nLab circle constant




Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)



There are two circle constants used in geometry, the more common circle constant π\pi representing the straight angle, and the less common circle constant τ=2π\tau = 2 \pi representing a full turn.


Leonard Euler original definition of the circle constant π\pi was the constant valued as 6.283…, which is equivalent in today’s notion as τ\tau. Euler’s definition was often used in competition with the definition 3.1415.. until the late 1700s, when mathematicians collectively decided on standardizing π\pi as 3.1415…

However, in 2010, Michael Hartl proposed that the constant 6.283… which equals 2π2 \pi is much better for mathematics than π\pi, as it would get rid of a significant number of factors of 2 from many equations, and is intuitively better for teaching and understanding subjects such as trigonometry and geometry, and proposed using the symbol τ\tau for 2π2 \pi. Since then, a few mathematicians such as Peter Harremoës have used τ\tau instead of π\pi for research mathematics, and τ\tau has been incorporated into a number of programming languages such as Python and Rust.

 See also


  • Euler, Leonard (1746). Nova theoria lucis et colorum. Opuscula varii argumenti, p.169-244.

  • Segner, Johann Andreas von (1761). Cursus Mathematicus: Elementorum Analyseos Infinitorum Elementorum Analyseos Infinitorvm (in Latin). Renger. p. 374. “Si autem π notet peripheriam circuli, cuius diameter eſt 2”

  • Hartl, Michael (2010). The Tau Manifesto

  • Tran, Andy (2017). Tau: The True Circle Constant. (slides)

  • Harremoës, Peter (2017). “Bounds on tail probabilities for negative binomial distributions”. Kybernetika. 52 (6): 943–966. arXiv:1601.05179. doi:10.14736/kyb-2016-6-0943. S2CID 119126029.

Last revised on May 16, 2022 at 21:58:07. See the history of this page for a list of all contributions to it.