I've been looking at Monte Carlo simulation recently, and have been using it to approximate constants such as π (circle inside a rectangle, proportionate area).
However, I'm unable to think of a corresponding method of approximating the value of Φ [Golden ratio ] using Monte Carlo integration or any other simulation.
Do you have any pointers on how this can be done?
On
Another simple way that's along the lines of what you've done before, is just to monte-carlo evaluate the integral $$ \ln \phi = \int_0^\frac{1}{2} \frac{1}{\sqrt{x^2+1}} \ \text{d}x$$
For instance, sample uniformly at random from the rectangle $[0, \frac{1}{2}] \times [0, 1]$, compute the proportion of times $y_n^2 (x_n^2+1) \leq 1$ double it (to account for using a rectangle of area $\frac{1}{2}$) and exponentiate it in order to get an estimate for $\phi$
The golden triangle is an acute isosceles triangle with a vertex angle of $36^{\circ}$ and base angles of $72^{\circ}$. The ratio of the leg to the base is the golden ratio, $\Phi$, so the area of the triangle is $A=bh/2=\Phi\cos(\pi/10)/2$. This should work nicely in a Monte Carlo simulation.