Happy Tau Day 2025

TL;DR of "The Tau Manifesto."

The original text defines $τ = 2 π = 6.283 \dots$ to replace the current circle constant $π$ 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 $2 π$ , 180° is defined as $π$ , and a quarter circle is defined as $\frac{π}{2}$ , which is counterintuitive.

If we define $τ = 2 π$ , then $\frac{τ}{n}$ 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 $τ$ , 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} d r = τ r d r = \frac{1}{2} τ r^{2}

In classical physics, formulas of the form $\frac{1}{2} {variable}^{2}$ 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 $τ$ were used instead of $π$ , 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 $π$ , the sum of the exterior angles is always $τ$ (and the curvature integral of any simple closed loop also equals $τ$ ). Therefore, $τ$ 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{π}{n} \cos \frac{π}{n}

But we can apply the double-angle identity:

A_{n} = \frac{n}{2} \sin \frac{τ}{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{τ}{2} \frac{\sin \frac{τ}{n}}{\frac{τ}{n}}

Here, you may notice a familiar pattern: $\frac{\sin x}{x}$ . Now, let $n \to \infty$ . In this limit, $A_{\infty}$ 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_{\infty} = lim_{n \to \infty} \frac{τ}{2} \frac{\sin \frac{τ}{n}}{\frac{τ}{n}} = \frac{τ}{2}

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

Euler's Formula

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

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

If we let $x = τ$ , then we have:

e^{i τ} = 1

This is more elegant than $e^{i π} = - 1$ .

Other Important Formulas

There are many commonly used formulas, such as:

Polar coordinate integration formula:

\int_{0}^{2 π} \int_{0}^{\infty} f (r, θ) r d r d θ

Normal distribution function:

f (x) = \frac{1}{σ \sqrt{2 π}} e^{- \frac{(x - μ)^{2}}{2 σ^{2}}}

Fourier transform formulas:

f (x) = \int_{- \infty}^{\infty} F (k) e^{2 π i k x} d k

F (k) = \int_{- \infty}^{\infty} f (x) e^{- 2 π i k x} d x

Cauchy integral formula:

f (a) = \frac{1}{2 π i} \oint_{γ} \frac{f (z)}{z - a} d z

Roots of unity formula:

z^{n} = 1 \Rightarrow z = e^{2 π i / n}

All of these contain the constant $2 π$ , which demonstrates that $τ$ 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 ( $\sum_{i = 1}^{n} x_{i}^{2} \leq 1$ ). Therefore:

The volume formula is:

V_{n} = \frac{{\sqrt{π}}^{n}}{Γ (1 + \frac{n}{2})}

The surface area formula is:

S_{n} = \frac{2 {\sqrt{π}}^{n}}{Γ (\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{τ^{⌊ \frac{n}{2} ⌋}}{n!!} \times ((n % 2) + 1)

S_{n} = \frac{τ^{⌊ \frac{n}{2} ⌋}}{(n - 2)!!} \times ((n % 2) + 1)

Although this case analysis can be offset by introducing a new constant $λ = \frac{τ}{4}$ to form:
$V_{n} = \frac{2^{n} λ^{⌊ \frac{n}{2} ⌋}}{n!!}$ $S_{n} = \frac{2^{n} λ^{⌊ \frac{n}{2} ⌋}}{(n - 2)!!}$
This, however, introduces an additional symbol $λ$ , which seems to deviate from the theme of using $τ$ to replace $π$ . 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{τ}{n}

\frac{S_{n}}{S_{n - 2}} = \frac{τ}{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 $π$ .

"Pi" in n-Dimensional Space

To demonstrate that $π = \frac{τ}{2}$ 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 $τ_{n}$ (the surface area constant family) and $σ_{n}$ (the volume constant family).

Thus, we have:

S_{n} = τ_{n} r^{n - 1}, V_{n} = σ_{n} r^{n}

Recall the earlier integral formula for a circle:

V_{n} (r) = \int S_{n} (r) d r

Therefore:

σ_{n} r^{n} = \frac{τ_{n}}{n} r^{n}

That is, $σ_{n} = \frac{τ_{n}}{n}$ . One can verify this equality using the earlier formula. Additionally, $τ_{n}$ 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 $π = \frac{C}{D}$ to $π_{n} = \frac{S_{n} (r)}{D^{n - 1}}$ , which simplifies to:

π_{n} = \frac{S_{n} (r)}{D^{n - 1}} = \frac{τ_{n} r^{n - 1}}{D^{n - 1}} = \frac{τ_{n}}{2^{n - 1}}

Hence, the author argues that $π = π_{2} = \frac{τ_{2}}{2}$ 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 $π^{2}$ 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 $τ$ had been popular back then instead of $π$ , probably no one would be wondering today whether $π^{2} = g$ is a coincidence.