Happy Pi Day (Only a Little Bit of Pi vs Tau Talk...)

2023-03-14 - [51] 12:29

Happy Pi Day to everyone who appreciates date formats where the day number comes after the month number!

Here is a photo of my Pi tattoo, so I feel like I'm obligated to mention Pi Day.

A photo of my Pi tattoo on my left hand/wrist, with the symbol for pi on my left hand just above my left wrist and the value 3.141592653589793 (15 places after the decimal point) wrapping around my left wrist

What is Pi?

It is likely that you already know what Pi is, but in short, it is the ratio of the length of the circumference of a circle (the circumference is the perimeter of a circle) compared to the length of the diameter of that same circle (the diameter is the straight line that starts and ends on the circumference of a circle and goes through the center of the circle). That ratio is about 3.141592653589793... to 1 and is one of the big constants of Mathematics. It is also sometimes referred to as the "circle constant".

Below is an image of a circle with its circumference and diameter.

A circle with the diameter line going through the circle

What makes today Pi Day?

Pi rounds to 3.14 and March 14th is the 3rd month of the year and 14th day of the month, so the date is 3/14 (in places where the day is written after the month). Today is also Albert Einstein's birthday, which makes the validity of today being Pi Day even greater.

But what about Tau? (Yes, I know I'm ANOTHER "Vi" with an opinion on Pi vs Tau [I'm on team Pi by the way])

In case you are not familiar with Tau, it is Pi multiplied by 2. This is because there are 2 Pi radians in a circle. More information about Tau can be found at the following links.

[HTTPS] Tau Manifesto page on the Tau Day website titled "No, really, pi is wrong: The Tau Manifesto"

[HTTPS] Vi Hart's YouTube video "Pi Is (still) Wrong."

Tau can also be described as the ratio of the circumference of a circle compared to its radian (a straight line from the center of a circle to the circumference of the circle). So the big difference between Pi and Tau is that Pi uses the circle's diameter and Tau uses the circle's radius, which is half the length of the diameter.

The big useful feature of Tau is that the angle of a full circle (360 degrees) is equal to 2 times Pi radians. If Pi is the "circle constant", then why is a circle 2 times Pi? Wouldn't it be easier, especially in terms of education, if we say the circle constant is equal to 1 circle instead of half a circle?

Why I prefer Pi as opposed to Tau

Before mentioning my reasons, I just want to mention that there seems to be a response to the Tau Manifesto that is on "team Pi", which I will link below. I have not read through much of it, but I am including it here for the sake of "completeness".

[HTTP] The Proper Pi Manifesto

If we want to use Pi, we need the diameter of a circle, and if we want to use Tau, we need the radius. It turns out it's not easy to determine the radius of an completely arbitrary circle of unknown circumference before determining the diameter (or 2 radiuses) of the circle. There are a few ways to find the radius (and center) of a circle though.

My personal favorite way is to draw 2 different cords (a line that goes through a circle and intersects at 2 different points of the circumference of that circle) through a circle , then draw the perpendicular bisector (a perpendicular line that goes through a cord where each side of the split cord inside of the circle is the same length) through both cords. The point where the 2 perpendicular bisectors meet is the center of the circle. This can be seen in the image below.

A circle with 2 different cords that are perpendicular bisected. The 2 perpendicular bisectors cross at a single point, which is the center of the circle.

The thing about this way of getting the center is that you need to get 2 radiuses of the circle (which is equal to the diameter of the circle) by the point where you are able to know what the radius of the circle is.

The next way I know to find the center of a circle is a little bit more intuitive to me, even if I don't like it as much. First, create a rectangle where all 4 corners touch the circumference of the circle. Next, draw a straight line through the opposite corners of the rectangle, creating 2 straight lines that intersect each other. The point where the straight lines intersect is the center of the circle. This can be seen in the image below.

A circle with a rectangle inside of it that has all 4 corners touching perfectly inside of the circle. Lines connect opposite corners of the rectangle and the crossing point of the lines connecting the rectangle corners is the center of the circle.

This method figures out the diameter of the circle before figuring out the radius of the circle.

The last method I know is impressive to me and actually determines the radius before finding the center, even if it's a bit more complicated than the other methods. For this explanation, I'm going to call the initial circle we're trying to find the center of the "large circle". First, draw a small circle, with the center of that small circle on the circumference of the large circle. Next, draw another small circle that is the same size as the first small circle, with the center of the second small circle on one of the points where the circumference of the first small circle and the circumference of the large circle intersect. Now draw straight line that goes through the 2 points where the circumferences of the small circles intersect and continue that line through the circumference of the large circle. After that, draw a straight line that goes through the point where the circumference of the first small circle intersects the circumference of the large circle that is not inside of the second small circle and goes through where the circumferences of the 2 small circles intersect that is inside of the large circle and continue that straight line through the circumference of the large circle. At this point, we actually have the radius of the circle before we have the center of the circle or the diameter. The radius is the segment of that straight line that was just drawn starting from the intersection point of the circumferences of the 2 small circles that is inside the large circle and ending at the circumference of the large circle that's not intersecting with the circumference of the first small circle. Draw an arc whose radius is the radius that was just figured out and use the intersection point of the second straight line and the circumference of the large circle that does not intersect with the circumference of the first small circle as the center point for making the arc. The point on the arc that intersects the first straight line but does not intersect the circumferences of both small circles is the center of the large circle. This method can seen in the image below.

This method doesn't really bode well on the "Pi is better" side of things, but that's only 1 out of the 3 methods I showed. I figured I'd show this method for the sake of completeness and because I find this method interesting.

Even with those methods being shown, why should Pi or Tau be the "circle constant" anyways? There are plenty of other useful values that could be the "circle constant" instead.

Pi/2 is also incredibly useful in mathematics, including trig and geometry, as it represents a 90 degree angle. The massive utility of that could warrant it being the constant compared to Pi or Tau.

Going even further, 45 degree angles and 30 degree angles are incredibly useful in terms of geometry, so we could use Pi/4 and Tau/3 as better constants than either Pi or Tau. No more of that "57.29577951... degrees is a useful measurement on a circle" crap.

We could do this all day.

Is Pi Wrong?

Is Tau Wrong?

No. Is it needed, considering we also have Pi? Also no. I do think it's pretty fun to be "against the masses" in terms of something like this for Mathematics though, so there's that going for Tau.

Should we ignore Tau Day?

If you like pie, then I recommend against ignoring either Pi Day or Tau Day. No matter how much pie you eat on Pi Day, you have an excuse to eat twice that amount of pie on Tau Day, which allows us to have 3 times the amount of pie in total per year!

Ending thoughts

Celebrate Pi Day. Celebrate Tau Day. Eat many pies. I personally don't know why we should fight for changing the "circle constant" from Pi to Tau when Pi is perfectly reasonable already, but neither of those constants are "wrong". That is my Pi vs Tau rant and I will probaby just link to this blog post when someone annoys me enough about Pi vs Tau.