## Preface

On August 31st 2016, I made a mistake; I left to work during the peak of rush hour. With the college students back in town (the beginning of the academic year), the move in day, and the usual rush hour traffic, I was in a standstill vehicle. This article reflects on some streams of thoughts arose during this period—on path, goal, and intent.

1. Traffic
2. Light Speed
3. Fermat
4. Destination

## Traffic

It's a Wednesday morning 8:40 AM—on my way to work. The static vehicle contrasted with the ticking watch. I'm stuck in a rush hour traffic, which is precisely the reason why I try to leave before 7:30 AM. Every morning, countless urban workers hop on various modes of transportation to get from home to work. With urbanization, automobile, and suburbanization, an individual can now travel a great distance to work—either voluntarily or out of options. The act, on its surface and when it first emerged, seems ordinary and benign. However, after decades of study, researchers found that commuting, especially a long one, is economically expensive and detrimental to your health and happiness. Unfortunately, the incurred cost is often delayed and consequently invisible, like a malignant tumor yet to be discovered.

Even without all the research, people innately desire to minimize commute. Let's think about a hypothetical commute where you need to drive from a location within the city to the center downtown. One obvious option is the shortest path, but you need to drive directly through the traffic-infested urban streets with a deluge of other commuters. Thus, despite the shortest distance, your time won't be optimal. An alternative route includes the local expressway, which allows you to drive much faster. However, this comes with a cost since the expressways are in the outskirts of the city. If you drive more on the expressway, you can drive faster (shorter time) but further away from your destination (consequently more time). In order to minimize the commuting time, you (or smart engineers at Google) need to carefully balance the time conserved by driving on the expressway against the added distance.

The shortest direct route isn't always the quickest.

## Light Speed

For a second, I drifted away from waiting and thought that it'd be quite neat to drive at the speed of light. Anyway, when it comes to the above problem, humans are not alone. Light, in vacuum, travels at a stunning 3 x 10^8 m/s or about a foot per nanosecond, but it goes slower (simplified explanation) in a denser medium—more traffic to get through. Now, if light is traveling through water, it would slow down to about 2.3 x 10^8 m/s or ~1.33 times slower. In order to simplify this, scientists use the following notation:
$$n = \frac{c}{v}$$
where $$c$$ is the speed in vacuum and $$v$$ is the speed in another medium such as water. So for water, $$n$$ would be 1.33. For diamond, it is 2.42. We call $$n$$ the refractive index of the medium.

Now what if light has to travel through two different media—such as air and water?

Let's look at the figure below, where white is air and blue is water. Again, the most obvious way is to take the direct/straight line path, which is the shortest in distance:

However, just like the traffic example, this may not be the fastest since traveling through water would slow things down.

Alternatively, it can take the below route to minimize the distance traveled in water, the slower medium:

But now the total distance travelled gets longer to the point that it no longer conserves time. If you need to detour a great distance to use the expressways, it might not make sense to do so since the time saved by traveling at a higher speed will be diminished by the time added by traveling a longer distance.

After all, light finds the optimal (the fastest) route that balances the travel distance and the medium velocity:

But how does it do that?

## Fermat

In 1662, Pierre de Fermat—a French mathematician well known for his Last Theorem, which used to be the "most difficult mathematical problem"—came up with something called the principle of least time or Fermat's principle. The statement is simple: "light travels between two points along the path that requires the least time, as compared to other nearby paths."

Using the aforementioned principle, we now can find the "path" that light takes.

The same figure is labeled with some variables:

The total time it takes to travel from A to B is:
$$\sum{t} = \frac{\sqrt{x^2+h_1^2}}{c / n_{air}} + \frac{\sqrt{(d-x)^2 + h_2^2}}{c / n_{water}}$$

Since we'd like to minimize this function (the total travel time), let's look at the first order change:
$$\frac{\partial \sum{t}}{\partial x} = \frac{n_{air} \cdot x}{c \cdot \sqrt{h_1^2+x^2}} - \frac{n_{water} \cdot (d-x)}{c \cdot \sqrt{h_2^2 + (d-x)^2}}$$

which is equivalent to:
$$\frac{\partial \sum{t}}{\partial x} = \frac{n_{air} \cdot \sin(\theta_1)}{c} - \frac{n_{water} \cdot \sin(\theta_2)}{c}$$

Trying to find an extrema:
$$0 = \frac{n_{air} \cdot \sin(\theta_1)}{c} - \frac{n_{water} \cdot \sin(\theta_2)}{c}$$

$$n_{air} \cdot \sin(\theta_1) = n_{water} \cdot \sin(\theta_2)$$

Voila!, using Fermat's principle, we derive the Snell's law, which describes the relationship between the incident angle of the ray and refraction.

If you're curious about the details on the least time principle, especially on its more modern/appropriate definition and how it actually works, Feynman, as usual, does a superb job.

Let's take Fermat's principle at face value. When I first learned about the least time principle, it bothered me a lot. The refraction of light explained with the least time principle is not causal but rather goal oriented—minimizing the travel time in this case. Instead of the destination resulting from the refraction of ray due to the medium speed difference, light somehow seems to already know the destination even before getting there and do so by pursuing with an intent to minimize the travel time.

Does this imply that nature has any intent?

In fact the very question was raised by Clerselier, Fermat's contemporary and a Cartesian:

"... in my view, much more scientific, to say as Mr. Descartes does, that the speed and direction of this body are altered by the alteration which takes place in the force and the disposition of this force, which are the true causes of its movement, and not to say as you do, that they change by an intention which nature possesses of always taking the path it can pursue most quickly, an intention which it cannot have, since it acts without knowledge, and thus has no effect on this body."

Though Clerselier's cogent arguments against Fermat's principle are quite convincing, there is a whole method called variational principle and many examples that follow the core nature of the least time principle—relying on extrema to define.

## Destination

I finally arrived, though it took ~3.5 times longer than my usual morning commute. Thinking about the traffic and subsequently the Fermat's principle made me reflect on goals and intentions.

Our modern society is obsessed with goals: the degree you've been working for, the car you want to buy, the job you strive to get, the race you desire to win, and the retirement you dream of. Indeed most people start off a new year by making a list of goals to reach. In grade school, we teach our kids how to make and achieve goals. We, more than often, focus on the destinations without carefully examining our intentions and true motivations.

Picture: Europe 1916, Boardman Robinson

People often compare a monumental challenge in life to climbing a tall mountain, like Mt. Everest, which has the highest peak in the planet. For the climbers who take this dangerous high risk challenge, the path matters as much as or perhaps even more than the destination, the goal. Simply flying there in a helicopter would strip away all the meaning and the overwhelming joy of finally conquering the peak—well unless you're the pilot landing it there.

Goals are very useful and important tools, but one should be able to see beyond that. The paths we take and the intents we possess matter.

Tonight, as I set myself a goal to get up early to beat the traffic, I simply wish that my intents will guide me to the destination—hopefully in the least-time way.