Everyone loves $pi$. It’s usually the first irrational number someone encounters. $pi$ is conceptually simple enough that it can be explained with basic geometry.
A circle with diameter 1 has a circumference of $pi$. The circumference can be measured by slicing the circle and unwrapping it. The more slices that are added by moving the slider, the more accurate the measurement becomes. (Adapted from here)
$pi$ is the ratio between the circumference and the diameter. Usually, written as:
$$ C = 2pi r $$
where $C$ is the circumference, $r$ is the radius (half of the diameter) and $pi$ is famously $3.14159..$.
But why does $pi$ have to have that value? Could it have some other value? The answer is yes! But to figure that out, we first have to talk about circles which are closely tied to the definition of $pi$.
Circle
Since the definition of $pi$ depends on two properties of a circle (circumference, radius), it’s good to figure out what a circle even is. Mathematically, a circle is the collection of all points that are an equal distance from the center. So if the radius of a circle is 1, then the circle is the collection of all the points that are 1 unit distance away from the center.
In practical terms, a circle tells you all the points that have an equal “cost”. For example:
- If you start running from the center, then the circle represents all the farthest points you can reach in a given amount of time. Here the distance is measured in units of time.
- If you start driving from the center, then the circle represents all the farthest points you can reach in a given amount of fuel. Here the distance is measured in units of fuel.
But not all constant-cost functions will create the same shape. For example, suppose you are sailing on a windy day. Traveling in the direction of the wind will be easy but traveling orthogonal to the wind will require more effort and traveling against the wind will require significant effort. So for fixed effort, the farthest points you can travel will create an ellipse shifted against the direction of the wind.
If there is strong wind to the right, you can sail much farther to the right than to the left with the same effort. The red ellipse shows the points you can reach with equal effort. The black circle represents the farthest you can travel with equal effort when traveling on a day without any wind or current.
[graph]
But does this cost-function (effort needed to sail) define a proper distance? Can we use it to measure radius and measure circumference? Seems kinda arbitrary to say yes or no. If time and fuel can all be “distances” in some situations then why couldn’t effort be a distance in this situation? Fortunately, we don’t have to make an arbitrary decision here since we can rely on a preexisting concept that defines what kinds of cost functions are valid distances.
Metrics
Mathematics can be seen as a logic game. You start with a set of assumptions and you come up with all the logical conclusions you can from that. Then, if someone else finds a situation that fits those assumptions, they can benefit from the pre-discovered logical conclusions. This means that if some conclusions require fewer assumptions, then those conclusions are more generally applicable.
As a result, mathematics goes through continuous cycles where mathematicians go back and trim down the assumptions needed for any mathematical system. For example, a lot of geometry from the times of the Greeks used only the single definition o
