Step Back: A Net Neutrality Lowdown

In February 2015, the FCC released a document stating the rules they wish to apply to protect the Open Internet, a term used to refer to an internet where the users can go where they want, when they want. The fight for these rules enacting “net neutrality” has been going on for longer than there’s been a name for it, but with no sign of the internet’s influence slowing down, many people began to push for the FCC to make a stand for this principle. The FCC’s document notes three bright-line rules:

  • No Blocking: broadband providers may not block access to legal content, applications, services, or non-harmful devices.
  • No Throttling: broadband providers may not impair or degrade lawful Internet traffic on the basis of content, applications, services, or non-harmful devices.
  • No Paid Prioritization: broadband providers may not favor some lawful Internet traffic over other lawful traffic in exchange for consideration of any kind — in other words, no fast lanes. This rule also bans ISPs from prioritizing content and services of their affiliates.

No blocking.

This part is the less debated of the three. It specifies that the content is legal, differentiating it from the push against SOPA (the Stop Online Piracy Act) three years ago. SOPA was so disputed because it would authorize the censoring of any internet sites that foster illegal content in any way, granting law enforcement unsettling power in censoring and filtering the internet. Proponents of the Open Internet lashed back, believing that in no way should the internet be censored, that another solution should be sought. On January 18, 2012, many major internet websites participated in a planned protest against the bill. Google’s logo had a black box placed over it, linking to information about SOPA. According to Google, that day, they got 4.5 million signatures on a petition against the bill. Two days later, congressman Lamar Smith, chairman of the Judiciary Committee, called off plans to formally draft SOPA.

Google's logo in protest of SOPA / PIPA
Google’s logo in protest of SOPA / PIPA

The discontinuation of SOPA was a huge victory for internet activists that proved that they could have an impactful effect on the regulation of the internet. About a year later, protests started once again to assist the FCC with their net neutrality propositions, and opposition has become more prominent as the regulations are being written up.

No throttling.

On March 20, 2014, Netflix’s CEO, Reed Hastings, published a blog post on Netflix’s website, “Internet Tolls And The Case For Strong Net Neutrality”. Hastings claims that major internet service providers (ISPs) throttled Netflix’s connection to it’s customers significantly. Since Netflix’s service relies on constant data being streamed to its users, these ISPs are able to extract a toll from Netflix in order for their service to be treated equal to other internet traffic. He concludes, “Though they have the scale and power to do this, they should realize it is in their long term interest to back strong net neutrality. While in the short term Netflix will in cases reluctantly pay large ISPs to ensure a high quality member experience, we will continue to fight for the Internet the world needs and deserves.” Hastings’ post no doubt fueled the fire even more, calling to attention that the dangers of what can happen without net neutrality are happening to Netflix already.

No paid prioritization.

This goes hand in hand with the proposal of no throttling. The same toll-generating technique can be used the other way around. In addition to banning throttling (inhibiting the connection of select services / websites), the ban of paid prioritization prevents ISPs from boosting the connection of select services and websites (be it ones that pay for the boost, or that they are buddy-buddy with).

Opponents of the FCC’s deal tend to be unconvinced of the last two because they believe in enterprise culture. Companies such as Comcast see an opportunity to make more money and therefore naturally make use of it. If you rent out spaces in a mall, and then the mall suddenly becomes extremely popular, wouldn’t you want to charge more for the rent? After all, it take more for the mall to manage with the influx of visitors. It also takes Comcast a lot of resources to make the biggest streaming service in the world available to millions. Makes sense right?

However, the difference between a mall and the miracle of the internet is not to be overlooked. The Internet has been a catalyst for innovation in the past decades. Its expansion is undoubtedly responsible for a level of globalization our ancestors could only dream of. It’s created a forum for the greatest minds to share and collaborate on innovations and new information. Limiting the user that uses more than the average is like limiting a cancer researcher to one experiment per day because they use too many resources. Throttling is like making new slow lanes on the major highways, costing you extra to use the normal lanes where it doesn’t take two hours to drive to work.

The invention of the internet is destined to lead to another great age in society. Many parallels can be drawn to the invention of the printing press. The dissemination of ideas that came as a result of the press, among other things, left the Medieval period behind and ushered in The Renaissance, and ultimately, the modern age. Through the power of the press, Martin Luther’s translation of the Bible became widespread, allowing simple people to read the entire bible for themselves, something that was before done only by educated Catholic priests and scholars. The printing press turned the world on its head in a way that is viewed today as crucial to where our society is now, how we improved from the fifteenth century to get to where we are now.
The internet right now is really in an infant state in the grand scope of things. With its protocols being invented just thirty years ago, it needs protection so that it can reach its full potential in globalizing society and bringing together people as a whole. See what the printing press fostered over five hundred years? The internet can do even more than that in an even shorter time. The internet is the key tool in advancing society and that is why it deserves these unalienable rights.

By The Opte Project [CC BY 2.5], via Wikimedia Commons

The Syrian Civil War: A Modern Conflict

The following was the main project of my Global Studies class, and was written in January of 2015. The year that has followed seems to be full of more anguish and little resolution. The Kobanî massacre resulted in 200 Kurds killed by ISIL attacks and suicide missions, and the Douma massacre resulted in 100 killed by Syrian government airstrikes. Regardless of numbers, truly no one is winning.

Evolution of Conflict

The conflict in Syria is one of epic proportions.  At least two million Syrian citizens have fled the country, and another four and a half million are internally displaced (The Political Science of Syria’s War). The equivalent of the population of Tennessee all being expelled from their homes. The campaign has evolved from an oppressed population protesting in the wake of successful Arab Spring protests (in countries such as Egypt and Libya) to a disjointed mess of rebels, extremists, governments, and terrorists.  Syrian President Bashar al-Assad has drawn disapproval from many by grouping together rebels who have taken up arms with terrorists that are piggybacking off of the mayhem to promote their own interests (President Bashar al-Assad Interview…). With no clear solution, a multitude of belligerents, and an uncooperative government, the impacts of the Syrian Civil War will be felt for decades to come.

Syria is a geographic nightmare. Bordered by Iraq to the east, the Islamic State of Iraq and the Levant has continually advanced through northeast Syria. In the north, Kurdish fighters take a stand for their own rights as an ethnic group that spreads across several countries. In the larger cities in the west, the Syria Battlelinesgovernment and opposition face off, while occasionally facing off with ISIS forces on the eastern front (Syrian Civil War). In addition, Lebanon and Turkey directly border and both have interests in the region. The government is often viewed as the voice of the Alawite minority. The Sunni versus Alawite perception is really an oversimplified version of what’s going on. The Sunni muslims are made up of Arabs as well as Kurds. The Islamic state adds a whole new dynamic to the situation. These four main factions have formed in a small area, equipped with weapons and support from other countries that wish to have an influence on the fighting, including, but not limited to the United States, Turkey, Qatar, Iran, Hezbollah, and Russia (The Political Science of Syria’s War).

With all this potential built up, protests in 2011 put everything into action. As said by Rami Nakhla,  a Syrian cyberactivist, “We want what what everyone in the region wants: an end to corruption, the ability to choose and dismiss our leaders, freedom of speech, and freedom from fear,” (Baker, Aryn. “Has Assad Won?”). Syria’s majority population’s discontent is not a recent development. It’s deeply rooted over half a century’s worth of mass detention, corruption, and oppression of intellectuals and politicians (Baker, Aryn, “Has Assad Won?”). These kinds of conditions inevitably lead to drastic actions.

The story of Joseph Dzhugashvili (later known as Joseph Stalin) is a fine example of such. As a young boy, Dzhugashvili lived the life of the lower class. Owning land was hard for the peasantry and when he was offered the schooling of a priest, he became more and more educated. Him and many colleagues developed a passion for Vladimir Lenin’s works and Stalin found that his way to materialize his beliefs was through the pen. He wrote for newspapers and soon became an enemy of the Tsar’s government. Although Dzhugashvili himself was exiled to Siberia several times, and lived a life as a fugitive, he and his associates, with the backing of the people, overtook the monarchy. (Service, Robert) With preconditions like this, many feel entitled to use force to get the rights that they deserve. As John Locke insisted, when government violates individual rights, the people may rebel. This ideology has been behind many comparable movements, such as the American and French revolutions (Powell, Jim). Syria’s situation, while adulterated by many other factors, is not much different, and should be expected to be fought by the opposition with full force because, as they see it, their livelihoods are on the line.

The war has extended longer than anyone had expected. Initial hopes were for a swift revolution like other Arab Spring events, but it has by now evolved into a monstrous, unmanageable mess. The situation is captured perfectly in the words of Mohammad, a protester who only wanted to be identified by his first name, “We thought it would work, that it would be quick, like Tunisia and Egypt. But our revolution was stolen. They turned it from a fight for freedom into an Islamic revolution. I don’t want to have to choose between the extremists and the government, they are both killers,” (Aryn, Baker; BUQAYA, LEBANON). The general consensus is that the conflict is much more than it originally intended to be, it has truly become an interstate conflict when its roots have it as a push for human rights.

Proxy WarWhy wasn’t it quick like Tunisia or Egypt? Because the Syrian Civil War has taken the form of a proxy war, where countries promote their interests in another sovereign country’s land. Syria receives support from Iran, Hezbollah, and Russia. The opposition does from the West, Saudi Arabia, Turkey, Qatar,  and jihadi groups. This leads to longer lasting-conflicts, as one side does not need to commit their own soldiers to a cause, they just need money. If one side starts losing, add a big chunk of money. When this happens on both sides, a sort of stalemate is not uncommon. (The Political Science of Syria’s War).

As for more intrastate matters, the conflict is rooted in a variety of sources. In an interview with Charlie Rose, president Bashar al-Assad says, when questioned whether he believes this is becoming a religious war, that “it started partly as a sectarian war in some areas, but now it’s not, because when you talk about sectarian war or religious war, you should have a very clear line between the sects and religions in Syria according to the geography and the demography in Syria, something we don’t have. So it’s not religious war, but al-Qaeda always uses religions – Islam, actually – as a pretext and as a cover and as a mantle for their war and for their terrorism and for their killing and beheading and so on” (“President Bashar al-Assad Interview…”). Whether Assad is right or not about just how much of the opposition are terrorists and how much are those same Syrian protesters that are simply fighting for human rights, the conflict has become a breeding ground for Islamist militants and propagandists. It is, however, still sectarian in other areas that are not dominated by ISIS. It’s also a bit of an ethnic conflict (versus a civil conflict) as Kurdish nationalism has shone through amidst all the hubbub of rebellion. Syrian Kurds have joined with their Iraqi cousins to protest the borders established by Europeans after World War I (Klein, Joe). At this point, “civil war” is hardly an adequate term to describe the conflict, it’s an absolute Middle Eastern fiasco; ethno-territorial, sectarian, radical Islamist, and political.

The Syrian government is dominating the opposition vis-à-vis military and intimidation strategies. The government forces generally have the upper hand. As President Assad said, “They went to every part there’s no army in it, and the army went to clean and get rid of them. They don’t go to attack the army in an area where the army occupied that area and took it from it … what the army is doing is cleaning those areas, and the indication that the army is strong is that it’s making advancement in that area” (“President Bashar al-Assad Interview…”). Although the president attempts to downplay the rebels (by making it seem like Syria does not care about the cities they occupy at all), the fact of the matter is they are generally not on the offensive, they have taken up cities and are being bombarded by effective, modern weaponry. Even worse, studies suggest that aerial bombing as a counterinsurgency tactic suggests that it only further drives the civilians caught in the crossfire to support the insurgents (The Political Science of Syria’s War). It’s a self-sustaining fire.

To add to the mayhem, chemical weapons — something as taboo in the military like any other weapon of mass destruction (biological, radiological, nuclear) — were used against civilians in Damascus, killing an estimated 1,400 people. UN-conducted investigations uncovered a lot about the attack, which used sarin, a schedule 1 substance. Since it is a nerve agent, it can cause permanent neurological damage (granted the victim is not killed by the attack). While the government still denies involvement, they are agreeing to cease production of chemical weapons and to get rid of the ones they have now. Some of his opponents believe he is delaying this process, and/or continuing to use less deadly chlorine gas (Baker, Aryn. “Has Assad Won?”).

With all this going on, Assad’s power has not truly wavered like Muammar Gaddafi’s or Hosni Mubarak’s. There haven’t been any signs of the government wishing to give in to demands, as they think they can handle the rebels forces (and are doing a good job at it). Again, the rebels wanted a quick shift in power, the government wanted to quickly put down a rebellion. Neither of the parties got what they really wanted. In the meantime, ISIS has entered the forefront and taken whatever power they can get amidst the chaos (Baker, Aryn. “Has Assad Won?”).

All of this tumult has taken its toll on the people. On average, over three thousands refugees arrived in a neighboring country every day in 2014. Nearly three million refugees are in neighboring countries Jordan, Lebanon, and Iraq. Not only syria_refugeesis this conflict clearly affecting these citizens, but it is also taking an economic toll on neighboring countries. Microeconomic impacts are heavy with the sharp increase in supply of low-skilled labor (Zetter, Roger). Despite refugees not having permission to work, undocumented jobs will always thrive, and as they do, market prices for basic commodities increase. Low income households are even more negatively affected, “Even before the crisis 25% of the Lebanese population lived below the upper poverty line of US$4 per day and the influx of refugees was projected to push an additional 170,000 Lebanese into poverty and to double unemployment to above 20% by 2014” (Zetter, Roger). Macroeconomically, regional trading patterns have been severely disrupted. Lebanon’s economic growth rate has been reduced by 2.9% from a predicted 4.4% in 2012–2014. The end user also feels the trade disruption in the form of increased commodity prices. Lebanese revenue has been decreased by $1.5 billion while expenditures have increased by $1.1 billion as the demand for public services grows. All the while, Syria is losing millions of its own workers and suffers from similar problems. If the war were to end, it’s estimated to take thirty years for the economy to recover to a pre-2010 stage (Baker, Aryn. “Has Assad Won?”). Exports from Lebanon to Syria have increased significantly, which is about the only way the conflict positively affects countries such as Lebanon (Zetter, Roger).

Internal and External Attempts on Peace

Attempts on peace have been well thought out, but not well-executed. When asked if he has a plan, President Bashar al-Assad replied, “When you have these terrorists, the first part of the same plan which is political should start with stopping the smuggling of terrorists coming from abroad, stopping the logistic support, the money, all kinds of support coming to these terrorists. This is the first part. Second, we can have national dialogue where different Syrian parties sit and discuss the future of Syria. Third, you can have interim government or transitional government. Then you have final elections, parliamentary elections, and you’re going to have presidential elections” (“President Bashar al-Assad Interview…”). However effective as this plan sounds, it isn’t executed due to unreasonable prerequisites on Assad’s part, as stated in his interview with Charlie Rose:

Charlie Rose: But the question is: would you meet with rebels today to discuss a negotiated settlement?

President al-Assad: In the initiative that we issued at the beginning of this year we said every party with no exceptions as long as they give up their armaments.

Charlie Rose: But you’ll meet with the rebels and anybody who’s fighting against you if they give up their weapons?

President al-Assad: We don’t have a problem.

Charlie Rose : Then they will say “you are not giving up your weapons, why should we give up our weapons?”

President al-Assad: Does a government give up its weapons? Have you heard about that before?

Charlie Rose: No, but rebels don’t normally give up their weapons either during the negotiations; they do that after a successful…

President al-Assad: The armament of the government is legal armament. Any other armament is not legal. So how can you compare? It’s completely different (“President Bashar al-Assad Interview…”).

The problem is that Assad creates a double standard. These people want their say in government, but the president does not want to treat them as equals. He does not want to acknowledge, in any way, that they are a strong force. Less symbolically and more literally, the rebels have no reason to trust that they will get what they want but have all the reason to believe that the government will play Judas at their weakest moment. There is a perfectly good protocol to follow to try and negotiate a power sharing agreement, but it’s hard to reach when Assad makes it hard to reach.

Syria OldIn the meanwhile, de facto partitioning is taking place, especially in the case of the Kurds. World War I resulted in the collapse of the Ottoman Empire followed by its partitioning by France and Britain. France mandated the area that became Syria and Lebanon today. It wasn’t until 1942 that the Alawite state became an official part of greater Syria. In 1936 six leading Alawites, including Bashar al-Assad’s grandfather Sulayman al-Assad, wrote a letter to the French powers in charge of the French Mandate for Syria and the Lebanon, pleading that the Alawite state not be made a part of Syria, as Alawites were and often still are discriminated against by Sunni muslims (Mackey, Robert). The point is, just because Europeans drew lines in the sand nearly 100 years ago doesn’t mean Kurdistanthat everyone in the same border will get along, or that they will agree with exactly where the lines are placed. Such is the case with the Kurds, who span across five nations. Turkey, Syria, and Iraq were all a part of the Ottoman empire pre World War I (Pichon, Eric). So while no one likes to talk about partitioning, it is still a reasonable proposition as a partial solution, and is especially appealing to Kurdish nationalists who strive for autonomy.

While the United Nations has chartered ceasefires and conducted investigations, they appear to have had little effect on the current situation. A ceasefire only lasts so long, and the president denies the conclusions made by the UN investigators at Damascus. The chemical weapons attack has brought a lot of scrutiny upon Assad, and he was questioned heavily about it in Charlie Rose’s interview:

Charlie Rose : So you’re against the use of chemical warfare?

President al-Assad : Yes, not only me. As a state, as a government, in 2001 we proposed to the United Nations to empty or to get rid of every WMD in the Middle East, and the United States stood against that proposal. This is our conviction and policy.

Charlie Rose: But you’re not a signatory to the chemical warfare agreement.

President al-Assad: Not yet.

Charlie Rose: Why not?

President al-Assad: Because Israel has WMD, and it has to sign, and Israel is occupying our land, so that’s we talked about the Middle East, not Syria, not Israel; it should be comprehensive.

Whether Assad’s point is reasonable or not (some might say that possessing chemical weapons does not really directly reason for others to have chemical weapons, others may see it necessary, like other weapons of mass destruction), he always has some excuse to not work with the United Nations’ or others’ proposals, be it allowing for talks without rebels giving up their arms, or to completely get rid of their chemical weapons (“President Bashar al-Assad Interview…”).

Recommended Strategies for Resolution

The ideal solution is vexed by hindrances such as Assad’s, but it would generally consist of demilitarization and power sharing from both sides. The first step should be to eliminate chemical weapons from the picture, as they pose an immediate threat to both parties. Few argue that they should be used, especially after World War I. Stockpiling and not using is not acceptable, and if it takes disarming Israel of their chemical weapons, so be it. The events in Damascus are a clear reason as to why this needs to be top priority.

Ceasefires should be next on the list, as they are easier to achieve than demilitarization, but still effective at inhibiting further violence. While this does not directly solve the problem, the goal is to reduce violence and take stepping stones towards demilitarization. Even though this is a relatively small act from a military standpoint, it can have a more profound cognitive effect on the people fighting, on either side. To give the belligerents a taste for the reduced fighting.

Attempts should be made by the United Nations to push for intrastate peace talks and power sharing negotiations.  The obstacle they need to get around is President Assad’s unwillingness to negotiate with rebels unless they lay down their arms.  It’s an unrealistic request that appears to be the main obstacle in further progress towards peace. Ideally, both rebel and government forces should demilitarize to reduce tensions which would make it easier to negotiate compromises. If the government, the rebels, and the Kurds are able to make compromises and to begin to work together, they can shift their focus to pushing out ISIS—a belligerent widely condemned—from the region.

Apace with these steps should be attempts to start fixing the social and physical issues brought on by the conflict. The ultimate goal is to restore Syria’s economy and infrastructure to pre-2011 levels and set them up to continue growing. Immediate focus should be placed on returning refugees and helping them, especially financially, to adjust back to life in Syria. The United Nations High Commissioner for Refugees, which staffs people in more than 125 countries, already has a presence in Syria helping refugees. Their continued assistance would be imperative to recovering the labor market that has been so hindered by the refugee crisis (About Us). Human Rights Watch could help lobby for solutions to issues with chemical weapons, and other, conventional methods of indiscriminate killing (which accounts for the majority of the death toll so far (Bolopion, Philippe).

Member nations should be careful not to clout the intrastate negotiations too much. World War I is over, and it’s time for the post-Ottoman people to take their own stands. Instead of Europeans drawing lines, people native to that region should be making the decisions. What the people want is their voices to be heard, so let it be so. Guide Syria to move away from civil violence and towards what Syria, as a whole, wants itself to be.

Works Cited

“About Us.” The UN Refugee Agency. N.p., n.d. Web. 21 Jan. 2015.

Baker, Aryn. “Has Assad Won?.” Time 183.22 (2014): 20-30. Academic Search Premier. Web. 10 Dec. 2014.

Baker, Aryn, LEBANON BUQAYA, and RAMI AYSHA. “Deepening Divide.” Time 177.24 (2011): 24-27. Academic Search Premier. Web. 10 Dec. 2014.

Bolopion, Philippe. “Dispatches: What About Justice for Syria’s Victims?” Human Rights Watch. N.p., 10 Sept. 2013. Web. 22 Jan. 2015.

Burrows, Lindsey. “Maps to Help You Understand the Syrian War.” The American Interest. The American Interest, 26 Feb. 2014. Web. 10 Jan. 2015.

Klein, Joe. “Number Four Bashar Assad The Lethal Tyrant.” Time 182.26 (2013): 118. MasterFILE Elite. Web. 10 Dec. 2014.

Latuff, Carlos. Proxy war on Syria Opera Mundi. N.p., n.d. Web.

Mackey, Robert. “Syria’s Ruling Alawite Sect.” The New York Times. The New York Times, 14 June 2011. Web. 10 Jan. 2015.

Pichon, Eric. “Iraq: Towards an Independent Kurdistan?” European Parliamentary Research Service. European Parliamentary Research Service, 10 July 2014. Web. 10 Jan. 2015.

“The Political Science of Syria’s War.” The Political Science of Syria’s War (n.d.): n. pag. Pomemps, 18 Dec. 2013. Web. 10 Dec. 2014.

Powell, Jim. “John Locke: Natural Rights to Life, Liberty, and Property.” Foundation for Economic Education. N.p., n.d. Web. 16 Jan. 2015.

“President Bashar Al-Assad Interview With CBS News.” Le Grand Soir. N.p., 13 Sept. 2013. Web. 10 Dec. 2014.

Service, Robert. Stalin: A Biography. Cambridge, MA: Belknap of Harvard UP, 2005. Print.

Zetter, Roger, and Héloïse Ruaudel. “Development And Protection Challenges Of The Syrian Refugee Crisis.” Revista Migraciones Forzadas 47 (2014): 6-10. Academic Search Premier. Web. 10 Dec. 2014.

Solving the Rainbow

The rainbow is undoubtedly one of the most spectacular light shows on earth. It’s hard to miss, cutting across the sky as an arc of seemingly unnatural light. Although a bit more complex than other optical phenomena such as sunrises and sunsets, the common rainbow is still a simple little thing.

What makes a rainbow?

A rainbow is a result of two types of events, refraction and reflection. Refraction occurs when a propagating wave (light in this case) moves from one medium to another. Most people have probably experienced refraction in one form or another. If you place a straw in a glass of water, it appears bent due to some light refracting before it reaches your eyes. In the case of the rainbow, refraction occurs when sunlight interacts with drops of water in the air.

Rainbow rays refracting in a raindrop

This of course does not happen before our very eyes, and raindrops in the air aren’t melon-sized. Instead, it occurs high up in the sky with very small droplets of water.

The sun relative to the raindrop

Don’t be fooled by the diagram though. I assure you that looking up at the sky when raindrops are present will not redirect the full strength of the sun into your eyes. In addition to light energy being absorbed by the atmosphere, the first diagram only pictured the rays responsible for the rainbow phenomenon we humans see down here. In actuality, some light waves reflect and others continue through. Most of these rays do not generate anything that is visible from where we stand; only the first one pictured hits our eyes in the form of the typical rainbow.

Rainbow reflection events

What about the different colors?

When we consider a source of light that produces many different wavelengths of light, such as the sun, a single beam of light is split up into its constituent wavelengths due to dispersion. Meaning when light passes between air and water, the refraction angle is different depending on its color.

Optical dispersion

Breaking it down

To get a good understanding of how refraction actually works, we need to understand Snell’s law. When waves change mediums in which their speed differs, Snell’s law relates the angle of incidence (the angle at which the light strikes the rainbow compared to the normal) to the angle of refraction  (the angle the light goes after refraction, still compared to the normal). In the case of a raindrop, we have a light wave moving from air to water, and the normal line is really like an extended radius since the normal is defined as being perpendicular to the tangent.

First refraction event

The law tells us that the ratio of the sines of the two angles (i, the angle of incidence, and r, the angle of refraction) is equal to the ratio of the phase velocities, v_{1} and v_{2}, of light in the respective media:

\frac{\sin i}{\sin r} = \frac{v_1}{v_2}

Instead of using the velocities of light as our variables, however, we’d like to describe this relationship using the refractive indices of the mediums: the ratio of the speed of light in a vacuum, c, to the speed of light in the specified medium, v

n = \frac{c}{v}

With a little math magic we see that the ratio of the two sines is equal to the inverse of the ratio of the two refractive indices.

\frac{\sin i}{\sin r} = \frac{v_1}{v_2} = \frac{\frac{c}{v_2}}{\frac{c}{v_1}} = \frac{n_2}{n_1}

We’d like to solve for r , so we rewrite it as:

\sin r = \frac{n_{1}}{n_{2}}\sin i

Luckily, for our purposes, we can make some simplifications before we go further. The refractive index of light in air at standard temperature and pressure is about 1.00029, but when it comes to refractive indices, we’ll find out later that we only have sufficient data for four significant figures, so we can just use 1.000 instead. We can also rename our refractive index of light in water as n_{w} :

\sin r = \frac{1}{n_{w}}\sin i

Solving for r, we find that

r = \sin^{-1} \left(\frac{1}{n_{w}}\sin i\right)

Where the colors separate

The key to the separation of colors is the phenomenon of optical dispersion, which states that the phase velocity of a wave is dependent on its frequency. In light-specific terms, dispersion tells us that the phase velocity v of a light wave is dependent on its color \lambda, which means our refraction angle r is ultimately a function of color!

We can test out this relationship with an example graph. Plotting the refraction angle r as a function of n_{w} between n_{w} = 1 and n_{w}= 2  for a 45° angle of incidence i we see the refraction angle change:


In this particular case, we see that the refraction angle r changes an entire 0.42 radians, or about 24°, effectively as a function of color. However, we haven’t truly related color to refraction angle, we’ve only related refractive index to refraction angle. We haven’t actually found a way to transitively relate refraction angle, r, to color, \lambda. We know that the relationship exists, but what are the details?

In other words, we know that

r\left(n, i\right) = \sin^{-1}\left( \frac{1}{n_{w}}\sin i \right)

but we need to understand n_{w}\left(\lambda\right) because we want to know r\left(\lambda, i\right)

r\left(\lambda, i\right) = \sin^{-1}\left( \frac{1}{n_{w}\left(\lambda\right)}\sin i \right)

For now, we are just worried about wavelength. We won’t worry about angle of incidence i until the end.

Modeling n as a function of λ

Finding a relationship between the refractive index and the wavelength of light for many optical materials is a topic that has been of interest to scientists for hundreds of years. Augustin-Louis Cauchy defined an empirical equation in 1836 relating the two:

n\left(\lambda\right) = B + \frac{C}{\lambda^{2}} + \frac{D}{\lambda^{4}} + \ldots

where B, C, and so on were empirically derived coefficients for different media. However, this model was only good for representing the visible spectrum. It became more inaccurate for infrared and ultraviolet waves. Wilhelm Sellmeier improved upon the errors of this equation in 1871, introducing a new equation more components and coefficients:


For our purposes, however, we don’t need a complex model to accurately approximate indices of refraction. We can fit experimental data instead. The light wavelengths that concern the rainbow are part of the visible spectrum, a range of only about about 300 nanometers. RefractiveIndex.INFO offers a great, public domain database for refractive indices of all sorts. Using the database, I easily found experimental data from Optical Constants of Water in the 200-nm to 200-μm Wavelength Region by George M. Hale and Marvin R. Querry. The curve seems rather daunting at first look:


But this experiment is for wavelengths of 0.2–200 micrometers. We only plan on modeling 0.390 to 0.700 µm, so we don’t have to try and model an equation for this funky graph.

Instead, the relevant data points could be downloaded as a raw comma-seperated value file:

Wavelength (nm) Refractive Index
375 1.341
400 1.339
425 1.338
450 1.337
475 1.336
500 1.335
525 1.334
550 1.333
575 1.333
600 1.332
625 1.332
650 1.331
675 1.331
700 1.331

These numbers all are noticeably very close. In fact, because there are only 4 significant digits, the refractive index from 650 to 700 nanometers does not “change” at all. However, we only need a reasonable estimate of how the index changes for this simulation, so we don’t need to worry much about misrepresenting the precision we actually know. Using a second degree polynomial fit, a good function to approximately model our n_{w}\left(\lambda\right) in the range of 390–700 nanometers would be:

n(\lambda, i) = \left(8.1318681 \times 10^{-8} \right)\lambda^{2}-\left (1.1757142\times 10^{-4}\right)\lambda + 1.3733752

Finally we can properly make use of our function for r:

r\left(\lambda, i\right) = \sin^{-1}\left( \frac{1}{n_{w}\left(\lambda\right)}\sin i \right)

Relating r and i to the geometry of the event

Ultimately, the rainbow is going to be comprised of a countless number of light waves of different wavelengths \lambda hitting the raindrop at different angles of incidence i. The outputted rays of light will be quantified as rays of wavelength/color \lambda at angle D.

Drawing auxiliary lines onto our original diagram will help solve this geometry problem:


We already know much about the situation from a small amount of information. We also already know r as a function of i and \lambda. We have enough information geometrically to represent every angle on the diagram, so we can fill out the important ones:


  1. Because opposite angles are equal, r + \left(i-r\right) has to be equal to i.
  2. Because all radii in a circle are congruent, we know that the triangles created by two of them are isosceles.
  3. The first half of the reflection angle is r because the two base angles of an isosceles triangle are equal.
  4. The second half of the reflection angle is r because the definition of a specular reflection is that the incident ray and the reflected ray make the same angle with respect to the normal (the radius).
  5. Again, the refraction angle of the second refraction event is r because it must be equal to the other base angle of the isosceles triangle.
  6. Ultimately, because this is the same wavelength ray changing mediums in the same way, it is undergoing the same refraction event, so therefore the other angle must be i - r like the first one.

After solving for these angles, the total deflection angle can simply be found by adding up 3 three net deflections, as animated below:


Simplified, the total deflection angle D as a function of i and r would be

D\left(i, r\right) = 180^{\circ} + 2i - 4r

Since we already know the reflection angle r as a function of i and \lambda, we can substitute to find D as a function of i and \lambda.

D\left(i, \lambda\right) = 180^{\circ} + 2i - 4sin^{-1}\left(\frac{1}{n_{w}\left(\lambda\right)}\sin i\right)

Analyzing this angle as a function of two variables

So far, we’ve only seen these events while ignoring the variability of the angle of incidence, i. In reality, rays will strike the raindrop from all angles between –90° and 90°. However, for the angles of incidence less than zero (on the bottom half), the ray will end up refracting another way, basically doing the same thing but mirrored over the x axis. For now, we’ll only worry about the angles from 0° to 90°. Let’s see how i affects the graph of D when n_{w} is held constant at 1.333:

As you go over all the angles of incidence on the raindrop, there is a certain minimum output angle. Because the graph of D flattens our near this minimum point, the rays cluster up and form a caustic. So in this case, when n_{w}\left(\lambda\right) =  1.333, there will be a caustic of rays of wavelength \lambda at angle 137.9°. For different wavelengths, there are different n_{w}\left(\lambda\right) values, different local minima, and different deflection angles D where the specific wavelength is most clustered.

Some examples using some n_{w}\left(\lambda\right) values from our empirically-derived function.

Wavelength (nm) Refractive Index Deflection Angle
390 1.33989098927173 138.91°
500 1.334919231025 138.20°
600 1.3321071431079998 137.79°
700 1.330921428817 137.62°

Simulating the rainbow

After doing a few more calculations for things such as coordinate points on the raindrop, the slope of the rays, and the color perceived by the human eye, we can run a simulation to test out this multivariate situation. This simulation will:

  1. Have rays of sunlight strike the raindrop in the northern hemisphere.
  2. Have each ray be made up of many different wavelengths of light equally spread throughout the visible spectrum.

I created the simulation using my own JavaSim framework. For more info on the simulation itself or to run it on your own computer, see the Raindrop Rainbow main page.

The simulation yields a beautiful cluster of rays:

Simulation1Most importantly, at the bottom you can see where the small differences in deflection angle start to form different caustics for different wavelengths. The rainbow truly comes to life. A stretched image of some of the bottommost pixels gives us a sample of the rainbow we’ve created:


Locating the newly-defined rainbow

These reflection angles for the minimum and maximum visible wavelengths tell us where the rainbow begins and where it ends! To locate the rainbow now, we need to make a small modification to our old diagram:


The sun is first of all, not right above you. In this diagram, the sun would actually be setting on the horizon because the rays in question are all completely parallel to the ground. We need to consider the position of the sun in the sky and how it changes where the rainbow ends up. If we redraw the diagram with a variable angle of depression and now include our shadow, we get what seems to be a more complex situation:


Therefore the true angle of the rainbow also depends on the height of the observer (specifically, the location of the eyes). Fortunately, since the rays that shine down from the sun to strike the raindrop are parallel to the rays that strike down on your back to make your shadow, we can view the problem from a different angle, literally:


This diagram is the same as before, except instead of being relative to the ground, it is relative to the line from your eyes to the head of your shadow is (or more specifically, where your shadow’s “eyes” are). This point on the shadow directly opposite the sun is consequently named the antisolar point. Some small geometry fill ins reveal that the angle made between here and the ray of angle D is the supplement of D. For the two ends of the visible spectrum, we wish to calculate these supplements.

Wavelength (nm) Deflection Angle Angle From Antisolar Point
390 138.91° 41.09°
700 137.62° 42.38°

Before we walk outside and look up 41.5° and wonder where the rainbow is, we need to remember the preconditions. First and foremost, water droplets must be present, whether from rain, mist, or a garden hose. Also, the sun cannot be high in the sky. It must be either the beginning or the end of the day, with few exceptions. In an area of completely flat ground, during the middle of the day, let’s just say you would need some… underground raindrops in order to see a rainbow: Diagram11So the full rainbow-hunter’s checklist would go something like:

  1. Choose a rainy, misty day or environment.
  2. Choose a time early or late in the day (late tends to have better weather).
  3. Locate the head of your shadow on the ground.
  4. Look up about 41°.

With any meteorological luck, you should see the world-renowned bow!

Extras: variations and unusual cases

Of course, when it comes to something as funky as a rainbow, there will be some more funky things that come along with it. Behold!

The “underground” rainbow

When the sun is too high in the sky, the diagrams tell us that raindrops would theoretically have to be underground for us to actually be able to see the phenomenon, but from the edge of a cliff, we can effectively make this happen:

Diagram12A great example of the “underground” rainbow taken midday at the Grand Canyon:


The secondary rainbow

Commonly referred to as a double rainbow, a secondary rainbow is not all that different from a typical rainbow. It is a result of the same processes with two key differences: the rays in question strike the southern hemisphere of the raindrops and there are two reflection events instead of one. The rays striking in the southern hemisphere cause the color order to be reversed and the extra reflection event causes more light to be lost.

Diagram13This sister phenomenon is more difficult to see due to another event in which more light is lost. The double rainbow is often seen in conjunction with already-spectaular rainbows because it already needs good conditions to be clearly visible.

An example of a secondary rainbow taken by myself in Stone Harbor, New Jersey on an August afternoon after a rainstorm:



Special thanks to Professor Halpin-Healy for a great lecture on rainbow physics that covered most of the content in this post and inspired me to learn and experiment more.

Thanks to Marvelous Marv for providing the great Grand Canyon rainbow photo and his wife for taking it.


“Cauchy’s Equation.” Wikipedia. Wikimedia Foundation, 25 Aug. 2014. Web. 29 Aug. 2015.


Halpin-Healy, Timothy. “Rainbow Physics.” Barnard College, New York City. 6 Aug. 2015. Lecture.


Nave, Carl R. “Rainbow Concepts.” HyperPhysics. Georgia State University, n.d. Web. 29 Aug. 2015.


“Plot.” Wolfram Alpha. Wolfram, n.d. Web. 29 Aug. 2015. <>.


“Plot.” Wolfram Alpha. Wolfram, n.d. Web. 29 Aug. 2015. <*sin%28i%29%29+


Polyanskiy, Mikhail. “Optical Constants of H2O, D2O (Water, Heavy Water, Ice).” RefractiveIndex.INFO. Mikhail Polyanskiy, n.d. Web. 29 Aug. 2015.


“Sellmeier Equation.” Wikipedia. Wikimedia Foundation, 16 Jan. 2015. Web. 29 Aug. 2015.

Simulating Monopoly For Statistics


I decided to make it my goal this summer to try to link my computer science capabilities to my math knowledge. I stumbled upon a website in which someone tried to calculate the percentages of landing on spots of the monopoly board with pure mathematics. I thought to myself, “I wonder how correct this data is, it doesn’t seem they tested it out.” So I set off to write a computer program to simulate the game itself and output the data to the comma separated value file (CSV). Here is some sample output of my computer program (Figure 1):

Sample CSV file opened as raw text in Notepad.
Figure 1. Sample CSV file opened as raw text in Notepad.

This data can be opened by Microsoft Excel with ease to look like this (Figure 2):

Sample CSV file opened in Microsoft Excel.
Figure 2. Sample CSV file opened in Microsoft Excel.

This then allows for all sorts of analysis of the data, and allows me to simulate a billion rolls of the dice!


The first and undoubtedly most important part of this project is to create the code that simulates a game of monopoly. Luckily, one simple fact about monopoly allows simulating it much, much easier: players cannot affect the position of other players. That means that only one player has to be simulated, removing big variables from the equation that exist in the simulation of many other games.

The program is designed to track the following data throughout the simulation: the number of rolls of the dice, number of moves (rolling doubles and going again counts as one move), position on the board (can be from 0 to 39, as there are 40 spaces on the board), number of doubles rolled, number of times go has been passed, total distance moved, the values of both dice, whether or not the player is in jail, and the number of times each of the forty spaces have been landed on.

It was coded in Java using Eclipse on Mac OSX as seen below (Figure 3):

Monopoly Simulator project opened in Eclipse.
Figure 3. Monopoly Simulator project opened in Eclipse.

The source code in its entirety can be found online on GitHub at

The final result ended up being 1,741 lines: 1,189 of code, 307 of comments, and 237 blank. There were several features that the skeleton was laid out for that were actually not completed. I designed the program to be extensible so it could be added later. The main feature that the groundwork is laid out for but is not coded is a visual representation of the board. It is set up in the same way that the data-collecting code is set up. It would output an image of a monopoly board with information on the current statistics (whereas the data-collecting code outputs a line of comma-separated values with information on the current statistics).

The program takes three important inputs from the user:

  1. Number of rolls to simulate.
  2. Save interval (in rolls)
  3. Location to save CSV (Comma-separated value) file.

The first two shape the data of the CSV file. If the number of rolls to simulate was 500 and the save interval was 10, there would be a total of 50 lines of data saved, which leads to 50 rows of data to work with in Microsoft Excel. I used this to simulate one billion rolls of the dice, and set the save interval to one billion as well. This only outputs one line of data, but this line of data can be used to make interesting graphs. I ran another that simulated one hundred thousand rolls with a save interval of one hundred, which output one hundred lines of data which I used to graph with the number of rolls as the independent variable (logically similar to graphing with time on the x-axis). For the last and most complex type of data I wanted, another short program was written to extend off the working program. It runs the code of the original program a certain number of times, only saving the last line of data. For example, I ran one million rolls one thousand times. The collected data looked like so (Figure 4):

Sample data opened in Microsoft Excel. (Note: some have 1,000,001 or 1,000,002 rolls because they rolled one or two doubles in their final turn)
Figure 4. Sample data opened in Microsoft Excel.
(Note: some have 1,000,001 or 1,000,002 rolls because they rolled one or two doubles in their final turn)

Generating this required another short program (see Figure 5) which made use of my previous program, allowing it to be done in only about one hundred lines, making use of a loop within a loop.

Code snippet from the auxiliary program.
Figure 5. Code snippet from the auxiliary program.

Now that all the data has been generated, it’s time to analyze it with Excel.


The analysis is where the main goals of the program are reached. The program can only do so much without reinventing the wheel. Instead of trying to analyze data and/or create graphs from within the java program, it makes much more sense to output data that can be read by a program that specializes in just that. Microsoft Excel is the industry standard for a vast majority of data-analyzing tasks.

The first and most obvious thing to analyze is how certain spaces were landed on more or less often than other spaces. This can be represented accurately using a single line of data after a billion rolls. Dividing the number of times each space has been landed on by 1 billion yields a percentage representative of how often that space was landed on. A good way to tell if a space is more or less popular is by subtracting the average from the percentages. Since there are forty spaces on the board, the average is 1/40 or 2.5%. Some interesting trends are revealed by the graph (Figure 6).

Deviation from average (2.5%) for all forty spaces.
Figure 6. Deviation from average (2.5%) for all forty spaces.

The most obvious outliers are jail, chance, and community chest spaces. There are many reasons that jail by far has the most hits.

  1. One of the forty spaces on the board sends the players directly to jail.
  2. One of the sixteen chance and one of the sixteen community chest cards send the player to jail.
  3. Rolling three doubles sends the player to jail.
  4. If the player does not roll doubles to get out of jail, it counts as another turn spent in jail.

Some less obvious ones are chance and community chest. The reason chance cards are seldom landed on is because eight of the sixteen chance cards actually move the player to another space, and the latter space is counted as the space landed on. The same goes for community chest, except only two of the sixteen cards actually change the location of the player.

“Go To Jail” is landed on 0% of the time because in reality, Jail is the space that is “landed” by the end of the roll.

The trends when it comes to other spaces are clearly shown in the graph: Mediterranean and Baltic Avenue are very unpopular while Illinois is very popular. The averages for the different types of space can be seen in Figure 7 below:

Average deviation from 2.5% for several categories.
Figure 7. Average deviation from 2.5% for several categories.

While Illinois is the most popular space, the orange properties, on average, are landed on considerably more than the red ones. The brown properties are far below those popular properties, almost 0.5% below the 2.5% average. It’s easy to tell by this ranking that certain properties are much more valuable than others.

The blue property category, which includes the infamous Boardwalk, is actually in the bottom three. This probably comes as a surprise to a lot of players because it is a very contested property as a result of its high payouts once the game has been going on for a while. It would be lower, actually. However, one of the chance cards, “Take a walk on the boardwalk”, advances the player to the boardwalk space, making it worth quite a bit more than its neighbor, Park Place.

This was only one test, however. I use the auxiliary program to collect a thousand trials of games with a million rolls. With this data, a box plot can be made to represent the minimums, maximums, first, second, and third quartiles for each property (Figure 8).

Box plot using 1,000 games of 1,000,000 rolls.
Figure 8. Box plot using 1,000 games of 1,000,000 rolls.

This proves beyond a reasonable doubt that Illinois is nearly always the most popular space to land on, only possibly being beaten by Go, as the maximum of Go is higher than the minimum of Illinois. It also shows that Park Place could sometimes be the single least popular property!

A lot of other small, interesting graphs can be analyzed with the miscellaneous data that the program keeps track of. A graph can be made to show the randomness of the dice by comparing it with expected, mathematically-calculated data (Figure 9).

Percentage of rolls that are doubles after X rolls compared to the mathematically expected 16.666%.
Figure 9. Percentage of rolls that are doubles after X rolls compared to the mathematically expected 16.666%.

Small tests like these confirm that the simulation is using good numbers that don’t stray from mathematical probabilities.

Another test, which failed several times during the creation of the computer program, is making sure that the average of the hit percentages from each space is equal to 2.5%. If it is lower, some hits are not being accounted for, and if it is higher, some are being counted twice. In the beginning, the percentage was lower, which showed that there were certain times where the computer program was not counting the hits. It turned out that the bug had to do with rolling doubles, and was eventually confirmed to be fixed when the average was exactly 2.5%.

The average of all these numbers (Figure 10): 0.025, 1/40, or 2.5%.

Hit percentages for 40 spaces which averages out to 2.5%
Figure 10. Hit percentages for 40 spaces which averages out to 2.5%


So why did the data come out the way it did? There is one extreme, obvious spike in the graphs. A spike so great that it is cut off or not included in all the graphs.


Jail is by far the most popular space on the board. As discussed before, there are many ways that a player can end up in jail.

When rolling two dice, the most common sum is a seven. The space seven spaces after jail is community chest (between the orange spaces) and seven spaces after that community chest is none other than Illinois!


The reason Illinois is higher than the orange spaces, though, is because there is also a chance card that advances the player to Illinois, instantly spiking up the chances of landing there. However, the oranges properties average a higher hit percentage than the red properties. This is because they are six, eight, and nine spaces away from jail, all of which are close to seven (the rolls are more and more common as they get closer and closer to seven).

However, none of the chance or community chest cards advance the player to an orange space, which is why none of them hold the title of most popular space. On the other hand, there is both a chance and community chest card that advances the player to Go, which is why, excluding Jail, it is the second most popular space to land on.

Bottom line? The saying applies in Monopoly as much as it does in real life:

Location, location, location.
(although in real life the value of a property usually goes up because of beaches, not jails…)