Overview
The quantity represented by the Greek letter tau (6.283185…) is the proportion of the circumference of a circle to its radius. The use of tau simplifies the relationships between angles and circles. These relationships underlie other branches of mathematics and applications to science.
Special Angles and Tau
The relationship of tau and the circle is such that tau is one turn of the circle. In fact, the Greek letter tau is the first letter in the word tornos, which the English word “turn” is from. Recall the special angles of 30o, 45o, 60o, 90o, 135o, 180o, and 270o with relationship to the circle. Those special angles can be represented in radians using tau τ, in such a way that their measure is expressed as a fraction of the entire circle.
Tau and the Unit Circle
The unit circle has a radius as 1, and the most fundamental relationships in trigonometry set sin θ and cos θ as points on the unit circle. Therefore, if the sine wave starts at zero, the value in Cartesian coordinates is (1, 0). At the highest point of the wave, sin θ has coordinates of (0, 1). The sine wave is again at equilibrium at coordinates (0, 0) and at its lowest point at coordinates (-1, 0). The cosine wave is the opposite of the sine wave, but it also has a period equal to the circle constant.
The Tau Manifesto
Mathematician and physicist Michael Hartl is one of the leaders of the Tau Day movement, to show the advantages of a widespread adoption of tau as the circle constant. The constant τ is related to the circle in a more intimate way than π, and it is more natural to use, in special angles, the unit circle, and other mathematical expressions.
Tau Is Easier than Pi
The tau constant simplifies special angles, sines, and cosines. Tau is exactly equal to 2π, so it can be used in any equation or mathematical expression that uses 2π. For example, the normal distribution has a factor of 2π in its definition. Some advanced mathematics equations that refer to space, time, and polar coordinates as integrals go from a limit of 0 to a limit of 2π, or τ. Even Euler’s formula, eiθ =cos θ + i sin θ, becomes Euler’s identity when the value of θ is set at τ. Then, eiτ =1. Without going into the very complex mathematical argument, the small letter e stands for the natural logarithm and the small i for the imaginary factor in complex numbers. The number 1 is unity, and τ brings it all together.
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