On 4/22/2023 7:37 PM, Ben Bacarisse wrote:
> no@no.no (James Waldby) writes:
>
>> Ben Bacarisse <ben.usenet@bsb.me.uk> wrote:
>>> "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
>>>> On 4/22/2023 2:57 PM, Ben Bacarisse wrote:
>>>>> Dan joyce <danj4084@gmail.com> writes:
>>>>>> Okay, on the same page but now thinking of a walk with 10 directions
>>>>>> going clockwise for each digit of pi and at each step.
>>>>> I wondered about that too, so here's a quick picture for the first
>>>>> million digits of pi:
>>>>> http://www.bsb.me.uk/tmp/pi-walk.png
>>>>> Cross-hairs show the origin, and the path is drawn without full opacity
>>>>> so you can see the more visited areas. I think it's quite pretty.
>>
>>>> Nice one! Looks like a brownian motion for sure. Has a fractal border.
>>>
>>> I made a colour version to show the stages of path in a spectrum. It
>>> starts red and ends blue:
>>> http://www.bsb.me.uk/tmp/pi-walk-col.png
>>
>> That's a different pattern than I get when plotting positions for
>> first million digits of pi.
>
> Ah, I'm doing something different. The digit only determines the
> direction in units of 36 degrees. All the steps are the same distance.
> I've not really been following the thread. It's just that Dan's comment
> reminded me of something I'd planned to do a while back and never got
> round to doing.
>

I remember doing one a while back where the directions were pi*2/10 for
a base 10 number. For instance, 0...9 each mapped to an angle. I only
traveled from the current point to the next point by a short distance.
It was something like the following pseudo-code:

plot digits...
___________________________
float distance = .001; // a short distance... ;^)
vec2 origin = { 0, 0 };
vec2 current = origin;

for each digit {
// get the mapped normalized direction of the digit
vec2 normal_direction = get_mapped_direction(digit);

// get the next point
vec2 next = current + normal_direction * distance;

// plot line from current to next...
plot_line(current, next);

// update the current position
current = next;
} ___________________________

It works for any 10-ary number system. So, the directions for a base 5
system would be pi*2/5, and we map the five symbols to these normalize
directions. This can be moved into 3d by gaining the mapped directions
from a sphere instead of a circle. The plot_line function can add some
color to the line when it draws it. Or, just use an alpha blend like you
did.

This is fun because each irrational should give a different plot.

On 4/22/2023 7:54 PM, Chris M. Thomasson wrote:
> On 4/22/2023 7:37 PM, Ben Bacarisse wrote:
>> no@no.no (James Waldby) writes:
>>
>>> Ben Bacarisse <ben.usenet@bsb.me.uk> wrote:
>>>> "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
>>>>> On 4/22/2023 2:57 PM, Ben Bacarisse wrote:
>>>>>> Dan joyce <danj4084@gmail.com> writes:
>>>>>>> Okay, on the same page but now thinking of a walk with 10 directions
>>>>>>> going clockwise for each digit of pi and at each step.
>>>>>> I wondered about that too, so here's a quick picture for the first
>>>>>> million digits of pi:
>>>>>> http://www.bsb.me.uk/tmp/pi-walk.png
>>>>>> Cross-hairs show the origin, and the path is drawn without full
>>>>>> opacity
>>>>>> so you can see the more visited areas.  I think it's quite pretty.
>>>
>>>>> Nice one! Looks like a brownian motion for sure. Has a fractal border.
>>>>
>>>> I made a colour version to show the stages of path in a spectrum.  It
>>>> starts red and ends blue:
>>>> http://www.bsb.me.uk/tmp/pi-walk-col.png
>>>
>>> That's a different pattern than I get when plotting positions for
>>> first million digits of pi.
>>
>> Ah, I'm doing something different.  The digit only determines the
>> direction in units of 36 degrees.  All the steps are the same distance.
>> I've not really been following the thread.  It's just that Dan's comment
>> reminded me of something I'd planned to do a while back and never got
>> round to doing.
>>
>
> I remember doing one a while back where the directions were pi*2/10 for
> a base 10 number. For instance, 0...9 each mapped to an angle. I only
> traveled from the current point to the next point by a short distance.
> It was something like the following pseudo-code:
>
> plot digits...
> ___________________________
> float distance = .001; // a short distance... ;^)
> vec2 origin = { 0, 0 };
> vec2 current = origin;
>
> for each digit {
>    // get the mapped normalized direction of the digit
>    vec2 normal_direction = get_mapped_direction(digit);
>
>    // get the next point
>    vec2 next = current + normal_direction * distance;
>
>    // plot line from current to next...
>    plot_line(current, next);
>
>    // update the current position
>    current = next;
> }
> ___________________________
>
> It works for any 10-ary number system. So, the directions for a base 5
> system would be pi*2/5, and we map the five symbols to these normalize
> directions. This can be moved into 3d by gaining the mapped directions
> from a sphere instead of a circle. The plot_line function can add some
> color to the line when it draws it. Or, just use an alpha blend like you
> did.
>
> This is fun because each irrational should give a different plot.

One can easily map a byte to an angle my pi*2/256 where 0...255 maps to
a unique angle.

On Saturday, April 22, 2023 at 4:00:34 PM UTC-4, James Waldby wrote:
> Dan joyce <danj...@gmail.com> wrote:
> ...
> > Okay, on the same page but now thinking of a walk with 10 directions going clockwise for each digit
> > of pi and at each step. x and -x and y and -y will have in most cases rational values but at some
> > point the value x,-x or y,-y will be an integer. I picture a larger and larger but not so perfect spiral probably
> > never to cross the x=0 or y=0 axis again. Thus expanding outward in a jagged edge expansion depicting
> > an imperfect spiral.
> > 3 y=3 x=0
> > 1 y=3.7???.. x = 0.5???..
> > 4 y=4.6???.. x =4.???..each with a 36 degree offset .
> > Probably a programming nightmare.
> Actually easy; see program spinner1.jl at <https://gitlab.com/ghjwp7/piwalkstats>,
> plus the text outputs from several runs of that program, in file
> results-piWalk1 (at lines 377-635). I haven't graphed any output,
> instead printing stats which include counts of near-the-origin, NearX,
> NearY, CrossX, CrossY, plus values of Xmin, Xmax, Ymin, Ymax. The
> program defines "near an axis" as within 0.002 units of it. Here are
> the counts and values when the input number is pi, taken to 1000000
> digits, 30000000 digits, and 100000000 digits, respectively, which
> suggest that during the 70 million digits after the first 30 million,
> the walk was stuck in a single quadrant:
>
> NearZZ NearX NearY Xmin Xmax Ymin Ymax CrossX CrossY
> 0 2 2 -596.31 1351.72 -1915.33 2177.78 1324 475
> 0 9 16 -5553.86 26758.82 -9535.72 10675.78 6230 13358
> 0 9 16 -5553.86 26758.82 -9535.72 16835.60 6230 13358
>
> Below are walk coordinates (and the sizes of steps taken, where
> dx^2+dy^2=digit^2) for the first 35 digits of pi; you can see the
> coordinates for the first 80 digits of pi near the end of file
> results-piWalk1, or can run the spinner1.jl program to produce as long
> a list as you like. .81
>
> D# Dig. X Y dx dy #digits=100000000
> 1. 3 3.00 0.00 3.00 0.00
> 2. 1 3.81 0.59 0.81 0.59
> 3. 4 5.05 4.39 1.24 3.80
> 4. 1 4.74 5.34 -0.31 0.95
> 5. 5 0.69 8.28 -4.05 2.94
> 6. 9 -8.31 8.28 -9.00 0.00
> 7. 2 -9.93 7.11 -1.62 -1.18
> 8. 6 -11.78 1.40 -1.85 -5.71
> 9. 5 -10.24 -3.36 1.55 -4.76
> 10. 3 -7.81 -5.12 2.43 -1.76
> 11. 5 -2.81 -5.12 5.00 0.00
> 12. 8 3.66 -0.42 6.47 4.70
> 13. 9 6.44 8.14 2.78 8.56
> 14. 7 4.28 14.80 -2.16 6.66
> 15. 9 -3.00 20.09 -7.28 5.29
> 16. 3 -6.00 20.09 -3.00 0.00
> 17. 2 -7.62 18.92 -1.62 -1.18
> 18. 3 -8.55 16.06 -0.93 -2.85
> 19. 8 -6.07 8.45 2.47 -7.61
> 20. 4 -2.84 6.10 3.24 -2.35
> 21. 6 3.16 6.10 6.00 0.00
> 22. 2 4.78 7.28 1.62 1.18
> 23. 6 6.64 12.98 1.85 5.71
> 24. 4 5.40 16.79 -1.24 3.80
> 25. 3 2.97 18.55 -2.43 1.76
> 26. 3 -0.03 18.55 -3.00 0.00
> 27. 8 -6.50 13.85 -6.47 -4.70
> 28. 3 -7.43 11.00 -0.93 -2.85
> 29. 2 -6.81 9.09 0.62 -1.90
> 30. 7 -1.15 4.98 5.66 -4.11
> 31. 9 7.85 4.98 9.00 0.00
> 32. 5 11.90 7.92 4.05 2.94
> 33. 0 11.90 7.92 0.00 0.00
> 34. 2 11.28 9.82 -0.62 1.90
> 35. 8 4.81 14.52 -6.47 4.70
> ...
So basically you are starting at 3.00 o'clock traveling right x=3 and y=0
Then moving counter clockwise by 36 degree offset where x+ 0.81 is
the second digit of pi =1 and y=0.59 that also represents the 1.
So-on until you rotate counterclockwise 360 degrees that starts the
11th digit of pi. Getting the full value of x added on to the rotated value of
x.
Nice

On Saturday, April 22, 2023 at 5:57:25 PM UTC-4, Ben Bacarisse wrote:
> Dan joyce <danj...@gmail.com> writes:
>
> > Okay, on the same page but now thinking of a walk with 10 directions
> > going clockwise for each digit of pi and at each step.
> I wondered about that too, so here's a quick picture for the first
> million digits of pi:
>
> http://www.bsb.me.uk/tmp/pi-walk.png
>
> Cross-hairs show the origin, and the path is drawn without full opacity
> so you can see the more visited areas. I think it's quite pretty.
>
> --
> Ben.
Nice.
What is the pixel map size?
Did you shrink down the pixel size to keep it on screen?

Ben Bacarisse <ben.usenet@bsb.me.uk> wrote:
> no@no.no (James Waldby) writes:
>> Ben Bacarisse <ben.usenet@bsb.me.uk> wrote:
>>> "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
>>>> On 4/22/2023 2:57 PM, Ben Bacarisse wrote:
>>>>> Dan joyce <danj4084@gmail.com> writes:
>>>>>> Okay, on the same page but now thinking of a walk with 10 directions
>>>>>> going clockwise for each digit of pi and at each step.
>>>>> I wondered about that too, so here's a quick picture for the first
>>>>> million digits of pi:
>>>>> http://www.bsb.me.uk/tmp/pi-walk.png
>>>>> Cross-hairs show the origin, and the path is drawn without full opacity
>>>>> so you can see the more visited areas. I think it's quite pretty.

>>>> Nice one! Looks like a brownian motion for sure. Has a fractal border.

>>> I made a colour version to show the stages of path in a spectrum.
>>> It starts red and ends blue: http://www.bsb.me.uk/tmp/pi-walk-col.png

>> That's a different pattern than I get when plotting positions for
>> first million digits of pi.

> Ah, I'm doing something different. The digit only determines the
> direction in units of 36 degrees. All the steps are the same distance.
> I've not really been following the thread. It's just that Dan's comment
> reminded me of something I'd planned to do a while back and never got
> round to doing.

I added a walker for that pattern to my xyPlot1.jl program, and now
<https://gitlab.com/ghjwp7/piwalkstats/-/blob/master/walkPlot1000000U.png>
is like your plot -- it replicated ok. (I'm now using a color
gradient red-green-blue, and selected alpha=0.3 in xyPlot1.jl on page
<https://gitlab.com/ghjwp7/piwalkstats>)

Barry Schwarz <schwarzb@delq.com> wrote:
> On Sun, 16 Apr 2023 14:00:00 -0700 (PDT), Dan joyce
> <danj4084@gmail.com> wrote:
>
>>Each digit of pi treated as an integer,
>>Starting with 3 and x=0 and y=0.
>>
>>3 down x=0 y=-3
>>1 left x=-1 y=-3
>>4 up x=-1 y=1
>>1 right x=0 y=1
>>5 down x=0 y=-4
....
>>Repeat that order of directions with each digit of pi.
>>What will be the x/y coordinates on the Cartesian coordinate plain
>>after 1,000,000 digits of pi?
>>How many times will it cross the x=0 axis and y=0 axis or where an
>>actual digit of pi ends up on x=0 and y=0?
>>Above the 10th and 11th digit of pi x=0 but y=-7 and y=-2 respectfully
....
> After the first billion digits in pi, we end up at (-22065,-140854).
> The x position reaches extremes at -46684 and 20240.
> The y position reaches extremes at -183016 and 12484.
> We land on the x axis (y=0) 1636 times.
> We land on the y axis (x=0) 20240 times.
> The least frequent digit in pi is 3 with 99,986,912 occurrences.
> The most frequent digit is 4 with 100,011,958 occurrences.

My program's results agree with those max and min values, but both of
my on-axis counts (Xon, Yon) are somewhat different (see top row of
numbers below - other lines are for smaller numbers of digits, for
possibly easier comparison of results). (The `Cross` numbers in
following are lower bounds for numbers of axis crossings.)

Stats for a billion digits of pi, a million, 100K, 10K, 1K, 100 --
Xmin Xmax Cross Xon Ymin Ymax Cross Yon
-46684 20240 58685 29687 -183016 12484 6953 3590
-2076 439 1074 563 -752 1842 753 388
-247 439 606 331 -526 588 176 66
-142 135 111 73 -65 195 62 22
-70 64 10 9 -39 40 49 18
-1 30 1 5 -11 40 5 0

Above stats snipped from output of variedExtremes1.jl but can also be
gotten from output of piWalk1.jl, among programs appearing at links on
page <https://gitlab.com/ghjwp7/piwalkstats> - jiw

On Sun, 23 Apr 2023 06:25:37 -0000 (UTC), no@no.no (James Waldby)
wrote:

>Barry Schwarz <schwarzb@delq.com> wrote:
>> On Sun, 16 Apr 2023 14:00:00 -0700 (PDT), Dan joyce
>> <danj4084@gmail.com> wrote:
>>
>>>Each digit of pi treated as an integer,
>>>Starting with 3 and x=0 and y=0.
>>>
>>>3 down x=0 y=-3
>>>1 left x=-1 y=-3
>>>4 up x=-1 y=1
>>>1 right x=0 y=1
>>>5 down x=0 y=-4
>...
>>>Repeat that order of directions with each digit of pi.
>>>What will be the x/y coordinates on the Cartesian coordinate plain
>>>after 1,000,000 digits of pi?
>>>How many times will it cross the x=0 axis and y=0 axis or where an
>>>actual digit of pi ends up on x=0 and y=0?
>>>Above the 10th and 11th digit of pi x=0 but y=-7 and y=-2 respectfully
>...
>> After the first billion digits in pi, we end up at (-22065,-140854).
>> The x position reaches extremes at -46684 and 20240.
>> The y position reaches extremes at -183016 and 12484.
>> We land on the x axis (y=0) 1636 times.
>> We land on the y axis (x=0) 20240 times.
>> The least frequent digit in pi is 3 with 99,986,912 occurrences.
>> The most frequent digit is 4 with 100,011,958 occurrences.
>
>My program's results agree with those max and min values, but both of
>my on-axis counts (Xon, Yon) are somewhat different (see top row of
>numbers below - other lines are for smaller numbers of digits, for
>possibly easier comparison of results). (The `Cross` numbers in
>following are lower bounds for numbers of axis crossings.)
>
>Stats for a billion digits of pi, a million, 100K, 10K, 1K, 100 --
> Xmin Xmax Cross Xon Ymin Ymax Cross Yon
> -46684 20240 58685 29687 -183016 12484 6953 3590
> -2076 439 1074 563 -752 1842 753 388
> -247 439 606 331 -526 588 176 66
> -142 135 111 73 -65 195 62 22
> -70 64 10 9 -39 40 49 18
> -1 30 1 5 -11 40 5 0
>
>Above stats snipped from output of variedExtremes1.jl but can also be
>gotten from output of piWalk1.jl, among programs appearing at links on
>page <https://gitlab.com/ghjwp7/piwalkstats> - jiw

I had a slightly more restrictive interpretation. I only considered
the movement as landing on the axis if we got there from a position
off the axis. For example, the first movement from (0,0) to (0,-3)
was not counted because x was already 0 before the move. Similarly, if
the digit in pi was 0, no movement occurred so nothing was counted.

When I removed these exceptions from my code, I duplicated your
values. I did not try to count axis crossings.

--
Remove del for email

"Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:

> On 4/22/2023 5:19 PM, Ben Bacarisse wrote:
>> "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
>>
>>> On 4/22/2023 4:04 PM, Ben Bacarisse wrote:
>>>> "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
>>>>
>>>>> On 4/22/2023 2:57 PM, Ben Bacarisse wrote:
>>>>>> Dan joyce <danj4084@gmail.com> writes:
>>>>>>
>>>>>>> Okay, on the same page but now thinking of a walk with 10 directions
>>>>>>> going clockwise for each digit of pi and at each step.
>>>>>> I wondered about that too, so here's a quick picture for the first
>>>>>> million digits of pi:
>>>>>> http://www.bsb.me.uk/tmp/pi-walk.png
>>>>>> Cross-hairs show the origin, and the path is drawn without full opacity
>>>>>> so you can see the more visited areas. I think it's quite pretty.
>>>>>>
>>>>>
>>>>> Nice one! Looks like a brownian motion for sure. Has a fractal border.
>>>> I made a colour version to show the stages of path in a spectrum. It
>>>> starts red and ends blue:
>>>> http://www.bsb.me.uk/tmp/pi-walk-col.png
>>>
>>> Nice! It's also fun to add a little color to a pixel. It creates a sort of
>>> weight map. Pixels that get visited more than once during iteration will be
>>> "heavier", so to speak.
>> Both images do a version of that by using opacity.
>
> Ahhh. I missed that. Using an alpha blend will work as well. Fwiw, have you
> ever tried to do a 3d plot from the weight map where the "denser" pixels are
> "raised up" along the z-axis?

No.

--
Ben.

Dan joyce <danj4084@gmail.com> writes:

> On Saturday, April 22, 2023 at 5:57:25 PM UTC-4, Ben Bacarisse wrote:
>> Dan joyce <danj...@gmail.com> writes:
>>
>> > Okay, on the same page but now thinking of a walk with 10 directions
>> > going clockwise for each digit of pi and at each step.
>> I wondered about that too, so here's a quick picture for the first
>> million digits of pi:
>>
>> http://www.bsb.me.uk/tmp/pi-walk.png
>>
>> Cross-hairs show the origin, and the path is drawn without full opacity
>> so you can see the more visited areas. I think it's quite pretty.
>>
>> --
>> Ben.
> Nice.
> What is the pixel map size?
> Did you shrink down the pixel size to keep it on screen?

It's drawn with a system that does not use pixels until rendering (and
not at all if the output is a PDF). So the plot it not "pixel perfect"
-- it's a series of thin lines.

--
Ben.

On Sunday, April 23, 2023 at 8:15:40 AM UTC-4, Ben Bacarisse wrote:
> Dan joyce <danj...@gmail.com> writes:
>
> > On Saturday, April 22, 2023 at 5:57:25 PM UTC-4, Ben Bacarisse wrote:
> >> Dan joyce <danj...@gmail.com> writes:
> >>
> >> > Okay, on the same page but now thinking of a walk with 10 directions
> >> > going clockwise for each digit of pi and at each step.
> >> I wondered about that too, so here's a quick picture for the first
> >> million digits of pi:
> >>
> >> http://www.bsb.me.uk/tmp/pi-walk.png
> >>
> >> Cross-hairs show the origin, and the path is drawn without full opacity
> >> so you can see the more visited areas. I think it's quite pretty.
> >>
> >> --
> >> Ben.
> > Nice.
> > What is the pixel map size?
> > Did you shrink down the pixel size to keep it on screen?
> It's drawn with a system that does not use pixels until rendering (and
> not at all if the output is a PDF). So the plot it not "pixel perfect"
> -- it's a series of thin lines.
>
> --
> Ben.
Interesting how it interprets one mapping (different length lines) and transposing it to pixels,
another mapping when viewed as a rendering.
Have I got that explanation right, or am I missing something?
Dan

On 4/23/2023 10:54 AM, Dan joyce wrote:
> On Sunday, April 23, 2023 at 8:15:40 AM UTC-4, Ben Bacarisse wrote:
>> Dan joyce <danj...@gmail.com> writes:
>>
>>> On Saturday, April 22, 2023 at 5:57:25 PM UTC-4, Ben Bacarisse wrote:
>>>> Dan joyce <danj...@gmail.com> writes:
>>>>
>>>>> Okay, on the same page but now thinking of a walk with 10 directions
>>>>> going clockwise for each digit of pi and at each step.
>>>> I wondered about that too, so here's a quick picture for the first
>>>> million digits of pi:
>>>>
>>>> http://www.bsb.me.uk/tmp/pi-walk.png
>>>>
>>>> Cross-hairs show the origin, and the path is drawn without full opacity
>>>> so you can see the more visited areas. I think it's quite pretty.
>>>>
>>>> --
>>>> Ben.
>>> Nice.
>>> What is the pixel map size?
>>> Did you shrink down the pixel size to keep it on screen?
>> It's drawn with a system that does not use pixels until rendering (and
>> not at all if the output is a PDF). So the plot it not "pixel perfect"
>> -- it's a series of thin lines.
>>
>> --
>> Ben.
> Interesting how it interprets one mapping (different length lines) and transposing it to pixels,
> another mapping when viewed as a rendering.
> Have I got that explanation right, or am I missing something?

Think of mapping point on a plane to its corresponding pixel on the
canvas. I have some older code you might want to take a look at. It
shows this type of mapping. Look at ct_plane_project and ct_plane_plot.

https://pastebin.com/raw/5UU6vrhP

On 4/16/2023 2:00 PM, Dan joyce wrote:
> Each digit of pi treated as an integer,
[...]

Fwiw, here is one I did where each 10-ary digit maps to a multiple of
the base angle of pi*2/10. So, I simply multiply the digit value by the
base angle. My distance is .001, the origin of the plot is at point (0,
0). The circle in white is the unit circle centered at (0, 0), with a
radius of one to give you a sense of scale.

https://i.ibb.co/zh8MBYF/image.png

I can show you some higher level code if you are interested.

On 4/23/2023 2:31 PM, Chris M. Thomasson wrote:
> On 4/16/2023 2:00 PM, Dan joyce wrote:
>> Each digit of pi treated as an integer,
> [...]
>
> Fwiw, here is one I did where each 10-ary digit maps to a multiple of
> the base angle of pi*2/10. So, I simply multiply the digit value by the
> base angle. My distance is .001, the origin of the plot is at point (0,
> 0). The circle in white is the unit circle centered at (0, 0), with a
> radius of one to give you a sense of scale.
>
> https://i.ibb.co/zh8MBYF/image.png
>
> I can show you some higher level code if you are interested.

A zoom. Well, this sure looks fractal!

https://i.ibb.co/FKmH54k/image.png

On Sunday, April 23, 2023 at 5:31:20 PM UTC-4, Chris M. Thomasson wrote:
> On 4/16/2023 2:00 PM, Dan joyce wrote:
> > Each digit of pi treated as an integer,
> [...]
>
> Fwiw, here is one I did where each 10-ary digit maps to a multiple of
> the base angle of pi*2/10. So, I simply multiply the digit value by the
> base angle. My distance is .001, the origin of the plot is at point (0,
> 0). The circle in white is the unit circle centered at (0, 0), with a
> radius of one to give you a sense of scale.
>
> https://i.ibb.co/zh8MBYF/image.png
>
> I can show you some higher level code if you are interested.
How many digits of pi?
Its got to be a lot!

On Sunday, April 23, 2023 at 5:31:20 PM UTC-4, Chris M. Thomasson wrote:
> On 4/16/2023 2:00 PM, Dan joyce wrote:
> > Each digit of pi treated as an integer,
> [...]
>
> Fwiw, here is one I did where each 10-ary digit maps to a multiple of
> the base angle of pi*2/10. So, I simply multiply the digit value by the
> base angle. My distance is .001, the origin of the plot is at point (0,
> 0). The circle in white is the unit circle centered at (0, 0), with a
> radius of one to give you a sense of scale.
>
> https://i.ibb.co/zh8MBYF/image.png
>
> I can show you some higher level code if you are interested.

So basically, the two unit dia. circle is scaled up to what value for dia. x

On Sunday, April 23, 2023 at 5:52:26 PM UTC-4, Chris M. Thomasson wrote:
> On 4/23/2023 2:31 PM, Chris M. Thomasson wrote:
> > On 4/16/2023 2:00 PM, Dan joyce wrote:
> >> Each digit of pi treated as an integer,
> > [...]
> >
> > Fwiw, here is one I did where each 10-ary digit maps to a multiple of
> > the base angle of pi*2/10. So, I simply multiply the digit value by the
> > base angle. My distance is .001, the origin of the plot is at point (0,
> > 0). The circle in white is the unit circle centered at (0, 0), with a
> > radius of one to give you a sense of scale.
> >
> > https://i.ibb.co/zh8MBYF/image.png
> >
> > I can show you some higher level code if you are interested.
> A zoom. Well, this sure looks fractal!
>
> https://i.ibb.co/FKmH54k/image.png

Nice'.
How many digits of pi?
What direction criteria did you use for each digit of pi?

On 4/23/2023 3:56 PM, Dan joyce wrote:
> On Sunday, April 23, 2023 at 5:31:20 PM UTC-4, Chris M. Thomasson wrote:
>> On 4/16/2023 2:00 PM, Dan joyce wrote:
>>> Each digit of pi treated as an integer,
>> [...]
>>
>> Fwiw, here is one I did where each 10-ary digit maps to a multiple of
>> the base angle of pi*2/10. So, I simply multiply the digit value by the
>> base angle. My distance is .001, the origin of the plot is at point (0,
>> 0). The circle in white is the unit circle centered at (0, 0), with a
>> radius of one to give you a sense of scale.
>>
>> https://i.ibb.co/zh8MBYF/image.png
>>
>> I can show you some higher level code if you are interested.
> How many digits of pi?
> Its got to be a lot!

I think it was 10000000 / 2 digits. Fwiw, here is one with 10000000
digits of pi:

https://i.ibb.co/5jXYMK1/image.png

Notice how we can overlay it with the lower iteration plot?

On 4/23/2023 4:11 PM, Dan joyce wrote:
> On Sunday, April 23, 2023 at 5:52:26 PM UTC-4, Chris M. Thomasson wrote:
>> On 4/23/2023 2:31 PM, Chris M. Thomasson wrote:
>>> On 4/16/2023 2:00 PM, Dan joyce wrote:
>>>> Each digit of pi treated as an integer,
>>> [...]
>>>
>>> Fwiw, here is one I did where each 10-ary digit maps to a multiple of
>>> the base angle of pi*2/10. So, I simply multiply the digit value by the
>>> base angle. My distance is .001, the origin of the plot is at point (0,
>>> 0). The circle in white is the unit circle centered at (0, 0), with a
>>> radius of one to give you a sense of scale.
>>>
>>> https://i.ibb.co/zh8MBYF/image.png
>>>
>>> I can show you some higher level code if you are interested.
>> A zoom. Well, this sure looks fractal!
>>
>> https://i.ibb.co/FKmH54k/image.png
>
> Nice'.
> How many digits of pi?
> What direction criteria did you use for each digit of pi?

Each digit is mapped to pi*2/10, so:
__________________
0 = pi*2/10 * 0
1 = pi*2/10 * 1
2 = pi*2/10 * 2
3 = pi*2/10 * 3
4 = pi*2/10 * 4
5 = pi*2/10 * 5
6 = pi*2/10 * 6
7 = pi*2/10 * 7
8 = pi*2/10 * 8
9 = pi*2/10 * 9
__________________

Dan joyce <danj4084@gmail.com> writes:

> On Sunday, April 23, 2023 at 8:15:40 AM UTC-4, Ben Bacarisse wrote:
>> Dan joyce <danj...@gmail.com> writes:
>>
>> > On Saturday, April 22, 2023 at 5:57:25 PM UTC-4, Ben Bacarisse wrote:
>> >> Dan joyce <danj...@gmail.com> writes:
>> >>
>> >> > Okay, on the same page but now thinking of a walk with 10 directions
>> >> > going clockwise for each digit of pi and at each step.
>> >> I wondered about that too, so here's a quick picture for the first
>> >> million digits of pi:
>> >>
>> >> http://www.bsb.me.uk/tmp/pi-walk.png
>> >>
>> >> Cross-hairs show the origin, and the path is drawn without full opacity
>> >> so you can see the more visited areas. I think it's quite pretty.
>> >>
>> >> --
>> >> Ben.
>> > Nice.
>> > What is the pixel map size?
>> > Did you shrink down the pixel size to keep it on screen?
>> It's drawn with a system that does not use pixels until rendering (and
>> not at all if the output is a PDF). So the plot it not "pixel perfect"
>> -- it's a series of thin lines.
>>
> Interesting how it interprets one mapping (different length lines) and
> transposing it to pixels, another mapping when viewed as a rendering.
> Have I got that explanation right, or am I missing something?

I don't get what you are saying, but I'm not sure that understanding the
picture will be very helpful. All the lines are of length 1. Each one
starts where the last left off, but is drawn at an angle determined by
the nth digit. Here's the same drawing but "zoomed in" to the origin
(the drawing is about 50 units square):

http://www.bsb.me.uk/tmp/pi-walk-detail.png

--
Ben.

On Sun, 23 Apr 2023 11:30:28 -0700 (PDT), Dan joyce
<danj4084@gmail.com> wrote:

>A new rendition of pi's digits on the x axis only, for now.
>3 R
>1 L
>4 R
>1 L
>5 R
>How long will this line be after 1000000000 digits?

Over the course of 1 billion digits, x ranges from -63650 to 95278 and
ends up at 94475.

>Always a finite length no matter how many digits of p1 --->oo.

It is true that for any finite number of digits, the line will have
finite length. Whether the length has an upper bound as you increase
the number of digits is unknown.

It is entirely possible for the digits of pi to form a very very long
sequence of values alternating between large ones and small ones, such
8,3,9,2,7,4,9,3,8,0,.... whcih would cause x to run off in one
direction or other.

As an example, if you only process the first 999 million digits, x
never gets past 94,950 (reached the first time at digit 997,855,651).
When you process the next million digits, it moves to 94,952 at digit
999,738,251 and eventually hits 95,278 for the first time at digit
999,791,361. If you were to expand the processing to the next 100
million, the maximum x might very well change again. There is nothing
that prevents the maximum x from growing every time you process
another 100 million or 100 billion.

The fact that some statement is true about the first billion digits of
pi tells you very little about the validity of extending the statement
to additional digits.

You might want to look at the youtube video about the Polya
Conjecture. It makes an excellent point about conclusions based on a
small sample size. Yes, 1 billion digits is a very small sample of
the digits in pi.

--
Remove del for email

On Sunday, April 23, 2023 at 7:31:45 PM UTC-4, Barry Schwarz wrote:
> On Sun, 23 Apr 2023 11:30:28 -0700 (PDT), Dan joyce
> <danj...@gmail.com> wrote:
>
> >A new rendition of pi's digits on the x axis only, for now.
> >3 R
> >1 L
> >4 R
> >1 L
> >5 R
> >How long will this line be after 1000000000 digits?
> Over the course of 1 billion digits, x ranges from -63650 to 95278 and
> ends up at 94475.
> >Always a finite length no matter how many digits of p1 --->oo.
> It is true that for any finite number of digits, the line will have
> finite length. Whether the length has an upper bound as you increase
> the number of digits is unknown.
>
> It is entirely possible for the digits of pi to form a very very long
> sequence of values alternating between large ones and small ones, such
> 8,3,9,2,7,4,9,3,8,0,.... whcih would cause x to run off in one
> direction or other.
>
> As an example, if you only process the first 999 million digits, x
> never gets past 94,950 (reached the first time at digit 997,855,651).
> When you process the next million digits, it moves to 94,952 at digit
> 999,738,251 and eventually hits 95,278 for the first time at digit
> 999,791,361. If you were to expand the processing to the next 100
> million, the maximum x might very well change again. There is nothing
> that prevents the maximum x from growing every time you process
> another 100 million or 100 billion.
>
> The fact that some statement is true about the first billion digits of
> pi tells you very little about the validity of extending the statement
> to additional digits.
>
> You might want to look at the youtube video about the Polya
> Conjecture. It makes an excellent point about conclusions based on a
> small sample size. Yes, 1 billion digits is a very small sample of
> the digits in pi.
> --
> Remove del for email

Nice!
This line will never stop growing in length as pi's digits --->oo,
So could the argument be made, this line also --->oo in length but at --->oo slow rate?
My point is, where does --->oo really begin.
A conundrum for sure.

On Sunday, April 23, 2023 at 1:54:51 PM UTC-4, Dan joyce wrote:
> On Sunday, April 23, 2023 at 8:15:40 AM UTC-4, Ben Bacarisse wrote:
> > Dan joyce <danj...@gmail.com> writes:
> >
> > > On Saturday, April 22, 2023 at 5:57:25 PM UTC-4, Ben Bacarisse wrote:
> > >> Dan joyce <danj...@gmail.com> writes:
> > >>
> > >> > Okay, on the same page but now thinking of a walk with 10 directions
> > >> > going clockwise for each digit of pi and at each step.
> > >> I wondered about that too, so here's a quick picture for the first
> > >> million digits of pi:
> > >>
> > >> http://www.bsb.me.uk/tmp/pi-walk.png
> > >>
> > >> Cross-hairs show the origin, and the path is drawn without full opacity
> > >> so you can see the more visited areas. I think it's quite pretty.
> > >>
> > >> --
> > >> Ben.
> > > Nice.
> > > What is the pixel map size?
> > > Did you shrink down the pixel size to keep it on screen?
> > It's drawn with a system that does not use pixels until rendering (and
> > not at all if the output is a PDF). So the plot it not "pixel perfect"
> > -- it's a series of thin lines.
> >
> > --
> > Ben.
> Interesting how it interprets one mapping (different length lines) and transposing it to pixels,
> another mapping when viewed as a rendering.
> Have I got that explanation right, or am I missing something?
>
> Dan
A new rendition of pi's digits on the x axis only, for now.
3 R
1 L
4 R
1 L
5 R
How long will this line be after 1000000000 digits?
Always a finite length no matter how many digits of p1 --->oo.

On Sunday, April 23, 2023 at 7:31:45 PM UTC-4, Barry Schwarz wrote:
> On Sun, 23 Apr 2023 11:30:28 -0700 (PDT), Dan joyce
> <danj...@gmail.com> wrote:
>
> >A new rendition of pi's digits on the x axis only, for now.
> >3 R
> >1 L
> >4 R
> >1 L
> >5 R
> >How long will this line be after 1000000000 digits?
> Over the course of 1 billion digits, x ranges from -63650 to 95278 and
> ends up at 94475.
> >Always a finite length no matter how many digits of p1 --->oo.
> It is true that for any finite number of digits, the line will have
> finite length. Whether the length has an upper bound as you increase
> the number of digits is unknown.
>
> It is entirely possible for the digits of pi to form a very very long
> sequence of values alternating between large ones and small ones, such
> 8,3,9,2,7,4,9,3,8,0,.... whcih would cause x to run off in one
> direction or other.
>
> As an example, if you only process the first 999 million digits, x
> never gets past 94,950 (reached the first time at digit 997,855,651).
> When you process the next million digits, it moves to 94,952 at digit
> 999,738,251 and eventually hits 95,278 for the first time at digit
> 999,791,361. If you were to expand the processing to the next 100
> million, the maximum x might very well change again. There is nothing
> that prevents the maximum x from growing every time you process
> another 100 million or 100 billion.
>
> The fact that some statement is true about the first billion digits of
> pi tells you very little about the validity of extending the statement
> to additional digits.
>
> You might want to look at the youtube video about the Polya
> Conjecture. It makes an excellent point about conclusions based on a
> small sample size. Yes, 1 billion digits is a very small sample of
> the digits in pi.
> --
> Remove del for email

Yes, the Polya Conjecture is a good example.
It's like proposing a Conjecture saying the line stops growing in length after
n amounts of pi's digits are processed from R,L,R,L,R,L,R... for each digit of pi.
Only possibly then to fail after >n amounts are tested.

On Sunday, April 23, 2023 at 2:25:47 AM UTC-4, James Waldby wrote:
> Barry Schwarz <schw...@delq.com> wrote:
> > On Sun, 16 Apr 2023 14:00:00 -0700 (PDT), Dan joyce
> > <danj...@gmail.com> wrote:
> >
> >>Each digit of pi treated as an integer,
> >>Starting with 3 and x=0 and y=0.
> >>
> >>3 down x=0 y=-3
> >>1 left x=-1 y=-3
> >>4 up x=-1 y=1
> >>1 right x=0 y=1
> >>5 down x=0 y=-4
> ...
> >>Repeat that order of directions with each digit of pi.
> >>What will be the x/y coordinates on the Cartesian coordinate plain
> >>after 1,000,000 digits of pi?
> >>How many times will it cross the x=0 axis and y=0 axis or where an
> >>actual digit of pi ends up on x=0 and y=0?
> >>Above the 10th and 11th digit of pi x=0 but y=-7 and y=-2 respectfully
> ...
> > After the first billion digits in pi, we end up at (-22065,-140854).
> > The x position reaches extremes at -46684 and 20240.
> > The y position reaches extremes at -183016 and 12484.
> > We land on the x axis (y=0) 1636 times.
> > We land on the y axis (x=0) 20240 times.
> > The least frequent digit in pi is 3 with 99,986,912 occurrences.
> > The most frequent digit is 4 with 100,011,958 occurrences.
> My program's results agree with those max and min values, but both of
> my on-axis counts (Xon, Yon) are somewhat different (see top row of
> numbers below - other lines are for smaller numbers of digits, for
> possibly easier comparison of results). (The `Cross` numbers in
> following are lower bounds for numbers of axis crossings.)
>
> Stats for a billion digits of pi, a million, 100K, 10K, 1K, 100 --
> Xmin Xmax Cross Xon Ymin Ymax Cross Yon
> -46684 20240 58685 29687 -183016 12484 6953 3590
> -2076 439 1074 563 -752 1842 753 388
> -247 439 606 331 -526 588 176 66
> -142 135 111 73 -65 195 62 22
> -70 64 10 9 -39 40 49 18
> -1 30 1 5 -11 40 5 0
>
> Above stats snipped from output of variedExtremes1.jl but can also be
> gotten from output of piWalk1.jl, among programs appearing at links on
> page <https://gitlab.com/ghjwp7/piwalkstats> - jiw

Nice work, James.

On Mon, 24 Apr 2023 07:56:41 -0700 (PDT), Dan joyce
<danj4084@gmail.com> wrote:

>On Sunday, April 23, 2023 at 7:31:45?PM UTC-4, Barry Schwarz wrote:
>> On Sun, 23 Apr 2023 11:30:28 -0700 (PDT), Dan joyce
>> <danj...@gmail.com> wrote:
>>
>> >A new rendition of pi's digits on the x axis only, for now.
>> >3 R
>> >1 L
>> >4 R
>> >1 L
>> >5 R
>> >How long will this line be after 1000000000 digits?
>> Over the course of 1 billion digits, x ranges from -63650 to 95278 and
>> ends up at 94475.
>> >Always a finite length no matter how many digits of p1 --->oo.
>> It is true that for any finite number of digits, the line will have
>> finite length. Whether the length has an upper bound as you increase
>> the number of digits is unknown.
>>
>> It is entirely possible for the digits of pi to form a very very long
>> sequence of values alternating between large ones and small ones, such
>> 8,3,9,2,7,4,9,3,8,0,.... whcih would cause x to run off in one
>> direction or other.
>>
>> As an example, if you only process the first 999 million digits, x
>> never gets past 94,950 (reached the first time at digit 997,855,651).
>> When you process the next million digits, it moves to 94,952 at digit
>> 999,738,251 and eventually hits 95,278 for the first time at digit
>> 999,791,361. If you were to expand the processing to the next 100
>> million, the maximum x might very well change again. There is nothing
>> that prevents the maximum x from growing every time you process
>> another 100 million or 100 billion.
>>
>> The fact that some statement is true about the first billion digits of
>> pi tells you very little about the validity of extending the statement
>> to additional digits.
>>
>> You might want to look at the youtube video about the Polya
>> Conjecture. It makes an excellent point about conclusions based on a
>> small sample size. Yes, 1 billion digits is a very small sample of
>> the digits in pi.
>> --
>> Remove del for email
>
>Nice!
>This line will never stop growing in length as pi's digits --->oo,

This conclusion is also unjustified. There is simply no way of
knowing what the next 100 billion digits of pi are like.

At some point, x could start to oscillate around some value. Consider
the irrational number 0.101001000100001... If we process these digits
using the same rule and, for ease of viewing, use s for a zero move
starboard (right) and S for a one move starboard and p and P for port
(left) moves, we have
sPsPspSpspSpspsPspspsPspspspS... Apologies to Jimmy Buffet but that
is two steps left, two steps right, and repeat. In terms of x, it
runs from 0 to -1 to -2 to -1 to 0 and repeats. Infinite sequences of
moves MAY or MAY NOT progress arbitrarily far from the origin.

There is absolutely nothing that prevents the digits of pi from
forming such a pattern at some point in the decimal expansion. As
Polya demonstrates, the fact that it doesn't do so in the first
trillion digits tells nothing about what happens later.

>So could the argument be made, this line also --->oo in length but at --->oo slow rate?

Nope. We cannot draw any conclusion about what the data looks like
that we have not processed.

Consider the fact that within the first billion digits, each digit
appears with a frequency between 9.998% and 10.002%. Yet we have no
reason to conclude that the same will be true with the next billion
digits.

>My point is, where does --->oo really begin.

Since it has no end, why should it have a beginning?

>A conundrum for sure.

Maybe for philosophers but mathematics has very practical definitions
of what it means for a value to approach infinity. These definitions
frequently include the phrase "increases without bounds."

--
Remove del for email