# nLab circle constant

Contents

### Context

Ingredients

Concepts

Constructions

Examples

Theorems

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

tangent cohesion

differential cohesion

singular cohesion

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

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## Idea

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

## History

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

However, in 2010, Michael Hartl proposed that the constant 6.283… which equals $2 \pi$ is much better for mathematics than $\pi$, 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 $\tau$ for $2 \pi$. Since then, a few mathematicians such as Peter Harremoës have used $\tau$ instead of $\pi$ for research mathematics, and $\tau$ has been incorporated into a number of programming languages such as Python and Rust.

## References

• 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.