Jonathan. Frech’s WebBlog

Tau Day MMXVII (#173)

Jonathan Frech,

Today it is June the 28th which means that it is 𝜏 day!
The irrational and transcendental con­stant 𝜏 is what defines $\pi=\frac{\tau}{2}$$\pi=\frac{\tau}{2}$$\pi=\frac{\tau}{2}$, which obviously makes it an important con­stant. To celebrate this day, I created a C pro­gram which calculates 𝜏 by ran­dom­ly creating 9-di­men­sion­al points inside the 9-di­men­sion­al hypercube and test­ing if they are inside the 9-di­men­sion­al hypersphere with its center located at $(0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5)$$(0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5)$$(0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5)$⁠¹.

Today’s 𝜏 time is 3:18:53 as $\tau=6.2831853\dots$$\tau=6.2831853\dots$$\tau=6.2831853\dots$. As one does not know if the time is specified as ante or post meridiem, there are actually two perfectly acceptable 𝜏 times.

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The formula used for cal­cu­lat­ing 𝜏 is derived from a 9-di­men­sion­al hypersphere’s hypervolume formula $V=\frac{32\cdot\pi^4}{945}\cdot R^9$$V=\frac{32\cdot\pi^4}{945}\cdot R^9$$V=\frac{32\cdot\pi^4}{945}\cdot R^9$ (see this Wiki­pe­dia article).

$$\begin{cases} V = \frac{2^5 \cdot \tau^4 \cdot R^9}{945 \cdot 2^4} = \frac{2 \cdot R^9}{945} \cdot \tau^4\\ \tau^4 = \frac{V \cdot 945}{2 \cdot R^9}; R = 0.5\\ \tau^4 = \frac{V \cdot 945 \cdot 2^9}{2}\\ \tau = \sqrt[4]{V \cdot 241920} \end{cases}$$$$\begin{cases} V = \frac{2^5 \cdot \tau^4 \cdot R^9}{945 \cdot 2^4} = \frac{2 \cdot R^9}{945} \cdot \tau^4\\ \tau^4 = \frac{V \cdot 945}{2 \cdot R^9}; R = 0.5\\ \tau^4 = \frac{V \cdot 945 \cdot 2^9}{2}\\ \tau = \sqrt[4]{V \cdot 241920} \end{cases}$$$$\begin{cases} V = \frac{2^5 \cdot \tau^4 \cdot R^9}{945 \cdot 2^4} = \frac{2 \cdot R^9}{945} \cdot \tau^4\\ \tau^4 = \frac{V \cdot 945}{2 \cdot R^9}; R = 0.5\\ \tau^4 = \frac{V \cdot 945 \cdot 2^9}{2}\\ \tau = \sqrt[4]{V \cdot 241920} \end{cases}$$

The con­stant gets calculated to $\tau^*=6.293700$$\tau^*=6.293700$$\tau^*=6.293700$. The real value is ap­prox­i­mate­ly 𝜏 = 6.283185…, which makes the percent error

$$\left|\frac{\tau^*-\tau}{\tau}\right|=\left|\frac{6.293700-6.283185}{6.283185}\right|=0.001673514 = 0.167\%.$$$$\left|\frac{\tau^*-\tau}{\tau}\right|=\left|\frac{6.293700-6.283185}{6.283185}\right|=0.001673514 = 0.167\%.$$$$\left|\frac{\tau^*-\tau}{\tau}\right|=\left|\frac{6.293700-6.283185}{6.283185}\right|=0.001673514 = 0.167\%.$$

Thereby, this C pro­gram’s approximation is not too far off.⁠² The source code is listed below and can also be downloaded here. In­struc­tions on how to com­pile it using GCC can be seen below or in the source code.

gcc tau.c -o tau -lm; ./tau
tau = 6.293700

Resources worth checking out re­gard­ing 𝜏 are The Tau Manifesto and 2 Pi or Not 2 Pi? I wish everybody a happy 𝜏 day.

Source code: tau-day-mmxvii.c

[1][2020-07-30] More concisely, $\frac 12\sum_je_j\in\mathbb{R}^9$$\frac 12\sum_je_j\in\mathbb{R}^9$$\frac 12\sum_je_j\in\mathbb{R}^9$.
[2][2020-07-30] Whatever that means …