Fancy a play??

[tenochtitlanuk]

I produced this graphic which uses 10 colours to represent digits 0 to 9. The finely detailed one is the original; the other blows up the view of the start top-lefthand-corner.

I did NOT use rnd() or any function that generates pseudo-random numbers.

Can you show, by using getpixel or reading the bmp ( save the bmp shown first), whether all digits are represented equally? Or that any chosen one is followed with equal probability by any of the ten? Perhaps even do a chi-squared test...?

Bonus gold star to anyone who can guess where I got these ( ?random?) numbers from?

[tsh73]

to anyone who can guess where I got these ( ?random?) numbers from?

Pi digits?

[logical1]

Jul 7, 2021 16:28:39 GMT -5 tsh73 said:

to anyone who can guess where I got these ( ?random?) numbers from?

Pi digits?

Yeah Pi. 

[tenochtitlanuk]

I knew that was easy- but has anyone either stated how to decode it, or done any of the suggested 'tests of randomness'?

[tsh73]

I sjst read on from top left horisontally
see
blue red yellow red
hints to 3.141
3 is blue
Then I googled some Pi digits
3.1415926535 8979323846 2643383279 5028841971
        535
"535" with blue center seems fit so I said "Pi"

[Brandon Parker]

Well, I was going to work on a small example, but I am visiting family at the moment and have just not had the time. Here is how I would go about it.

I also noticed the same thing as Anatoly, Since we know that it takes 33 digits (including the 3) to "see" all numbers from 0 - 9, I would traverse the first 33 points of color using GetPixel() and assign that color to the appropriate digit in PI's sequence. Then, you can just continue on down the line and spit out the corresponding number from our color to number pair matching we did with the first 33 digits of PI. In that way, you could then print out all of the digits of PI represented within the image.

If you look at it long enough, all sorts of interesting color patterns start dancing around...


{:0)

Brandon Parker

[tsh73]

Since we know that it takes 33 digits (including the 3) to "see" all numbers from 0 - 9,

Ha! I didn't knew
but I just happen to save some 40 didgits of Pi so it happens to be enough

My first attempt ends in fail - I tried to read in BMP but forgot lines are stored top down
So I reverted to slow but working function GetPixelValue$(x, y, handle$)
'pi digits
'since I failed with eading BMP directly (lines are stored from below)
fileName$="origP.bmp"
call GetBmpDimensions fileName$, w,h
print w,h   '194           69
N=w*h
print N '13386
w=w-1
h=h-1   'for single gray line top left

'nomainwin
open "test" for graphics_nsb_nf as #gr
#gr "trapclose [quit]"
#gr "down"
'#gr "fill white; flush"
#gr "home; posxy cx cy"

loadbmp "bmp", fileName$
#gr, "drawbmp bmp 0 0"

#gr "flush"

pi0$="3.1415926535 8979323846 2643383279 5028841971"
pi$="31415926535897932384626433832795028841971"

dim d$(10), d(10)
for i = 1 to len(pi$)
   SCAN
   d=val(mid$(pi$,i,1))
   if d$(d)="" then
       d$(d)= GetPixelValue$(i, 1, "#gr")
       print i, d, d$(d), "new"
   else
       if d$(d)= GetPixelValue$(i, 1, "#gr") then
           print i, d, d$(d), "(OK)"
       else
          print i, d, d$(d), "NOT OK!!!": end
       end if
   end if

next
print

open "pi.txt" for output as #2

for j = 1 to h
   print "line ";j;" of ";h
   for i = 1 to w
       SCAN
       'if (j-1)*w+i>500 then exit for
       'print j, i, (j-1)*w+i, GetPixelValue$(i, j, "#gr")
       'print (j-1)*w+i,mid$(pi$,i,1), GetPixelValue$(i, j, "#gr")
       col$= GetPixelValue$(i, j, "#gr")
       for k = 0 to 9
           if col$=d$(k) then d(k)=d(k)+1: #2, k;:exit for
       next
   next
   'if (j-1)*w+i>500 then exit for
next
close #2

print
for k = 0 to 9
   print k, d(k)
next

wait

[quit]
timer 0
close #gr
end

'Custom functions and subs:

sub GetBmpDimensions fileName$, byref width, byref height
   open fileName$ for input as #gbd
   temp$ = input$(#gbd, 24)
   close #gbd
   width = asc(mid$(temp$, 19, 1))+asc(mid$(temp$, 20, 1))*256
   height = asc(mid$(temp$, 23, 1))+asc(mid$(temp$, 24, 1))*256
end sub

'*****************************************************
'GetPixelValue$ returns a string with the RGB values of the pixel
'in coordinates x and y in window/graphicbox names handle$ (e.g, "#main.graph")
function GetPixelValue$(x, y, handle$)

'Grab a 1*1 bitmap
   #handle$, "getbmp gpv "; x; " "; y; " "; 1; " "; 1

'Save in a bmp file
   bmpsave "gpv", "getpvaluetemp.bmp"

'Open the file for string input and get it's full contents
   open "getpvaluetemp.bmp" for input as #gpv
   s$ = input$(#gpv, lof(#gpv))
   close #gpv

'Check if user's display is 32-bit, and read the red-green-blue values
'If display 16 bit, then colors are masked. So some last (3 for red, 2 for green, 3 for blue) bits always 0
'That means that you did not get 255 255 255 for white - (248 252 248) instead. You have to experiment
'otherwise function returns nothing (support for other display types could be added (?))
   bpp =  asc(mid$(s$, 29, 1))
   select case bpp
   case 32
       red = asc(mid$(s$, 69, 1))
       green = asc(mid$(s$, 68, 1))
       blue = asc(mid$(s$, 67, 1))
   case 24        'LBB: 24 bit, no palette
       red = asc(mid$(s$, 57, 1))
       green = asc(mid$(s$, 56, 1))
       blue = asc(mid$(s$, 55, 1))
       'print red,green, blue
   case 16
       bytes = asc(mid$( s$, 67, 1)) + 256*asc(mid$( s$, 68, 1))
       red =  (bytes AND 63488) /256       '0xF800
       green =  (bytes AND 2016) / 32 * 4  '0x7E0
       blue =  (bytes AND 31) * 8          '0x1F
   end select

'concatenate the return value, delete temporary file and free memory
   GetPixelValue$ = using("###",red)+using("####",green)+using("####",blue)
   kill "getpvaluetemp.bmp"
   unloadbmp "gpv"
end function


Reading first 40+ pixels and checking it fits Pi
1             3               0   0 255   new
2             1             255   0   0   new
3             4             255 255   0   new
4             1             255   0   0   (OK)
5             5               0 255 255   new
6             9             150 150  80   new
7             2               0 255   0   new
8             6             255   0 255   new
9             5               0 255 255   (OK)
10            3               0   0 255   (OK)
11            5               0 255 255   (OK)
12            8              80  80  80   new
13            9             150 150  80   (OK)
14            7             255 255 255   new
15            9             150 150  80   (OK)
16            3               0   0 255   (OK)
17            2               0 255   0   (OK)
18            3               0   0 255   (OK)
19            8              80  80  80   (OK)
20            4             255 255   0   (OK)
21            6             255   0 255   (OK)
22            2               0 255   0   (OK)
23            6             255   0 255   (OK)
24            4             255 255   0   (OK)
25            3               0   0 255   (OK)
26            3               0   0 255   (OK)
27            8              80  80  80   (OK)
28            3               0   0 255   (OK)
29            2               0 255   0   (OK)
30            7             255 255 255   (OK)
31            9             150 150  80   (OK)
32            5               0 255 255   (OK)
33            0               0   0   0   new
34            2               0 255   0   (OK)
35            8              80  80  80   (OK)
36            8              80  80  80   (OK)
37            4             255 255   0   (OK)
38            1             255   0   0   (OK)
39            9             150 150  80   (OK)
40            7             255 255 255   (OK)
41            1             255   0   0   (OK)


Counting number of occurences for each digits
i0             1278
1             1294
2             1306
3             1286
4             1278
5             1352
6             1323
7             1344
8             1327
9             1336


Now, it runs slow, but it creates file pi.txt with just digits.
So for graphing I used it
In shades of green, 0..9 <=> black..green
source picture repainted (193 x 68)

remapped to square (114 x 114)

woven into square spiral, from center

into Archimedes spiral from center


Really, there is nothing special in base 10, so "10 shades of green" is pretty random.
I would like to see same on base 2 pi, just with black and white.

codes
'pi digits
'read from a text file
fileName$="pi.txt"
open fileName$ for input as #1
pi$=input$(#1, lof(#1))
close #1
print len(pi$)
'194           69

WindowWidth=1050:WindowHeight=650
nomainwin
open "test" for graphics_nsb_nf as #gr
#gr "trapclose [quit]"
#gr "down"
'#gr "fill white; flush"
#gr "home; posxy cx cy"
#gr "flush"

w=193
s=5
'uncomment to map on square
'w=int(sqr(193*68)):s=5
notice "w=";w
#gr "size ";s
for i = 1 to len(pi$)
   SCAN
   d=val(mid$(pi$,i,3))
   x=(i-1) mod w
   y=int((i-1)/w)
   c$= "0 ";int(255*d/9);" 0"'rgb
   #gr "color ";c$;"; set ";x*s+2/2;" ";y*s+s/2
next
#gr "flush"

wait

[quit]
timer 0
close #gr
end

'Custom functions and subs:

sub GetBmpDimensions fileName$, byref width, byref height
   open fileName$ for input as #gbd
   temp$ = input$(#gbd, 24)
   close #gbd
   width = asc(mid$(temp$, 19, 1))+asc(mid$(temp$, 20, 1))*256
   height = asc(mid$(temp$, 23, 1))+asc(mid$(temp$, 24, 1))*256
end sub

'*****************************************************
'GetPixelValue$ returns a string with the RGB values of the pixel
'in coordinates x and y in window/graphicbox names handle$ (e.g, "#main.graph")
function GetPixelValue$(x, y, handle$)

'Grab a 1*1 bitmap
   #handle$, "getbmp gpv "; x; " "; y; " "; 1; " "; 1

'Save in a bmp file
   bmpsave "gpv", "getpvaluetemp.bmp"

'Open the file for string input and get it's full contents
   open "getpvaluetemp.bmp" for input as #gpv
   s$ = input$(#gpv, lof(#gpv))
   close #gpv

'Check if user's display is 32-bit, and read the red-green-blue values
'If display 16 bit, then colors are masked. So some last (3 for red, 2 for green, 3 for blue) bits always 0
'That means that you did not get 255 255 255 for white - (248 252 248) instead. You have to experiment
'otherwise function returns nothing (support for other display types could be added (?))
   bpp =  asc(mid$(s$, 29, 1))
   select case bpp
   case 32
       red = asc(mid$(s$, 69, 1))
       green = asc(mid$(s$, 68, 1))
       blue = asc(mid$(s$, 67, 1))
   case 24        'LBB: 24 bit, no palette
       red = asc(mid$(s$, 57, 1))
       green = asc(mid$(s$, 56, 1))
       blue = asc(mid$(s$, 55, 1))
       'print red,green, blue
   case 16
       bytes = asc(mid$( s$, 67, 1)) + 256*asc(mid$( s$, 68, 1))
       red =  (bytes AND 63488) /256       '0xF800
       green =  (bytes AND 2016) / 32 * 4  '0x7E0
       blue =  (bytes AND 31) * 8          '0x1F
   end select

'concatenate the return value, delete temporary file and free memory
   GetPixelValue$ = using("###",red)+using("####",green)+using("####",blue)
   kill "getpvaluetemp.bmp"
   unloadbmp "gpv"
end function

'pi digits
'spiral
fileName$="pi.txt"
open fileName$ for input as #1
pi$=input$(#1, lof(#1))
close #1
print len(pi$)
'194           69

WindowWidth=650:WindowHeight=650
'nomainwin
open "test" for graphics_nsb_nf as #gr
#gr "trapclose [quit]"
#gr "down"
'#gr "fill white; flush"
#gr "home; posxy cx cy"
#gr "flush"

s=5
x=0:y=0
dir=0
l=0

#gr "size ";s

l=0
i=1
d=val(mid$(pi$,i,1))
c$= "0 ";int(255*d/9);" 0"'rgb
#gr "color ";c$;"; set ";cx+x*s;" ";cy+y*s
while i<len(pi$)
   SCAN
   i=i+1
   d=val(mid$(pi$,i,1))
   k=k+1
       select case dir mod 4
       case 0:x=x+1
       case 1:y=y-1
       case 2:x=x-1
       case 3:y=y+1
       end select
   if k>l then
       k=0
       dir=dir+1
       if dir mod 2 = 0 then l=l+1
   end if

   c$= "0 ";int(255*d/9);" 0"'rgb
'    print i, d, x, y
'    if i >20 then exit while
   #gr "color ";c$;"; set ";cx+x*s;" ";cy+y*s
wend
#gr "flush"

wait

end

for i = 1 to len(pi$)
   SCAN
   d=val(mid$(pi$,i,3))
   x=(i-1) mod w
   y=int((i-1)/w)
   c$= "0 ";int(255*d/9);" 0"'rgb
   #gr "color ";c$;"; set ";x*s;" ";y*s
next
#gr "flush"

wait

[quit]
timer 0
close #gr
end

'Custom functions and subs:

sub GetBmpDimensions fileName$, byref width, byref height
   open fileName$ for input as #gbd
   temp$ = input$(#gbd, 24)
   close #gbd
   width = asc(mid$(temp$, 19, 1))+asc(mid$(temp$, 20, 1))*256
   height = asc(mid$(temp$, 23, 1))+asc(mid$(temp$, 24, 1))*256
end sub

'*****************************************************
'GetPixelValue$ returns a string with the RGB values of the pixel
'in coordinates x and y in window/graphicbox names handle$ (e.g, "#main.graph")
function GetPixelValue$(x, y, handle$)

'Grab a 1*1 bitmap
   #handle$, "getbmp gpv "; x; " "; y; " "; 1; " "; 1

'Save in a bmp file
   bmpsave "gpv", "getpvaluetemp.bmp"

'Open the file for string input and get it's full contents
   open "getpvaluetemp.bmp" for input as #gpv
   s$ = input$(#gpv, lof(#gpv))
   close #gpv

'Check if user's display is 32-bit, and read the red-green-blue values
'If display 16 bit, then colors are masked. So some last (3 for red, 2 for green, 3 for blue) bits always 0
'That means that you did not get 255 255 255 for white - (248 252 248) instead. You have to experiment
'otherwise function returns nothing (support for other display types could be added (?))
   bpp =  asc(mid$(s$, 29, 1))
   select case bpp
   case 32
       red = asc(mid$(s$, 69, 1))
       green = asc(mid$(s$, 68, 1))
       blue = asc(mid$(s$, 67, 1))
   case 24        'LBB: 24 bit, no palette
       red = asc(mid$(s$, 57, 1))
       green = asc(mid$(s$, 56, 1))
       blue = asc(mid$(s$, 55, 1))
       'print red,green, blue
   case 16
       bytes = asc(mid$( s$, 67, 1)) + 256*asc(mid$( s$, 68, 1))
       red =  (bytes AND 63488) /256       '0xF800
       green =  (bytes AND 2016) / 32 * 4  '0x7E0
       blue =  (bytes AND 31) * 8          '0x1F
   end select

'concatenate the return value, delete temporary file and free memory
   GetPixelValue$ = using("###",red)+using("####",green)+using("####",blue)
   kill "getpvaluetemp.bmp"
   unloadbmp "gpv"
end function

'pi digits
'square spiral
fileName$="pi.txt"
open fileName$ for input as #1
pi$=input$(#1, lof(#1))
close #1
print len(pi$)
'194           69

WindowWidth=650:WindowHeight=650
nomainwin
open "Pi Archimedes spiral" for graphics_nsb_nf as #gr
#gr "trapclose [quit]"
#gr "down"
'#gr "fill white; flush"
#gr "home; posxy cx cy"
#gr "flush"

s=5
x=0:y=0
dir=0
l=0

#gr "size ";s
pi=acs(-1)
tau=2*pi

l=0
i=1
d=val(mid$(pi$,i,1))
c$= "0 ";int(255*d/9);" 0"'rgb
#gr "color ";c$;"; set ";cx+x*s;" ";cy+y*s
a=1
while i<len(pi$)
   SCAN
   i=i+1
   d=val(mid$(pi$,i,1))
   x=a*cos(a)/tau
   y=0-a*sin(a)/tau
   a=a+1/a*s

   c$= "0 ";int(255*d/9);" 0"'rgb
'    print i, d, x, y
'    if i >20 then exit while
   #gr "color ";c$;"; set ";cx+x*s;" ";cy+y*s
wend
#gr "flush"

wait

end

for i = 1 to len(pi$)
   SCAN
   d=val(mid$(pi$,i,3))
   x=(i-1) mod w
   y=int((i-1)/w)
   c$= "0 ";int(255*d/9);" 0"'rgb
   #gr "color ";c$;"; set ";x*s;" ";y*s
next
#gr "flush"

wait

[quit]
timer 0
close #gr
end

'Custom functions and subs:

sub GetBmpDimensions fileName$, byref width, byref height
   open fileName$ for input as #gbd
   temp$ = input$(#gbd, 24)
   close #gbd
   width = asc(mid$(temp$, 19, 1))+asc(mid$(temp$, 20, 1))*256
   height = asc(mid$(temp$, 23, 1))+asc(mid$(temp$, 24, 1))*256
end sub

'*****************************************************
'GetPixelValue$ returns a string with the RGB values of the pixel
'in coordinates x and y in window/graphicbox names handle$ (e.g, "#main.graph")
function GetPixelValue$(x, y, handle$)

'Grab a 1*1 bitmap
   #handle$, "getbmp gpv "; x; " "; y; " "; 1; " "; 1

'Save in a bmp file
   bmpsave "gpv", "getpvaluetemp.bmp"

'Open the file for string input and get it's full contents
   open "getpvaluetemp.bmp" for input as #gpv
   s$ = input$(#gpv, lof(#gpv))
   close #gpv

'Check if user's display is 32-bit, and read the red-green-blue values
'If display 16 bit, then colors are masked. So some last (3 for red, 2 for green, 3 for blue) bits always 0
'That means that you did not get 255 255 255 for white - (248 252 248) instead. You have to experiment
'otherwise function returns nothing (support for other display types could be added (?))
   bpp =  asc(mid$(s$, 29, 1))
   select case bpp
   case 32
       red = asc(mid$(s$, 69, 1))
       green = asc(mid$(s$, 68, 1))
       blue = asc(mid$(s$, 67, 1))
   case 24        'LBB: 24 bit, no palette
       red = asc(mid$(s$, 57, 1))
       green = asc(mid$(s$, 56, 1))
       blue = asc(mid$(s$, 55, 1))
       'print red,green, blue
   case 16
       bytes = asc(mid$( s$, 67, 1)) + 256*asc(mid$( s$, 68, 1))
       red =  (bytes AND 63488) /256       '0xF800
       green =  (bytes AND 2016) / 32 * 4  '0x7E0
       blue =  (bytes AND 31) * 8          '0x1F
   end select

'concatenate the return value, delete temporary file and free memory
   GetPixelValue$ = using("###",red)+using("####",green)+using("####",blue)
   kill "getpvaluetemp.bmp"
   unloadbmp "gpv"
end function


[tenochtitlanuk]

Good stuff Anatoly! Now I must try a Ulam spiral display..

I also was initially caught by the 'bottom first' nature of bmps.

Must have a go at the binary expansion- 11.001001000011111101101010100010001000010110100011000010001101001100010011000110011000101000101110000000110111000001110011010001...

Pi always fascinated me- especially once I understood complex numbers and 'e'.

[tenochtitlanuk]

Since pi is infinitely long in ANY base, it is fun to create lots of its digits to base 27, using A..Z and space to represent each digit. Somewhere in there you will find every novel that can be written, every secret government document, 666 names of God, and even- but more realistically- your own name. Must try it and perhaps discover the secret of 'Life and the Universe' in it...

[tsh73]

Since Pi digits is pretty random I wonder how it will work if used as direction change
Here is random walk with step "s" in radom direction "d=int(rnd(0)*10)"
I just wanted it to have visible "recent" part fading into "old" part, could be tweaked more.
(Smaller N = faster, bigger N - nice fading effect)
Now, only put next digit of Pi intstead of "d" (saved of another computer...)
nomainwin
'open "test" for graphics_nsb_nf as #gr
open "test" for graphics_nsb_fs as #gr
#gr "down; home; posxy cx cy"
#gr "trapclose [quit]"
#gr "size 2"

N=30
k = 1000
dim x(k), y(k)
pi = acs(-1)
' randomize 0.5
s = 15
x(0)=cx:y(0)=cy
for i = 1 to k
   d=int(rnd(0)*10)
   a=2*pi*d/10
   x(i)=x(i-1)+s*cos(a)
   y(i)=y(i-1)+s*sin(a)
next

for i = 1 to k-N
   SCAN
   for j = N-1 to 0 step -1
       'c = int(127+128*j/N)
       'N: bright. 0: almost white
       c = int(200-200*j/N)
       c$=c;" 254 ";c
       'print c, ">";c$;"<"
       #gr "color ";c$
       ' #gr "color 100 254 190"
       #gr "line ";x(i+j);" ";y(i+j);" ";x(i+j-1);" ";y(i+j-1)
   next
'wait
   call pause 10
next


wait

[quit]
   timer 0
   close #gr
   end

sub pause mil
   t=time$("ms")+mil
   while time$("ms")<t
       scan
   wend
[quit]
end sub


[tsh73]

+ is starting point, x is ending point.
13xxx points as provided by John.
A little bit like Americas on the globe, no?
Now just download 100000 of Pi and see if I got whole globe
. from say www.angio.net/pi/digits.html

[tsh73]

100 k digits.
1) I did not get whole globe
2) indeed picture is completely different.
That means digits extracted from picture from first post is off...
(I could mess it up, or picture could be a bit off - ?? )
Then I compare these files, I see they start differ at point 196 - and continue differently
Source picture was 194x69, with single gray line, so actual line length 193.

[tenochtitlanuk]

Interesting. I HOPE that published versions of Pi agree- perhaps we ought to check the web ones we are using agree with each other! I last did a random walk code using LB's turtle graphics, and the turtle's rounding errors distorted it. LB5 seems to have corrected this so I wouldn't need my turtle-replacement functions. Remember it gave us trouble on things like Sierpinski curves??

I love the idea that a random walk might generate visuals like a map of the world!

As an ( ex) physicist I realise it is little more than a century since Perrin and Einstein really 'proved' that discrete atoms exist with 'Brownian Motion' experiments. It would be decades later before electron microscopes could 'see' individual atoms.

Sad to report, my base-27 PI produced 'B.BUDYBUALKXYWLRVOQOHHIYGVDAFHYOHVS_' using '_ABCDEFGHIJKLMNOPQRSTUVWXYZ' to represent the base=27 digits, using the first 36 decimal terms of Pi. Hasn't shown any interesting words yet- got to try it on a million digits next.

[tenochtitlanuk]

You'll like this one Anatoly- from pi2e.ch/blog/2017/03/10/pi-digits-download/#inspirations

[tenochtitlanuk]

I've run my LB version with LB's rnd(); with a million digits of Pi; and with a million-digit web download of random digits.

You have to screensave/flush/redraw to avoid memory problems, but I'm happy with my colouring scheme- but not happy with the results using Pi nor LB's rnd().

Will put code up on my site soon- busy celebrating our Golden Wedding Anniversary at present!