Happy Tau Day 2025

TL;DR of "The Tau Manifesto."

The original text defines $\tau =2\pi =6.283\cdots$ to replace the current circle constant $\pi$ and lists several reasons for doing so. In addition, I have also included one more reason.

Definition of a Circle

The definition of a circle is:

The set of all points in a plane that are at a fixed distance from a given point.

Here, the fixed distance is the radius $r$. Therefore, defining the mathematical constant π as $\frac{C}{r}$ is more natural than defining it as $\frac{C}{d}$.

Trigonometry

In the radian system, a full circle is defined as $2\pi$, 180° is defined as $\pi$, and a quarter circle is defined as $\frac{\pi }{2}$, which is counterintuitive.

If we define $\tau =2\pi$, then $\frac{\tau }{n}$ corresponds exactly to $\frac{1}{n}$ of the circumference. At the same time, the periods of trigonometric functions such as $\sin$, $\cos$, $\sec$, and $\csc$ become $\tau$, which indeed appears more elegant.

Integration

We divide the circle into rings of various sizes and integrate from the inside out:

$$S={\int }_0^r{C}_{r}dr=\tau rdr=\frac{1}{2}\tau r^2$$

In classical physics, formulas of the form $\frac{1}{2}{\text{variable}}^2$ frequently appear. For example (integration steps omitted):

• Uniformly accelerated linear motion: $s=\frac{1}{2}v{t}^2$
  • Free fall: $h=\frac{1}{2}g{t}^2$
• Hooke's law: $E=\frac{1}{2}k{x}^2$
• Kinetic energy: $K=\frac{1}{2}m{v}^2$

The method for solving such "quadratic" formulas is related to integration. Therefore, if $\tau$ were used instead of $\pi$, you might have been able to derive the integral formula for the area of a circle on your own a few years earlier.

Interior and Exterior Angles of Regular Polygons

Although the sum of the interior angles of a regular polygon is always a multiple of $\pi$, the sum of the exterior angles is always $\tau$ (and the curvature integral of any simple closed loop also equals $\tau$). Therefore, $\tau$ still holds a slight advantage.

Area Formula for Regular Polygons

Let $A_n$ be the area of a regular $n$-gon with apothem $1$. Then:

$$A_n=n\sin \frac{\pi }{n}\cos \frac{\pi }{n}$$

But we can apply the double-angle identity:

$$A_n=\frac{n}{2}\sin \frac{\tau }{n}$$

This formula clearly appears more concise than the previous one and, like the integral formula above, includes the constant $\frac{1}{2}$. We continue simplifying:

$$=\frac{\tau }{2}\frac{\sin \frac{\tau }{n}}{\frac{\tau }{n}}$$

Here, you may notice a familiar pattern: $\frac{\sin x}{x}$. Now, let $n\to \mathrm{\infty }$. In this limit, $A_{\mathrm{\infty }}$ equals the area of a unit circle, while the right-hand side approaches the limit of $\frac{\sin x}{x}$ as $x\to 0$, which is $1$. Therefore:

$$A_{\mathrm{\infty }}=\underset{n\to \mathrm{\infty }}{lim}\frac{\tau }{2}\frac{\sin \frac{\tau }{n}}{\frac{\tau }{n}}=\frac{\tau }{2}$$

This is exactly the value of $\frac{1}{2}\tau r^2$ when $r=1$.

Euler's Formula

We know that the original form of Euler's formula is:

$${\mathrm{e}}^{\mathrm{i}x}=\cos x+\mathrm{i}\sin x$$

If we let $x=\tau$, then we have:

$${\mathrm{e}}^{\mathrm{i}\tau }=1$$

This is more elegant than ${\mathrm{e}}^{\mathrm{i}\pi }=-1$.

Other Important Formulas

There are many commonly used formulas, such as:

Polar coordinate integration formula:

$${\int }_0^{2\pi }{\int }_0^{\mathrm{\infty }}f(r,\theta )r\mathrm{d}r\mathrm{d}\theta$$

Normal distribution function:

$$f(x)=\frac{1}{\sigma \sqrt{2\pi }}{\mathrm{e}}^{-\frac{(x-\mu {)}^2}{2{\sigma }^2}}$$

Fourier transform formulas:

$$f(x)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}F(k)e^{2\pi ikx}\mathrm{d}k$$$$F(k)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)e^{-2\pi ikx}\mathrm{d}x$$

Cauchy integral formula:

$$f(a)=\frac{1}{2\pi i}{\oint }_{\gamma }\frac{f(z)}{z-a}\mathrm{d}z$$

Roots of unity formula:

$$z^n=1\Rightarrow z={\mathrm{e}}^{2\pi \mathrm{i}/n}$$

All of these contain the constant $2\pi$, which demonstrates that $\tau$ indeed has its value.

Surface Area and Volume of the n-Dimensional Unit Hypersphere

We define the $n$-dimensional unit hypersphere as the closed region bounded by the (hyper)surface:

$$\sum _{i=1}^n{x}_i^2=1$$

The surface area is a measure of its boundary ($\sum _{i=1}^n{x}_i^2=1$), while the volume is a measure of both its boundary and interior ($\sum _{i=1}^n{x}_i^2\le 1$). Therefore:

The volume formula is:

$$V_n=\frac{{\sqrt{\pi }}^n}{\mathrm{\Gamma }(1+\frac{n}{2})}$$

The surface area formula is:

$$S_n=\frac{2{\sqrt{\pi }}^n}{\mathrm{\Gamma }(\frac{n}{2})}$$

Since the Gamma function itself is non-trivial, the original author believed this gives the formula a "false appearance of simplicity" (I have reservations about this view) and replaced it with a double factorial form, introducing an additional case analysis:

$$V_n=\frac{{\tau }^{\lfloor \frac{n}{2}\rfloor }}{n!!}\times ((n\mathrm{\%}2)+1)$$$$S_n=\frac{{\tau }^{\lfloor \frac{n}{2}\rfloor }}{(n-2)!!}\times ((n\mathrm{\%}2)+1)$$

Although this case analysis can be offset by introducing a new constant $\lambda =\frac{\tau }{4}$ to form:

$$V_n=\frac{2^n{\lambda }^{\lfloor \frac{n}{2}\rfloor }}{n!!}$$$$S_n=\frac{2^n{\lambda }^{\lfloor \frac{n}{2}\rfloor }}{(n-2)!!}$$

This, however, introduces an additional symbol $\lambda$, which seems to deviate from the theme of using $\tau$ to replace $\pi$. We will not discuss this further for now.

An interesting property is that for any $n>2,n\in N_{+}$, we have:

$$\frac{V_n}{V_{n-2}}=\frac{\tau }{n}$$$$\frac{S_n}{S_{n-2}}=\frac{\tau }{n-2}$$

This is a rather elegant conclusion—at least it appears simpler by a constant factor of $2$ compared to the corresponding recurrence using $\pi$.

"Pi" in n-Dimensional Space

To demonstrate that $\pi =\frac{\tau }{2}$ is merely a coincidence, the original author attempted to generalize the concept of "pi" to $n$ dimensions.

The generalization is straightforward: by removing the radius $r=1$ from $S_n$ and $V_n$, they become two dimensionless constants, denoted as ${\tau }_n$ (the surface area constant family) and ${\sigma }_n$ (the volume constant family).

Thus, we have:

$$S_n={\tau }_n{r}^{n-1},V_n={\sigma }_n{r}^n$$

Recall the earlier integral formula for a circle:

$$V_n(r)=\int S_n(r)\mathrm{d}r$$

Therefore:

$${\sigma }_n{r}^n=\frac{{\tau }_n}{n}{r}^n$$

That is, ${\sigma }_n=\frac{{\tau }_n}{n}$. One can verify this equality using the earlier formula. Additionally, ${\tau }_n$ can be interpreted as a "hyper-angle measure" (referred to as "angle measure" in the original text, though its practical utility remains unclear).

Similarly, the author generalized $\pi =\frac{C}{D}$ to ${\pi }_n=\frac{S_n(r)}{D^{n-1}}$, which simplifies to:

$${\pi }_n=\frac{S_n(r)}{D^{n-1}}=\frac{{\tau }_n{r}^{n-1}}{D^{n-1}}=\frac{{\tau }_n}{2^{n-1}}$$

Hence, the author argues that $\pi ={\pi }_2=\frac{{\tau }_2}{2}$ is merely a coincidence, as $2^{n-1}$ equals $2$ only when $n=2$.

τ^2 ≠ g

At a certain point in history, the value of ${\pi }^2$ was numerically equal to the gravitational acceleration $g$. In 1668, John Wilkins proposed a definition of the meter: the length of a pendulum with a period of 2 seconds would be 1 meter. However, because the value of $g$ varies across different regions of the Earth, this definition was eventually abandoned.

If $\tau$ had been popular back then instead of $\pi$, probably no one would be wondering today whether ${\pi }^2=g$ is a coincidence.