Tau (constant)

The Greek letter τ, $\mathbf {\tau }$ (tau) is a suggested symbol for the circle constant representing the ratio between circumference and radius. The constant is equal to $2\pi$ (2 times pi), and approximately $6.28$ .

Derivation

While there are infinitely many shapes with constant diameter, the circle is unique in having a constant radius. Therefore, rather than set the circle constant as

\pi ={\frac {C}{d}}

where $C$ represents the circumference and $d$ represents the diameter, it would arguably be more natural to use $r$ to represent the radius. This gives the formula

\tau ={\frac {C}{r}}.

This new circle constant, $\tau$ , may then be solved for in terms of $\pi$ . Since

r={\frac {d}{2}},

the $\tau$ formula may be rewritten as

\tau ={\frac {2C}{d}}.

Then, substituting the $\pi$ formula, the result is:

\tau =2\pi \approx 6.28318530717958647692528676655900576839433879875021\cdots

Applications

Using $\tau$ simplifies many common expressions involving $\pi$ , due to the factor of $2$ that often accompanies $\pi$ . An elementary example is the circumference formula,

C=2\pi r,

which may be rewritten in a more wieldy form as

C=\tau r.

$\tau$ makes it easier to express angles measured in radians. The unit circle is $2\pi$ radians in circumference, leading to confusing multiplications and divisions by $2$ through. If $\tau$ were used, values in radians would accurately express the fraction travelled around the circle. For example ${\frac {\tau }{4}}$ would be ${\frac {1}{4}}$ of the way around the circle. $\tau$ radians represents "one full turn" around a circle. On the same principle, sine, cosine, and many other trigonometric functions have a period of $\tau$ .

Though experts may be comfortable using equations in terms of $\pi$ , the above facts make $\tau$ the less-confusing choice for teaching geometry, as students will be more directly able to visualize and apply concepts using the unit circle without the potential for confusion by factors of $2$ .

$\tau$ also simplifies Euler's identity. Applying Euler's formula,

e^{i\vartheta }=\cos \vartheta +i\sin \vartheta

with the substitution of $\vartheta =\tau$ , results in

e^{i\tau }=1.

$2\pi$ also appears in Cauchy's integral formula, the Fourier transform, and sometimes in the Riemann zeta function, among other equations, making $\tau$ a potentially useful substitution for those situations.

Geometric significance

An advanced argument may be made that $\tau$ has special geometric significance in hyperspheres in arbitrary dimensions, whereas $\pi$ is only significant in two-dimensional circles:

\pi _{n}\equiv {\frac {S_{n}}{D^{n-1}}}={\frac {S_{n}}{(2r)^{n-1}}}={\frac {S_{n}}{2^{n-1}r^{n-1}}}={\frac {\tau _{n}}{2^{n-1}}}

and with $n=2$

\pi _{2}=\pi ={\frac {\tau _{2}}{2^{2-1}}}={\frac {\tau }{2}}

For higher dimensions,

n\neq 2^{n-1},

giving $\pi$ no geometrical significance.

Criticism

$\tau$ has been criticized for potentially causing ambiguity in expressions, due to sharing a symbol with proper time, shear stress, and torque.

It can be argued from a perspective outside of pure math that since the diameter of a circle is easier to measure,

{\frac {C}{d}}

should remain the circle constant. Due to the circular area formula being a quadratic form, rewriting it in terms of $\tau$ introduces a factor of ${\frac {1}{2}}$ , resulting in the equation