Tau (2π): Difference between revisions

In mathematics, tau (τ) is a constant which has been proposed by Bob Palais, Peter Harremoes, Hermann Laurent, Fred Hoyle, Michael Hartl, and others,^[1][2][3][4] as a replacement for the familiar circle-constant π. Their main reason is that circles are more naturally defined by their radius than by their diameter.^[5] The value ${\displaystyle \tau =2\pi }$, or approximately 6.28,^[6] occurs very frequently in mathematics.

Many symbols have been suggested, including ${\displaystyle 2\pi }$ (Laurent), ${\displaystyle \pi \!\!\!\!\;\pi }$ (Palais), ${\displaystyle \varpi }$ (Harremoes), and ${\displaystyle \tau }$ (Hartl). The symbol τ was chosen as to stand for turn or τόρνος (tornos), since τ radians are equivalent to one full turn.

Possible advantages

Palais and Hartl claim a number of advantages of using τ instead of π.

• The so called "special angles", that need to be memorized when using π, simply become fractions of a whole circle, that is ${\displaystyle {\scriptstyle {\frac {1}{2}}}\tau }$, ${\displaystyle {\scriptstyle {\frac {1}{3}}}\tau }$, ${\displaystyle {\scriptstyle {\frac {1}{4}}}\tau }$, ${\displaystyle {\scriptstyle {\frac {1}{6}}}\tau }$ and ${\displaystyle {\scriptstyle {\frac {1}{12}}}\tau }$. It is easier to explain that one eighth of a pizza corresponds to ${\displaystyle {\scriptstyle {\frac {1}{8}}}\tau }$ than ${\displaystyle {\scriptstyle {\frac {1}{4}}}\pi }$.^[7] Hartl describes the use of pi in this context as a "pedagogical disaster."
• In many formulae, such as normal distribution and Fourier transforms, the factor 2π can be eliminated, thus simplifying them.^[4]
• The periodicity of the cosine and sine functions is τ instead of 2π, which is simpler and arguably more intuitive.^[4]
• The formula for the circumference of a circle becomes simply τr, without introducing a factor 2.
• The formula for the area of a circle falls in line with the power rule for integrals (e.g. kinetic energy ${\displaystyle K={\scriptstyle {\frac {1}{2}}}mv^{2}}$). Instead of ${\displaystyle A=\pi r^{2}}$, it becomes ${\displaystyle A={\scriptstyle {\frac {1}{2}}}\tau r^{2}}$.^[4]

Possible disadvantages

• The area of a circle is expressed as ${\displaystyle {\scriptstyle {\frac {1}{2}}}\tau r^{2}}$ rather than ${\displaystyle \pi r^{2}}$.
• Tau has many other unrelated mathematical meanings.
• Using τ brings simplification to only a handful of formulae, while bring no improvement to the rest. For example, see the formulae for volumes and surface areas of hyperspheres. The time, effort, money and confusion involved in changing centuries of mathematical convention and education far outweighs the benefits gained by a handful of simplified formulae.

Trivia

• Paul Laurent in Traité D'Algebra wrote equations using 2π as a single symbol.^[1]
• The famous Feynman point (six consecutive 9s early in the decimal digits of π) appears one digit earlier in τ, and is seven digits long instead of six (3.14...349999998... * 2 = 6.28...699999996...).^[8]

See also

• 2π theorem

References

Further reading

• Hartl, Michael (June 28, 2010). "The Tau Manifesto". Tau Day. Retrieved 2011-07-03.
• "The Pi Manifesto". July 4, 2011. Retrieved 2011-07-07.^{[dead link]}
• "On National Tau Day, Pi Under Attack". Fox News Channel. NewsCore. June 28, 2011. Retrieved 2011-07-03.
• Springmann, Alessondra (June 28, 2011). "Tau Day: An Even More Fundamental Holiday Than Pi Day". PC World. Retrieved 2011-07-03.
• Palmer, Jason (28 June 2011). "'Tau day' marked by opponents of maths constant pi". BBC News. Retrieved 2011-07-03.

External links

• Vi Hart (Author). Pi Is (still) Wrong (flv). YouTube.