# Simple approximation for $\pi$ by means of well known constant like $\pi \sim \dfrac{2e-\frac{3\sqrt{2}\gamma^2}{4}}{\phi}$? [closed]

For community wiki, On the occasion of the World day of $$\pi$$, I want to look for simple formulas which involves $$\pi$$ using known constants like $$e, \phi,\gamma,\cdots$$ , For instance just a simple attempt I got the following formula with approximation of $$10^{-5}$$ such that :$$\pi \sim \dfrac{2e-\frac{3\sqrt{2}\gamma^2}{4}}{\phi}$$ , Are there others ?

Note: $$\phi$$ is the golden ratio

• All these constants are irrational numbers, you can approximate $\pi$ pretty well, that's KAT. – rtybase Mar 6 at 0:08
• Another example: $$\pi \approx \dfrac{2\sqrt{10}}{\gamma^2 + e/\varphi} = \underline{3.14159265}\color{gray}{26267...}$$ – Oleg567 Mar 9 at 15:35
• Yet another approximation: $$\pi \approx \dfrac{50(75+\gamma)}{11(17^2-e)(2-\varphi)} = \underline{3.14159265358}\color{gray}{25265...}$$ – Oleg567 Mar 10 at 0:12
• All of these formulas does not look very "efficient". For example it takes $44$ characters to type out your formula and evaluating it would also require knowing these constants and still it only gives $\pi$ to 5 digits. The approximation $\pi \approx 3.14159265$ on the other hand is superior, just $10$ characters and correct to $8\cdot 10^{-9}$. It's just not very pretty. – Winther Mar 10 at 8:14
• I wrote a very simple python3 code to estimate the value of $\pi$ by linear combination of $e$, $\phi$, $\log 2$ and $\gamma$ with simple rational denominator. An example is $\pi\approx (10 e -7 \log 2 -14 \phi + 9\gamma)/5 = \underline{3.141592}7880168177\cdots$. Also, wolfram|alpha gave $(-3 e e! + 32 + 14 e + e^2)/(5 e) \approx \underline{3.141592}503375$. – dust05 Mar 10 at 9:53

Few examples (using relatively small integers and constants $$e$$, $$\varphi$$, $$\gamma$$) with increasing precision (and, of course, increasing size of the expression):

$$\pi \approx \dfrac{5}{7}\varphi e = \underline{3.141}\color{gray}{623...}\tag{1}$$

$$\pi \approx \dfrac{8}{e-1}-\dfrac{\sqrt{2}}{\varphi \gamma} = \underline{3.141592}\color{gray}{711...}\tag{2}$$

$$\pi \approx \dfrac{2\sqrt{10}}{\gamma^2 + e/\varphi} = \underline{3.14159265}\color{gray}{262...}\tag{3}$$

$$\pi \approx \dfrac{50(75+\gamma)}{11(17^2-e)(2-\varphi)} = \underline{3.14159265358}\color{gray}{252...}\tag{4}$$

$$\pi \approx \dfrac{28 + \left(e-\dfrac{7}{6}\right)\left(\dfrac{7}{1+\varphi}+\gamma + 3\right)}{12} = \underline{3.14159265358979}\color{gray}{773...}\tag{5}$$

The WolframAlpha link to check last approximation.