than 3kg) pee with an unnecessarily high-speed camera and found that they all generally pee for 21 \pm 13 seconds. Large mammals can push more pee through their pee-holes and gravity helps them out a bit. It’s harder for small animals to pee because they have smaller pee-holes. Surprisingly, pee-holes work for mammals with a range of sizes. We hope this study will help mammals with peeing problems.

I genuinely enjoyed reading their paper, and actually recommend it for a bit of fun. Here are some of the high-speed videos (which you may or may not want to watch) associated with the paper.

Please feel free to experiment with your own “translations” in the comments.