***Testing Normality of Pi, Console Example ***
   
This page is under development. Comments are
welcome, but please load any comments
in the comments section 
at the middle of the page. Thanks,[gold]
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[gold] Here is an eTCL  script  
on testing the normality of [pi] for the etcl console.
It has not been mathematically proven that [pi] is a normal number,
but numerical analysis with tcl can check for some aspects of normality in pi.
The normality of a number means
that the numbers (0,1,2..9) in the mantissa are equally common at infinity. In base 10, the 
relative frequencies of the N numbers in the manitssa approach 1/10  as N approaches infinity.
For example,   Kanada used a set of 6.4E9 decimal digits and found the frequency s 600009044/6442450000 or 
0.0931336749218077. One can paste statements into the etcl console.

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% puts [ expr {1.*600009044/6442450000 }]
0.0931336749218077
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In planning any software, there is 
a need to develop testcases.  
 
 Testcase 1.
  frequency of "7"  for collection of N digits of pi
%|quantity|frequency|N |digits|%
&|7| |20| digits |&
&|7| | 50 |digits|&
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***Screenshots Section***
  frequency of "7" in pi mantissa approaches limit of 0.1 as N>>1
  frequency versus set of numbers of total N. 
[http://img546.imageshack.us/img546/9785/aaarx.jpg]
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'''Comments Section'''

Please place any comments here, Thanks.

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***References:***
   * http://pi314.at/math/normal.html
   * http://mathworld.wolfram.com/NormalNumber.html
   * http://www.lbl.gov/Science-Articles/Archive/pi-random.html 
   * http://www.geom.uiuc.edu/~huberty/math5337/groupe/digits.html
   * http://www.eveandersson.com/pi/digits/
   * http://www.geom.uiuc.edu/~huberty/math5337/groupe/digits.html
   * http://www.super-computing.org/
   * http://www.piworld.de/pi-statistics/
   * citeseerx.ist.psu.edu/ viewdoc/ download?doi=10.1.1.133.7330 &rep=rep1 &type=pdf

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****Appendix TCL programs and scripts ****

**** Pretty Print Version*** 

======
               # Pretty print version from autoindent
               # and ased editor
               # written on Windowws XP on eTCL
               # working under TCL version 8.5.6 and eTCL 1.0.1
               # gold on TCL WIKI , 25may2011
               #rank of indiv. "throws" over all "throws".
               #pi mantissa used here
        package require Tk 
          console show
            proc calculation {  facen }  {
                # prob. subroutines for mimic sequence of bronze
                # prob. is throw combos of eg. "7" over all possible throws
                
                set lister [split {14159265358979323846} ""]
                
                set ee [llength  $lister ]
                set kk [ llength [ lsearch -all $lister $facen ] ]
                set prob [ expr { ($kk*1.) / $ee  } ]
                return $prob
            }
            set limit 12
            for { set i 0 } { $i <= $limit }  { incr i } {
                lappend listxxx $i
                lappend listxxx [ calculation  $i ]
                puts " $i [ calculation  $i ] "
            }
            #end
 results for first 20 numbers of pi mantissa
         
 0 0.0 
 1 0.1 
 2 0.1 
 3 0.15 
 4 0.1 
 5 0.15 
 6 0.1 
 7 0.05 
 8 0.1 
 9 0.15 
 10 0.0 
 11 0.0 
 12 0.0 
{14159265358979323846264338327950288419716939937}
 1 0.0851063829787234 
 2 0.10638297872340426 
 3 0.1702127659574468 
 4 0.0851063829787234 
 5 0.0851063829787234 
 6 0.0851063829787234 
 7 0.0851063829787234 
 8 0.10638297872340426 
 9 0.1702127659574468 
 10 0.0 
 11 0.0 
 12 0.0 

results for first 20 numbers of pi mantissa
 1 0.1 
 2 0.1 
 3 0.15 
 4 0.1 
 5 0.15 
 6 0.1 
 7 0.05 
 8 0.1 
 9 0.15 
results for small numbers of pi mantissa
 1 0.0851063829787234 
 2 0.10638297872340426 
 3 0.1702127659574468 
 4 0.0851063829787234 
 5 0.0851063829787234 
 6 0.0851063829787234 
 7 0.0851063829787234 
 8 0.10638297872340426 
 9 0.1702127659574468 
results for 100k numbers of pi mantissa
 1 0.1000947923455181 
 2 0.09782372573414697 
 3 0.0989790074451488 
 4 0.09843592629895136 
 5 0.09899875585046508 
 6 0.09899875585046508 
 7 0.09898888164780693 
 8 0.09852479412287458 
 9 0.09777435472085629 



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*** Code scraps, timing statements***
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        puts " loading [time {pixane load $p -file $fin} ]"        
        puts "Scaling ($sourcewidth x $sourceheight) => ($virtualwidth x $virtualheight)"
        puts " resizex [time {pixane resize $targetpix $virtualwidth $virtualheight } ]"
        puts " blanking [time {pixane blank $targetpix } ]"
        puts " rescale [time {pixane scale $targetpix $p } ]"
        puts " saving [time {pixane save $targetpix -file $fout -format jpg } ]"
        puts " proc time [time { jackpix } ]"
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