I continue to be impressed with how rich this subject is, and my understanding of \( \pi \) and \( \tau \) continues to evolve. On Half Tau Day, 2012, I believed I identified exactly what is wrong with \( \pi \). My argument hinged on an analysis of the surface area and volume of an \( n \)-dimensional sphere, which (as shown below) makes clear that \( \pi \) doesn’t have any fundamental geometric significance. My analysis was incomplete, though—a fact brought to my attention in a remarkable message from Tau Manifesto reader Jeff Cornell. As a result, this section is an attempt not only to definitively debunk \( \pi \), but also to articulate the truth about \( \tau \), a truth that is deeper and subtler than I had imagined.

Note: This section is more advanced than the rest of the manifesto and can be skipped without loss of continuity. If you find it confusing, I recommend proceeding directly to the conclusion in Section 6.

We start our investigations with the generalization of a circle to arbitrary dimensions. This object, called a hypersphere or an \( n \)-sphere, can be defined as follows. (For convenience, we assume that these spheres are centered on the origin.) A \( 0 \)-sphere is the empty set, and we define its “interior” to be a point. A \( 1 \)-sphere is the set of all points satisfying

\[ x^2 = r^2, \]

which consists of the two points \( \pm r \). Its interior, which satisfies

\[ x^2 \leq r^2, \]

is the line segment from \( -r \) to \( r \). A \( 2 \)-sphere is a circle, which is the set of all points satisfying

\[ x^2 + y^2 = r^2. \]

Its interior, which satisfies,

\[ x^2 + y^2 \leq r^2, \]

is a disk. Similarly, a \( 3 \)-sphere satisfies

\[ x^2 + y^2 + z^2 = r^2, \]

and its interior is a ball. The generalization to arbitrary \( n \), although difficult to visualize for \( n > 3 \), is straightforward: an \( n \)-sphere is the set of all points satisfying

\[ \sum_{i=1}^{n} x_i^2 = r^2. \]

The Pi Manifesto (discussed in Section 4.2) includes a formula for the volume of a unit \( n \)-sphere as an argument in favor of \( \pi \):

\begin{equation} \label{eq:unit_n_sphere_pi} \frac{\sqrt{\pi}^{n} }{\Gamma(1 + \frac{n}{2})}, \end{equation}

where the Gamma function is given by Eq. (11). Eq. (14) is a special case of the formula for general radius, which is also typically written in terms of \( \pi \):

\begin{equation} \label{eq:n_sphere_pi} V_n(r) = \frac{\pi^{n/2} r^n}{\Gamma(1 + \frac{n}{2})}. \end{equation}

Because \( V_n(r) = \int S_n(r)\,dr \), we have \( S_n(r) = dV_n(r)/dr \), which means that the surface area can be written as follows:

\begin{equation} \label{eq:n_sphere_pi_r} S_n(r) = \frac{n \pi^{n/2} r^{n-1}}{\Gamma(1 + \frac{n}{2})}. \end{equation}

Rather than simply take these formulas at face value, let’s see if we can untangle them to shed more light on the question of \( \pi \) vs. \( \tau \). We begin our analysis by noting that the apparent simplicity of the above formulas is an illusion: although the Gamma function is notationally simple, in fact it is an integral over a semi-infinite domain, which is not a simple idea at all. Fortunately, the Gamma function can be simplified in certain special cases. For example, when \( n \) is an integer, it is easy to show (using integration by parts) that

\[ \Gamma(n) = (n-1)(n-2)\ldots 2\cdot 1 = (n-1)! \]

Seen this way, \( \Gamma \) can be interpreted as a generalization of the factorial function to real-valued arguments.

In the \( n \)-dimensional surface area and volume formulas, the argument of \( \Gamma \) is not necessarily an integer, but rather is \( \left(1 + \frac{n}{2}\right) \), which is an integer when \( n \) is even and is a half-integer when \( n \) is odd. Taking this into account gives the following expression, which is taken from a standard reference, Wolfram MathWorld, and as usual is written in terms of \( \pi \):

\begin{equation} \label{eq:surface_area_mathworld} S_n(r) = \begin{cases} \displaystyle \frac{2\pi^{n/2}\,r^{n-1}}{(\frac{1}{2}n - 1)!} & n \text{ even}; \\ \\ \displaystyle \frac{2^{(n+1)/2}\pi^{(n-1)/2}\,r^{n-1}}{(n-2)!!} & n \text{ odd}. \end{cases} \end{equation}

Integrating with respect to \( r \) then gives

\begin{equation} \label{eq:volume_mathworld} V_n(r) = \begin{cases} \displaystyle \frac{\pi^{n/2}\,r^n}{(\frac{n}{2})!} & n \text{ even}; \\ \\ \displaystyle \frac{2^{(n+1)/2}\pi^{(n-1)/2}\,r^n}{n!!} & n \text{ odd}. \end{cases} \end{equation}

Let’s examine Eq. (18) in more detail. Notice first that MathWorld uses the double factorial function \( n!! \)—but, strangely, it uses it only in the odd case. (This is a hint of things to come.) The double factorial function, although rarely encountered in mathematics, is elementary: it’s like the normal factorial function, but involves subtracting \( 2 \) at a time instead of \( 1 \), so that, e.g., \( 5!! = 5 \cdot 3 \cdot 1 \) and \( 6!! = 6 \cdot 4 \cdot 2 \). In general, we have

\begin{equation} \label{eq:double_factorial} n!! = \begin{cases} n(n-2)\ldots6\cdot4\cdot2 & n \text{ even}; \\ \\ n(n-2)\ldots5\cdot3\cdot1 & n \text{ odd}. \end{cases} \end{equation}

(By definition, \( 0!! = 1!! = 1 \).) Note that Eq. (19) naturally divides into even and odd cases, making MathWorld’s decision to use it only in the odd case still more mysterious.

To solve this mystery, we’ll start by taking a closer look at the formula for odd \( n \) in Eq. (18):

\[ \frac{2^{(n+1)/2}\pi^{(n-1)/2}\,r^n}{n!!} \]

Upon examining the expression

\[ 2^{(n+1)/2}\pi^{(n-1)/2}, \]

we notice that it can be rewritten as

\[ 2(2\pi)^{(n-1)/2}, \]

and here we recognize our old friend \( 2\pi \).

Now let’s look at the even case in Eq. (18). We noted above how strange it is to use the ordinary factorial in the even case but the double factorial in the odd case. Indeed, because the double factorial is already defined piecewise, if we unified the formulas by using \( n!! \) in both cases we could pull it out as a common factor:

\[ V_n(r) = \frac{1}{n!!}\times \begin{cases} \ldots & n \text{ even}; \\ \\ \ldots & n \text{ odd}. \end{cases} \]

So, is there any connection between the factorial and the double factorial? Yes—when \( n \) is even, the two are related by the following identity:

\[ \left(\frac{n}{2}\right)! = \frac{n!!}{2^{n/2}}. \]

(This is easy to verify using mathematical induction.) Substituting this into the volume formula for even \( n \) then yields

\[ \frac{2^{n/2}\pi^{n/2}\,r^n}{n!!}, \]

which bears a striking resemblance to

\[ \frac{(2\pi)^{n/2}\,r^n}{n!!}, \]

and again we find a factor of \( 2\pi \).

Putting these results together, we see that Eq. (18) can be rewritten as

\begin{equation} \label{eq:volume_2pi} V_n(r) = \begin{cases} \displaystyle \frac{(2\pi)^{n/2}\,r^n}{n!!} & n \text{ even}; \\ \\ \displaystyle \frac{2(2\pi)^{(n-1)/2}\,r^n}{n!!} & n \text{ odd} \end{cases} \end{equation}

and Eq. (17) can be rewritten as

\begin{equation} \label{eq:surface_area_2pi} S_n(r) = \begin{cases} \displaystyle \frac{(2\pi)^{n/2}\,r^{n-1}}{(n-2)!!} & n \text{ even}; \\ \\ \displaystyle \frac{2(2\pi)^{(n-1)/2}\,r^{n-1}}{(n-2)!!} & n \text{ odd}. \end{cases} \end{equation}

Making the substitution \( \tau=2\pi \) in Eq. (21) then yields

\[ S_n(r) = \begin{cases} \displaystyle \frac{\tau^{n/2}\,r^{n-1}}{(n-2)!!} & n \text{ even}; \\ \\ \displaystyle \frac{2\tau^{(n-1)/2}\,r^{n-1}}{(n-2)!!} & n \text{ odd}. \end{cases} \]

To unify the formulas further, we can use the floor function \( \lfloor x \rfloor \), which is simply the largest integer less than or equal to \( x \) (equivalent to chopping off the fractional part, so that, e.g., \( \lfloor 3.7 \rfloor = \lfloor 3.2 \rfloor = 3 \)). This gives

\[ S_n(r) = \begin{cases} \displaystyle \frac{\tau^{\left\lfloor \frac{n}{2} \right\rfloor}\,r^{n-1}}{(n-2)!!} & n \text{ even}; \\ \\ \displaystyle \frac{2\tau^{\left\lfloor \frac{n}{2} \right\rfloor}\,r^{n-1}}{(n-2)!!} & n \text{ odd}, \end{cases} \]

which allows us to write the formula as follows:

\begin{equation} \label{eq:surface_area_tau} S_n(r) = \frac{\tau^{\left\lfloor \frac{n}{2} \right\rfloor}\,r^{n-1}}{(n-2)!!}\times \begin{cases} 1 & n \text{ even}; \\ \\ 2 & n \text{ odd}. \end{cases} \end{equation}

Integrating Eq. (22) with respect to \( r \) then yields

\begin{equation} \label{eq:volume_tau} V_n(r) = \frac{\tau^{\left\lfloor \frac{n}{2} \right\rfloor}\,r^n}{n!!}\times \begin{cases} 1 & n \text{ even}; \\ \\ 2 & n \text{ odd}. \end{cases} \end{equation}