Approximation of $e$ using $\pi$ and $\phi$? $$e \approx \frac{4 \phi +3 \pi-5}{4}$$
where $~\phi~$ is a Golden ratio .
Is it possible to construct better approximation of $e$ using $\pi$ , $\phi$ and integers ?
 A: Of course. $ \phi \approx 1.618033989$ and 
$ e \approx 2.718281828$ so $\displaystyle e \approx \frac{2718281828}{1618033989} \phi .$ Extend to whatever accuracy you desire. 
Edit in response to comment: If you require using $\pi$ as well, then we can do $$\displaystyle e \approx \frac{2718281828}{2\cdot 1618033989} \phi + \frac{2718281828}{2\cdot 3141692654} \pi.$$ 
A: Try: $${\frac {16+31\,\phi+6\,\pi +25\,{\pi }^{2}-14\,{\pi }^{3}}{11+25\,\phi
+4\,\pi -48\,{\pi }^{2}+12\,{\pi }^{3}}}
$$
A: You can approximate any number using only $\phi$ by expressing it in the Golden ratio base.
For example, $e \approx \phi^2 + \phi^{-5} + \phi^{-10} + \phi^{-13} = 100.0000100001001_\phi = 2.71825392998$
A: Since $\phi$ is algebraic, and $e$ is transcendent, it follows that $\frac{e}{\phi}$ is irrational.
Then, the set $\{ m\frac{\pi}{\phi} +n | m,n \in Z \}$ is dense in $R$ and thus is also 
$$ \{ m \pi +n\phi | m,n \in Z \}$$
Thus, for each $\epsilon >0$ there exists infinitely many pairs $(m,n)$ of integers so that 
$$e \approx  m \pi +n\phi \,,$$
with the error of the approximation less than $\epsilon$...
A: Try an integer relation algorithm, such as PSLQ, but you'll probably need high-precision approximations of $e$, $\pi$, and $\phi$.
A: We know that $\phi^2 = \phi + 1$ , from where we deduce that $\frac1\phi + \frac1{\phi\ +\ 1} = \frac1\phi + \frac1{\phi^2} = 1$ .
At the same time, we also know that $e = 2.718^{^+}$ , and $\phi = 1.618^{^+}$ , from where we deduce that $e \simeq \phi + 1.1 = \phi^2 + \frac1{10}$ .
Putting the two together, we get the following approximations:
$$\frac1e + \frac1\phi + \frac1{71} \simeq 1 \qquad,\qquad \frac1e + \frac1{\sqrt{e}} + \frac1{39} \simeq 1$$
$$\frac1{\sqrt{e}} + \frac1{\phi^2} + \frac1{87}\ =\ \frac1{\sqrt{e}} + \frac1{\phi + 1} + \frac1{87} \simeq 1$$
I would also like to recommend the following resources : I hope you'll find them helpful.
www.mrob.com/pub/ries/
en.wikipedia.org/wiki/Almost_integer
en.wikipedia.org/wiki/Mathematical_coincidence
A: I also once played with these famous constants- I came across to this following relation :
$e+\pi \cong 3\phi+1$ 
Approximation error:$|3\phi-(e+\pi-1)|\approx0,00577 < \frac{3}{500}$
Here we go:  $e\approx 3\phi+1-\pi$
