13.5 More complex: π generating π.....

What is a random integer? Is an integer choosen randomly between 1 and 1000000 really representative for all integers choosen randomly? We can see that our experience is only an approximation of an ideal model. Here, we’re going to modify the method for generating random integers... We won’t use the primitive random, we’re going to generate random integers with the π digits sequence.

π digits have always interested mathematicians:

• The numbers 0 to 9, do some appear more often than others?
• Is there some sequence of integers that appear frequently?

In reality, it seems that the π digit sequence is a really randomly sequence. (Result not demonstrated yet). It’s not possible to predict the following digit after the others, there’s no period.



Here is the method we’re going to use to generate integers randomly choosen:

• First, we need the first digit of π (For example, one billion)
  1. First way: some programs calculate the π digits. For example, PiFast in Windows environment and SchnellPi for Linux.
  2. Second way: you can download this file from XLOGO website:

     http://downloads.tuxfamily.org/xlogo/common/millionpi.txt

• To generate the integers, we’re going to read the digits sequence in packet of 7 digits:
  
  

  _{First number}

  _{Second number}

  _{Third number}338327950288419716939 etc
  
  
  
  I remove the point “. ” in 3.14 ....that will cause problem when we’re going to extract the digits

Let’s create now a new procedure called randompi and let’s modify the procedure test

 to gcd :a :b
 if (modulo :a :b)=0 [output :b][output gcd :b modulo :a :b]
 end
 
 to test :tries
 # We open a flow whose identifier is 1 towards the file  millionpi.txt
 # Here we suppose that millionpi.txt is in the current directory
 # Otherwise, fix it with changedirectory
                                                                                                  
                                                                                                  
 openflow 1 "millionpi.txt
 # Set the variable line to the first line of the file millionpi.text
 globalmake "line first readlineflow 1
 # Set the variable counter to  0
 globalmake "counter 0
 repeat :tries [
   if 1=gcd randompi 7 randompi 7 [globalmake "counter :counter+1]
 ]
 # Calculate frequency
 globalmake "f :counter/:tries
 # Display th pi approximation
 print sentence [ pi approximation:] squareroot (6/:f)
 closeflow 1
 end
 
 to randompi :n
 localmake "number "
 repeat :n [
 # If there’s no char yet on the line
 if 0=count :line [globalmake "line first readlineflow 1]
 # Set the variable char to the first character of the line
 globalmake "char first :line
 # Then remove  first character from the line.
 globalmake "line butfirst :line
 globalmake "number word :number :char
 ]
 output :number
 end
 
 test 10
 approximation de pi: 3.4641016151377544
 test 100
 approximation de pi: 3.1108550841912757
 test 1000
 approximation de pi: 3.081180112566604
 test 10000
 approximation de pi: 3.1403714651066386
 test 70000
 approximation de pi: 3.1361767950325627

We find a correct approximation of π with its own digits!



It’s still possible to imrove the program by indicating the time for the computation. We add on the first line of the procedure test:

Then we append before closeflow: