Recursive Sequences

$$\text{Circumference and Convenience}$$

=== $\\$

Tomorrow (June $28$) is Tau Day, a holiday celebrating the constant $\tau$, which is equal to $2\pi=6.2831853…$.

Many mathematicians prefer $\tau$ to $\pi$ because it simplifies several circle formulas; for example, the circumference $C$ of a circle can be represented as the product of its radius $R$ and $\tau$: $$C=\tau R$$ Entire manifestos have even been written about Tau.

Notably, $\tau$ is used for convenience in $2$ dimensional spaces (especially with circles, which are really just $2$ dimensional spheres). That’s pretty useful.

But why stop there? Let’s look at some more dimensions!

$$\begin{array}{c c c} \text{Number of Dimensions} & \text{Volume} & \text{Surface Area} \\ \hline 2 & \pi R^2 & 2\pi R \\ 3 & \frac{4}{3}\pi R^3 & 4\pi R^2 \\ 4 & \frac{1}{2}\pi^2 R^4 & 2\pi^2 R^3 \\ 5 & \frac{8}{15} \pi^2 R^5 & \frac{8}{3}\pi^2 R^4 \\ 6 & \frac{1}{6}\pi^3 R^6 & \pi^3 R^5 \\ 7 & \frac{16}{105}\pi^3 R^7 & \frac{16}{15}\pi^3 R^6 \\ 8 & \frac{1}{24}\pi^4 R^8 & \frac{1}{3}\pi^4 R^7 \end{array}$$

Note that we typically call the volume of a circle the area and the surface area of a circle the circumference. Of course, thinking of spheres in multiple dimensions is a little weird, but you don’t actually need to know what these objects look like to determine their properties!

We can look at the surface areas to see that there is always some constant multiplied by a power of $R$. In our beloved two dimensional sphere (a circle!), this constant is $2\pi$, which is $\tau$. So if one of the purposes of $\tau$ is convenience, let’s look at its analogs in other dimensions!

For example, the “tau” of the $5^{th}$ dimension is $\frac{8}{3}\pi^2$ because this is the constant that appears before the power of $R$ (which here is $R^4$). This is also the second such constant to be divisible by $\pi^2$.