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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 $

with respect to the radius.

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