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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+isin\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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