If you are reading this, you are probably a math geek who stands out in your class, takes advanced courses, and has memorized more than a few digits of π. But what if I told you that π is wrong? The concept of π is a horrible flaw in math that is taught in schools all over the world today, and here is why.
π is defined as the circumference of a circle divided by its diameter, approximately 3.14. It is used in many equations involving spheres, circles, cones, etc. However, why do we use the ratio of circumference to diameter? Where have you seen the diameter of a circle in all of those equations? The radius seems like the true defining feature of a circle, so why don't we use the circumference divided by the radius? That number would be equal to 2 times π, or 6.28. This number is called τ (tau), and there is currently a movement which aims to establish it as the circle constant.
First of all, the circumference of a circle is 2πr. If we use tau instead, it simplifies to a neat τr. You might say, "But what about the area of a circle, πr²? Isn't that so much more elegant than ½τr²?" In a way, yes. However, it hides the way that you actually get the area of a circle in the first place. Here's how, as explained by Minutephysics.
π is defined as the circumference of a circle divided by its diameter, approximately 3.14. It is used in many equations involving spheres, circles, cones, etc. However, why do we use the ratio of circumference to diameter? Where have you seen the diameter of a circle in all of those equations? The radius seems like the true defining feature of a circle, so why don't we use the circumference divided by the radius? That number would be equal to 2 times π, or 6.28. This number is called τ (tau), and there is currently a movement which aims to establish it as the circle constant.
First of all, the circumference of a circle is 2πr. If we use tau instead, it simplifies to a neat τr. You might say, "But what about the area of a circle, πr²? Isn't that so much more elegant than ½τr²?" In a way, yes. However, it hides the way that you actually get the area of a circle in the first place. Here's how, as explained by Minutephysics.
Notice how the ½ cancels out with the 2 in 2πr, hiding the fact that the area of a circle is in fact the area of a triangle with the circumference as its base and the radius as its height. If you use τ instead, it actually shows this method of deriving the area of a circle. Equivalently, if you're familiar with calculus, the area of a circle is the integral of the circumference with respect to the radius:
∫τrdr = ½τr²
which hearkens back to the formula so many calculus students have ingrained in their heads:
∫xdx = ½x²
For another thing, τ makes radians much easier. Instead of a circle measuring 2π radians around, it's just τ radians. Half of a circle is ½τ. A quarter of a circle is ¼τ, rather than ½π. With τ, you don't need to do that x2 conversion that you need with π. This makes trigonometric functions easier, too: sin(θ) is the y-coordinate of a point on the unit circle θ radians around, and cos(θ) is the x-coordinate. Using the simpler, more elegant τ-based radians, trig functions are easier to visualize in your head - no memorization required!
Your next defense of π may be, "What about Euler's identity, e^iπ+1=0? Surely e^½iτ+1=0 is uglier!" I agree, but there is a way to rewrite Euler's identity with τ so that it's even more elegant than the original. The formula for Euler's identity comes from the general case,
e^iθ = cos(θ) + i*sin(θ),
where θ is the angle around a unit circle on the complex plane. Substituting π into this yields e^iπ = -1, and the -1 is moved to the other side so that we get something a little more pleasant. It basically means that if you go halfway around a unit circle (π radians), then add 1, you're back at the center. But shouldn't it be something a little more... wholesome? Maybe going all the way around the circle? If you put τ into this formula, you get
e^iτ = 1
Truly, a triumph for the better circle constant!
If you're not convinced to use τ, that's fine. If you are, spread the word! Tell other people about it! Help us in this movement to make mathematics even more beautiful and elegant, because really, who doesn't like a little pumpkin τ during the holidays?
For more information, see the following links:
The Tau Manifesto, by Michael Hartl
The Wikipedia page on τ
Pi is (still) Wrong, also by Vihart
The Tau Manifesto, by Michael Hartl
The Wikipedia page on τ
Pi is (still) Wrong, also by Vihart