Today, June 28th, is Tau Day.  Someone restrain ‘Puter because we’re going to get into some math here and I’d like to keep my arms attached.

Pi (π) is a concept that everyone reading this (for the most part) should know well.  The transcendental (a number that is not algebraic – not the root of a non-constant polynomial equation with rational coefficients) number was termed in 1706 by William Jones as he described the ratio of the diameter of a circle to its circumference as 1 to 3.14159…and equated that number to π.  The definition (at least in Euclidean plane geometry) is:

 (see note)

The history of pi is well documented and discussed – Wikipedia has a decent summary.  However, even there we have evidence that pi might not be what we should be using.  Some of the earliest “evidence” of the use of pi are the Egyptian pyramids: the Pyramid of Meidum, the Great Pyramid at Giza and the pyramids at Abusir.  It was constructed with a perimeter of 1760 cubits and a height of 280 cubits.  1760/280 = 2π.  While there is no proof that this was calculated and measured to achieve the 2π value, it is clear that the results map to the value. However, why 2π?  Let me describe tau, . 

Some of you who remember your high school geometry and trigonometry classes may recall the following.  An angle in radians is defined as the ratio of the arc length of the circle inscribed by the angle to the radius of the circle.  This simplifies a number of trigonometric functions and leads us to a diagram of a circle as shown here:

Note carefully, that a half circle is π.  With the number of functions and equations in mathematics that utilize 2π, it begs the question why do we need to use a coefficient-loaded constant?  Enter tau.  Really, the angles above (and if you convert them into degrees, the same will hold true) are just fractions of a circle: 1/12, 1/8, 1/6, 1/4, 1/2, etc.

If we redefine the anglular measurement above and use a fraction of the full circumference of the circle, you get a resulting formula that is: angle = (fraction of circumference) * .  Now look at the resulting diagram:

I suggest going here and reading the premise in detail.  It makes a lot of sense to me.  I battled through numerous courses of calculus that provided the underpinnings that finally resulted in Fourier transforms.  Once I saw the transforms I (an engineer) slammed my fists and cried out how simple this was compared to the multiple integrals and series notations of the past.  So enjoy 6.28 – Tau Day.

(amended note: note that a circle is really defined by the radius.  If we take the diameter, d, from the equation above and replace it with 2r (two times the radius), and reduce it to C/r, we get the other side of the equation to be 2π which we can replace with .)