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The Greek letter τ (tau) is a suggested symbol for the circle constant representing the ratio between circumference and radius. The constant is equal to 2π (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, τ, may then be solved for in terms of π. Since

$r=\frac{d}{2}$ , the τ formula may be rewritten as

$\tau=\frac{2C}{d}$ . Then, substituting the π formula, the result is:

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

## Applications

Using τ simplifies many common expressions involving π, due to the factor of 2 that often accompanies π. An elementary example is the circumference formula,

$C=2\pi r$ , which may be rewritten in a more wieldy form as

$C=\tau r$ . τ makes it easier to express angles measured in radians. The unit circle is 2π radians in circumference, leading to confusing multiplications and divisions by 2 through. If τ were used, values in radians would accurately express the fraction traveled around the circle. For example,

$\frac{\tau}{4}$ would be

$\frac{1}{4}$ of the way around the circle. τ radians represents "one turn" around it. On the same principle, sine, cosine, and many other trigonometric functions have a period of τ.

Though experts may be comfortable using equations in terms of π, the above facts make τ 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.

τ also simplifies Euler's identity. Solving Euler's formula,

$e^{i\theta}=\cos\theta+i\sin\theta$ , with the substitution of τ for θ (theta), results in

$e^{i\tau}=1$ .

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

## Geometric significance

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

$\pi_2=\pi=\frac{\tau_2}{2^{2-1}}=\frac{\tau}{2}$, where$\pi_n\equiv\frac{S_n}{D^{n-1}}=\frac{S_n}{(2r)^{n-1}}=\frac{S_n}{2^{n-1}r^{n-1}}=\frac{\tau_n}{2^{n-1}}$ and with$n = 2$

For higher dimensions,

$n\neq2^{n-1}$ , giving π no geometrical significance.

## Criticism

τ 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 τ 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 π. There are other such formulas that are simpler using π than τ.

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