Happy Tau Day! (6.28)

June 28^th or 6.28 also known as Tau day is the lesser known brother to Pi Day or March 14^th (3.14).

The number tau (τ) is equal to 2π and is approximately 6.283185307179586. There is a (very) quiet movement afoot to get π replaced by this new constant τ and for good reasons which this post will get into.

_{Image credit: Michael Hartl link Public domain image.}

Tau Is Simply A More Natural Way To Describe Circles

How many degrees are in a circle? The answer is 360° are in a circle. Why is that? Few people really care including me. It might have something to do with the Babylonians but I can't be arsed to check into that because the number 360 is awkward and unnatural.

Okay, let's describe a circle more naturally.

The number π is defined as the circumference 'C' of a circle divided by its diameter 'd' as in π = C/d. In terms of the radius of the circle it is defined as π = C/2r.

If you start at a point on the circumference of a large circle and walk the distance of one radius around that circle the angle you end up at is defined as a radian.

If you walk all the way around that circle it turns out that the angle subtended will be 2π radians. Huh? Why two? Why did mathematicians not define it from the start as 1 π radians to go around a circle?

This is where tau (τ) comes in.

One full turn of a circle is 1 tau radians instead of 2π radians. A half turn of a circle is 0.5 tau radians instead of π radians . A quarter turn of a circle is 0.25 tau radians instead of 0.5 π radians.

1 tau is 1 turn. With τ it all comes out more naturally and intuitively.

Sine Waves Are Easier To Understand

The sine wave simply describes the height on a circle as something rotates around it.

If you are using π in your math, then π/2 radians (a 1/4 turn) around a circle brings you to full amplitude. If you were using τ then it would be τ/4 radians around a circle (a 1/4 turn) brings you to full amplitude.

If you are using π in your math then π radians (a 1/2 turn) around a circle brings you back to zero amplitude. If you were using τ then it would be τ/2 radians around a circle (a 1/2 turn) brings you to full amplitude.

Tau is simply much more natural way to describe circles in mathematics.

Sine waves a describe the motion of simple harmonic oscillators very well. Imagine a weight bobbing up and down on a spring, the position of that mass will follow the sine wave as a function of time.

Oscillators like this can be found all throughout physics, for example two atoms bound together in a molecule. Their vibratory motion can be approximated fairly well in this way and so the number 2π will pop up throughout physics and all of the other STEM fields over and over again. In fact physics textbooks are crammed full of 2π.

Closing Words

Will we ever replace π with τ?

No. Pi has been in use for far too long and is in too many textbooks for it to ever be replaced.

The best that can be done, and should be done, is to at least mention tau when pi is introduced to students. At this early point in their math education they will at least know that π is not the perfect natural descriptor for circles and that there is at least something else out there that is a better fit.

Happy tau day to you all and thank you for reading my post.

Post Sources

[1] Pi Is Wrong - Bob Palais
[2] Tau Day - Michael Hartl
[3] The Tau Manifesto - Michael Hartl
[4] Pi Is (still) Wrong - ViHart Video
[5] Tau replaces Pi - Numberphile Video