Instruction: What is the Tau Circle Constant?

Response: Tau is defined as the ratio of a circle's circumference to its radius.  Some 21st century mathematicians and physicists believe Tau is a better choice than Pi for use in science and mathematics.

Like Pi, Tau is an irrational and transcendental number.  In decimal form, the value of Tau is 6.283185…

Part of the motivation for Tau is the observation that the number 2*Pi or 2π occurs with surprising frequency throughout science and mathematics.  This raises the question of whether Pi is really an important number or something equal to two times Pi is actually more important.  Perhaps the implication is that Pi is exactly one half of some very important number!

Further consideration of the question leads to several compelling observations:

	- Using C/r as the key circle constant is better aligned with the way we define a circle: the set of all points a fixed distance (r) form a give point.
	- Radian angle measurements of fraction of a circle make much more sense when Tau is used as the base constant.  Using Pi, on quarter of a circle is Pi is Pi/2, one half is Pi, and the angle of a a full circle is 2*Pi.  With Tau, the angle measures are intuitively aligned to the fraction of a circle: a quarter circle is Tau/4, a half circle is Tau/2 and the angle of a full circle is Tau.  
	- Similarly, the radian zeroes, max, and min of the sin and cosine functions align to fractions of Tau in an intuitive fashion that aids, rather than confuses, understanding.
	- Key formulas that traditionally use Pi actually show more clear mathematical meaning when expressed using Tau.  For example, the area of a circle itself, when expressed with Tau, shows the same pattern of quadratic forms in distance fallen, spring energy, and kinetic energy.
	- Similarly, the tau-based formula for circular area actually confirms to the original proof by Archimedes showing that area of a circle equals the area of a triangle with base C (circumference) and height r (radius):  A = 1/2 b*h = 1/2 C*r = 1/2 (Tau*r)*r = 1/2 Tau*r^2.


Tau has seen significant adoption since it was originally proposed.  

The relationship between Tau and Pi is very simple.  Tau = 2*Pi.  Perhaps just as informative is the fact that Tau > Pi.