Different ways to define distance and the definition of $\pi$

I saw a post a couple of posts about "proving that $$\pi$$ = $$4$$", and I thought of using a naive way to say that it is not impossible.

We know that $$\pi$$ is commonly defined like this:

$$\pi = \frac{\text{Circumference}}{\text{Diameter}}.$$

How will you measure the circumference? That is equivalent to saying that "What metric I will choose to measure the length of this curved line?" The important point is that the metric we choose will affect what value of $$\pi$$ we will get.For example, we will get $$\pi = 4$$ if we choose the taxicab metric $$(|\Delta x|+|\Delta y|)$$ and we will get $$\pi = 3.1415...$$ if we choose the Euclidean metric $$(\sqrt{\Delta x^2 + \Delta y^2})$$.

In short, we will get different answers based on the underlying assumptions we are taking beforehand. Is my approach wrong?

I'd say that in the end it simply depends on how you define $$\pi$$. Most people (especially non-mathematicians) know the definition you mention, i.e. $$\pi$$ is the ratio between the circumference and the diameter of a unit circle. Implicitly we use the euclidean metric to define what a unit circle is (because who wouldn't think of the euclidean metric in everyday life first, right?).

Now I think what you are doing is exactly the 'easiest' way to justify an assessment like $$\pi = 4$$. But we'll have to deviate from the 'everyday life definition' above in the following way. Let's take a different metric, e.g. $$d(x,y) = \max(|x|,|y|)$$. What is the unit circle $$S$$ in this metric? It's

$$S = \{(x,y)\in \mathbb{R}^2 | d(x,y) = \max(|x|,|y|) = 1\}$$

You can easily draw this circle, and you will see that it turns out to be the square of length 2 centered around the origin, which has of course circumference 8. If we now apply our definition of $$\pi$$ being equal to the circumference divided by the diameter, we get $$\pi = \frac{8}{2} = 4$$. (Small remark: you get a different result when using the taxicab metric as you proposed. You can also draw the unit circle with respect to the taxicab metric to figure it out!).

So as often: It depends on your definition.

I hope this helps!

• I guess the definition of pi we choose beforehand is also a assumption. – Prithu biswas Feb 2 at 8:17
• Sure, I mean in principle $\pi$ is just a greek letter. A lot of people think of $\pi$ as the mentioned ratio just because it has been defined that way a long time ago, and has been used consistently enough that it stuck. So if you don't say anything, it's ok to assume that by $\pi$ we mean this ratio. But $\pi$ (the letter) can also be used for plenty of other definitions. – noam.szyfer Feb 2 at 8:55