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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:

$\tau=2\pi\approx6.28318530717958647692528676655900576839433879875021\cdots$

## 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,

$n\neq2^{n-1},$

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

$A=\frac{\tau r^2}{2}$

which is less elegant than the one involving $\pi$, which is

$A=\pi r^2.$

There are other such formulas that are simpler using $\pi$ than $\tau$. However,

$A=\frac{\tau r^2}{2}$

more easily represents how the area is the integral of the circumference

$C=\tau r$