π in Different Universes | Azeem Bande-Ali

Everybody loves $pi$. It’s often the primary irrational quantity somebody encounters. $pi$ is conceptually easy sufficient that it may be defined with primary geometry.

$pi$ is the ratio between the circumference and the diameter. Normally, written as:

$$ C = 2pi r $$

the place $C$ is the circumference, $r$ is the radius (half of the diameter) and $pi$ is famously $3.14159..$.

However why does $pi$ should have that worth? May it have another worth? The reply is sure! However to determine that out, we first have to speak about circles that are carefully tied to the definition of $pi$.

Circle

Because the definition of $pi$ will depend on two properties of a circle (circumference, radius), it’s good to determine what a circle even is. Mathematically, a circle is the gathering of all factors which might be an equal distance from the middle. So if the radius of a circle is 1, then the circle is the gathering of all of the factors which might be 1 unit distance away from the middle.

In sensible phrases, a circle tells you all of the factors which have an equal “price”. For instance:

• Should you begin working from the middle, then the circle represents all of the farthest factors you may attain in a given period of time. Right here the space is measured in models of time.
• Should you begin driving from the middle, then the circle represents all of the farthest factors you may attain in a given quantity of gas. Right here the space is measured in models of gas.

However not all constant-cost capabilities will create the identical form. For instance, suppose you’re crusing on a windy day. Touring within the route of the wind will probably be straightforward however touring orthogonal to the wind would require extra effort and touring towards the wind would require important effort. So for mounted effort, the farthest factors you may journey will create an ellipse shifted towards the route of the wind.

However does this cost-function (effort wanted to sail) outline a correct distance? Can we use it to measure radius and measure circumference? Appears kinda arbitrary to say sure or no. If time and gas can all be “distances” in some conditions then why couldn’t effort be a distance on this scenario? Happily, we don’t should make an arbitrary resolution right here since we will depend on a preexisting idea that defines what sorts of price capabilities are legitimate distances.

Metrics

Arithmetic could be seen as a logic sport. You begin with a set of assumptions and also you provide you with all of the logical conclusions you may from that. Then, if another person finds a scenario that matches these assumptions, they will profit from the pre-discovered logical conclusions. Which means if some conclusions require fewer assumptions, then these conclusions are extra typically relevant.

Consequently, arithmetic goes via steady cycles the place mathematicians return and trim down the assumptions wanted for any mathematical system. For instance, a whole lot of geometry from the instances of the Greeks used solely the one definition of distance: the Euclidean distance ($d = sqrt{x^2 + y^2}$). We even named it after the Greek mathematician. Even Newton relied solely on Euclidean distance when he invented Calculus. Then, within the early twentieth century, mathematicians realized that any operate can be utilized as a distance operate, so long as it adopted some primary necessities. So long as the space operate adopted these necessities, a whole lot of the established math would nonetheless work. So you could possibly nonetheless do geometry and calculus and topology with simply minor tweaks. Capabilities that match these necessities are referred to as metrics.

A operate is a metric if it follows the next guidelines:

1. The space between some extent and itself is all the time 0. Should you don’t go anyplace, the space traveled is 0.
2. The space between any two completely different factors is all the time optimistic
3. The space from $a$ to $b$ is similar as the space from $b$ to $a$
4. Going immediately from $a$ to $c$ is a minimum of as quick as going from $a$ to $b$ after which from $b$ to $c$

Now, with these necessities, does “effort to sail” outline a metric? The reply isn’t any. We are able to most likely debate 1 and a pair of however 3 could be very clearly not true. If $b$ is downwind of $a$, then the hassle in a single route is smaller than the hassle in the wrong way.

Okay, so what capabilities are a metric? There are two basic examples: Manhattan distance and most distance.

Manhattan Distance

If you find yourself driving in a metropolis grid, you may’t drive diagonally. You need to drive in a single grid route after which the opposite. Whenever you drive like this, the space between two factors is simply $d = x + y$. That is popularly referred to as the Manhattan distance or the taxicab distance.

One utility of this metric is measuring accuracy. Suppose you’re requested to foretell the full inhabitants change of two cities. Should you guess inside 1,000 individuals, then you definately win a prize. You may visualize this by charting the inhabitants change of 1 metropolis on the x-axis and the inhabitants change of the opposite metropolis on the y-axis. Then a “circle” round your guess with a radius of 1,000 tells you the vary the place you may nonetheless win the prize.

The circle appears to be like like a rotated sq.!

The circle has radius of 1,000 however what’s its circumference? If we use the Manhattan distance to measure the circumference then every line has a distance of two,000 ($x + y = 1000 + 1000$) and since there are 4 traces, the circumference is 8,000. This implies:

$$ C = 2 pi r $$
$$ 8,000 = 2 pi (1,000)$$
$$ 4 = pi $$

Within the universe the place you measure distance utilizing the Manhattan distance, the worth of $pi$ is 4!

Maximal Distance

One other distance operate that could be a legitimate metric is the maximal distance: $d = max(x,y)$. So as a substitute of mixing x and y, we use the bigger of the 2 as the space.

Plenty of instances when you find yourself doing a number of issues on the similar time, it solely issues how lengthy the longest merchandise takes. For instance, suppose you want to put together two elements for a dish, and you’ll put together the elements in parallel. The period of time you want to end your dish is so long as the slowest ingredient.

Suppose for a cooking competitors, you’re required to complete cooking in precisely 60 minutes plus/minus 5 minutes. Then how a lot time can every ingredient take? If we use the x-axis to signify the time ingredient 1 takes and the y-axis to signify the time ingredient 2 takes, then we will draw a 5 minutes-wide circle that tells us how lengthy every ingredient can take.

This circle appears to be like like an everyday sq.! The circle has radius 5 however what’s its circumference? Every line has a distance of 10 and there are 4 traces so the circumference is 40.

$$ C = 2 pi r $$
$$ 40 = 2 pi 5 $$
$$ 4 = pi $$

And once more, within the universe the place distance is measured utilizing the maximal distance, the worth of $pi$ is 4!

p-norm

Until now now we have lined three distances: Euclidean, Manhattan, and Maximal. What different examples can we have a look at? Nicely, now we have the p-norm metric which is a group of infinite metrics outlined as:

$$ d = left(x^p + y^pright)^{1/p}$$

the place $p$ could be any quantity higher than or equal to 1.

The cool factor about p-norms are that they’re a generalization of the metrics we lined earlier than.
Euclidean, Manhattan, and maximal distances are particular examples of p-norms.
When $p$ is 1, now we have the Manhattan distance. When $p$ is 2, now we have the Euclidean distance. When $p$ is $infty$, now we have the maximal distance. So Manhattan distance and maximal distance are the extremes of the p-norms.

Listed here are what circles seem like in numerous p-norms:

Simply as we did earlier than, the circumference and the worth of $pi$ could be calculated for the completely different p-norms. In earlier examples, we have been capable of calculate the circumference simply by it, since they have been straight traces however that technique doesn’t work typically. So, for the remainder of the shapes we have to caculate the circumferences computationally. We are able to write a program to inform a pc to stroll across the circle and monitor the space traveled. Happily, another person has already finished this work so we will confer with their outcomes^.

Along with calculating these values, the paper linked above additionally proves that 3.14159 is the smallest worth of $pi$ attainable for all of the p-norms. Our common $pi$ is the smallest attainable $pi$ within the household of p-norms!

All Metrics

Whereas there are infinitely many p-norm metrics, there are infinitely extra metrics that aren’t p-norms. What are the values of $pi$ in all the opposite metrics?

This article by Sahoo proves that $pi$ is between 3 and 4 for all metrics. We’ve got seen the metrics that give us $pi = 4$. What’s the metric that provides us $pi = 3$?

The answer is a little bit gnarly

$$ d = frac{1}{2sqrt{3}}sum_{n=1}^6left|xsinleft(frac{pi n}{3}proper)+ycosleft(frac{pi n}{3}proper)proper| $$

However drawing out the circle for this metric provides us a hexagon.

Through the use of the space equation outlined above, we will calculate the size of every line of the hexagon^{. This provides us a size of 1 for every line of the hexagon which implies that the circumference is 6.}

$$ 6 = C = 2 pi r = 2 pi (1) $$
$$ 3 = pi $$

So subsequent March, as a substitute of simply celebrating $pi$-day on March 14th (3/14), be at liberty to rejoice $pi$-month all via March. You simply should do the work of discovering the suitable metric for every day of the month.