Jonathan. Frech’s WebBlog

Pi Day MMXVII (#163)

Jonathan Frech

Every year on March the 14th, for one day the world gets irrationally excited about the famous con­stant 𝜋. As is tradition, you try to calculate 𝜋 in unusual ways, demonstrating the con­stant’s ubiquity as it crops up in the most unexpected circumstances.

           lnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnn
       nJ$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$w
    `v$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ 
   n$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$z 
  u$$mnn:       Y$$$i        .@$$$$$,             
 $$n           )$$$*         W$$$$$m              
-n[            $$$$.        ]$$$$$$               
              h$$$w         $$$$$$Y               
             [$$$$         X$$$$$$                
            "$$$$n        '$$$$$${                
           .$$$$$         8$$$$$$                 
           *$$$$}        :$$$$$$+                 
          #$$$$u         $$$$$$%                  
        t$$$$$$         ;$$$$$$`                  
       u$$$$$$!         $$$$$$W                   
      Y$$$$$$M         .$$$$$$,                   
    f$$$$$$$$.         Z$$$$$Z          nn        
  `w$$$$$$$$|          $$$$$$(         v$z        
 n$$$$$$$$$W           $$$$$$$1      'X$8         
Y$$$$$$$$$$            *$$$$$$$8nnnn$$$p          
$$$$$$$$$@.             W$$$$$$$$$$$$$n           
_$$$$$$${                x$$$$$$$$$0>             
   -n{.                     !|nt_.

A fairly well-known way to ap­prox­i­mate 𝜋 is to ran­dom­ly choose points in a square (often thought of as throwing darts at a square piece of cardboard), de­ter­mine their distance to a circle’s center and do a division, as I did in my 𝜋 Generator post.

However, 𝜋 does not on­ly appear in the formula for a circle’s area, $A=\pi\,r^2$$A=\pi\,r^2$, but also in the formula for a sphere’s volume, $V=\frac{4}{3}\,\pi\,r^3$$V=\frac{4}{3}\,\pi\,r^3$, and for all the infinite hyperspheres above di­men­sion three (view this Wikipedia article for more about volumes of higher-di­men­sion­al spheres).

In particular, the formula for the hypervolume of a hypersphere in four dimensions is defined as being $V=\frac{\pi^2}{2} \cdot r^4$$V=\frac{\pi^2}{2} \cdot r^4$. Using this formula, my Python script ran­dom­ly chooses four-di­men­sion­al points (each in the interval $\left[0,1\right)$$\left[0,1\right)$), calculates their distance to the point $\left(0.5,0.5,0.5,0.5\right)$$\left(0.5,0.5,0.5,0.5\right)$ and determines if they are in the hypersphere around that point with radius 0.5.
By dividing the num­ber of ran­dom points which lie in the hypersphere by the num­ber of iterations used ($10^6$$10^6$ in the example below), the script approximates the hypersphere’s hypervolume. By then rearranging the equation $V=\frac{\pi^2}{2}\cdot r^4$$V=\frac{\pi^2}{2}\cdot r^4$ with 𝑟 = 0.5 to $\pi=\sqrt{V\cdot 32}$$\pi=\sqrt{V\cdot 32}$, the desired con­stant can be approximated.

$ python pi.py
3.14196371717

Source code: pi-day-mmxvii.py