> > nonprimes = <3 5 7 9 15 17 25 ..>
> that sequence looks familiar ...
>
>
>
> >
> > http://www.new-math.com/
>
> ahhh its my sequence
> >
> > non prime means non prime right? Or, is that a typo?

no i dont mean composites

a random distribution in 1/log(n)
adds to all evens (80% of time)

a random distribution in 2/log(n) AND odd(n)
adds to all evens (98% of time)

On Wednesday, April 19, 2023 at 11:00:32 PM UTC-4, Chris M. Thomasson wrote:
> On 4/19/2023 4:14 PM, Dan joyce wrote:
> > On Wednesday, April 19, 2023 at 6:56:26 PM UTC-4, Chris M. Thomasson wrote:
> >> On 4/19/2023 3:51 PM, Dan joyce wrote:
> >>> On Wednesday, April 19, 2023 at 6:43:43 PM UTC-4, Chris M. Thomasson wrote:
> >>>> On 4/19/2023 3:39 PM, Chris M. Thomasson wrote:
> >>>>> On 4/19/2023 3:23 PM, Dan joyce wrote:
> >>>>>> On Wednesday, April 19, 2023 at 6:07:35 PM UTC-4, Chris M. Thomasson
> >>>>>> wrote:
> >>>>>>> On 4/19/2023 2:34 PM, Dan joyce wrote:
> >>>>>>>> On Wednesday, April 19, 2023 at 4:10:57 PM UTC-4, Chris M. Thomasson
> >>>>>>>> wrote:
> >>>>>>>>> On 4/18/2023 7:11 PM, Chris M. Thomasson wrote:
> >>>>>>>>>> On 4/16/2023 2:00 PM, Dan joyce 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
> >>>>>>>>>>> 9 right x=9 y=-4
> >>>>>>>>>>> 2 up x=9 y=-2
> >>>>>>>>>>> 6 left x=3 y=-2
> >>>>>>>>>>> 5 D x=3 y=-7
> >>>>>>>>>>> 3 L x=0 y=-7
> >>>>>>>>>>> 5 U x=0 y=-2
> >>>>>>>>>>> 8 R x=8 y=-2
> >>>>>>>>>>> 9 D x=8 y=-11
> >>>>>>>>>>> 7 R x=15 y=-11
> >>>>>>>>>>> 9 U x=15 y=-2
> >>>>>>>>>>> 3 L x=12 y=-2
> >>>>>>>>>>> 2
> >>>>>>>>>>>
> >>>>>>>>>>> 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
> >>>>>>>>>>>
> >>>>>>>>>>> We know pi's digits --->oo but the Cartesian coordinate plain
> >>>>>>>>>>> will not
> >>>>>>>>>>> --->oo in any direction, in fact using the above method it will
> >>>>>>>>>>> cross
> >>>>>>>>>>> or land on the x=0 or y=0 --->oo.
> >>>>>>>>>>> I used this repeated order D,L,U,R,D,R,U,L to accommodate all the
> >>>>>>>>>>> single
> >>>>>>>>>>> numbers of pi 0,1,2,3,4,5,6,7,8,9 that gives a small repeated
> >>>>>>>>>>> pattern of joining lines but it takes 140 iterations to complete the
> >>>>>>>>>>> pattern.
> >>>>>>>>>>> Then it just retraces the lines in the next 140 iterations and so on
> >>>>>>>>>>> --->oo.
> >>>>>>>>>>>
> >>>>>>>>>>> etc.
> >>>>>>>>>>> The numbers above from 0-9 repeated for each direction to 140
> >>>>>>>>>>> iterations produces a unique 140 x\y coordinates and then repeats
> >>>>>>>>>>> that
> >>>>>>>>>>> same unique x\y coordinates in the next 140 iterations and so-on.
> >>>>>>>>>>> When a zero or zeros like in pi are encountered it does not draw a
> >>>>>>>>>>> line but uses only a direction change.
> >>>>>>>>> Wait a minute. I am missing where it "changes direction" when a zero
> >>>>>>>>> digit is encountered. You say x=0, y=0 so are you resetting the
> >>>>>>>>> origin,
> >>>>>>>>> or just treat the zero as a non-movement like encounter? In other
> >>>>>>>>> words
> >>>>>>>>> the current point is left alone when a zero digit is observed?
> >>>>>>>>
> >>>>>>>> Say a zero lands on the up direction after a digit >0 draws a line
> >>>>>>>> on the left direction but
> >>>>>>>> now a line is NOT drawn on the up direction because of the zero. So
> >>>>>>>> say the next integer
> >>>>>>>> after the zero is >o it just retraces the left direction but now
> >>>>>>>> going in the right direction over the left direction over laying
> >>>>>>>> either partially or same length or longer (a larger digit that
> >>>>>>>> traced over the left direction).
> >>>>>>>> When you consider the order of directions --- D,L,U,R,D,R,U,L repeat
> >>>>>>>> --->oo
> >>>>>>> [...]
> >>>>>>>
> >>>>>>> Is this close? Using only a 5-ary symbol system for simplicity:
> >>>>>>>
> >>>>>>> Say we define in a 2d plane the following normalized directions:
> >>>>>>>
> >>>>>>> neutral = { 0, 0 }
> >>>>>>> up = { 0, 1 }
> >>>>>>> down = { 0, -1 }
> >>>>>>> left = { -1, 0 }
> >>>>>>> right = { 1, 0 }
> >>>>>>>
> >>>>>>> where:
> >>>>>>>
> >>>>>>> 0 = neutral
> >>>>>>> 1 = up
> >>>>>>> 2 = down
> >>>>>>> 3 = left
> >>>>>>> 4 = right
> >>>>>>>
> >>>>>>>
> >>>>>>> So, a number
> >>>>>>>
> >>>>>>> 10133402
> >>>>>>>
> >>>>>>> If we started with a point z at point { 0, 0 },
> >>>>>>>
> >>>>>>> it would be:
> >>>>>>>
> >>>>>>> z = { 0, 0 }
> >>>>>>>
> >>>>>>> // iterate the digits
> >>>>>>>
> >>>>>>> 1: z = z + up = { 0, 1 }
> >>>>>>> 0: z = z + neutral = { 0, 1 }
> >>>>>>> 1: z = z + up = { 0, 2 }
> >>>>>>> 3: z = z + left = { -1, 2 }
> >>>>>>> 3: z = z + left = { -2, 2 }
> >>>>>>> 4: z = z + right = { -1, 2 }
> >>>>>>> 0: z = z + neutral = { -1, 2 }
> >>>>>>> 2: z = z + down = { -1, 1 }
> >>>>>>>
> >>>>>>> I only used a 5-ary set of symbols, but it would work.
> >>>>>>
> >>>>>> You are using the digit length as a direction where I am using the
> >>>>>> order of direction
> >>>>>> to point that next integer to ---- D,LU,R,D,R,U,L
> >>>>>
> >>>>> What is the LU? I think left and up. So z = z + left + up
> >>>>>
> >>>>> LU seems to be treated as a single digit with the commas in your
> >>>>> direction list. Or is LU a normal vector like:
> >>>>>
> >>>>
> >>>>
> >>>>> north_east = normalize({-1, 1}) ?
> >>>> ^^^^^^^^^^^^^^^^^^^^^^^^^^
> >>>> [..]
> >>>>
> >>>> Humm.. I don't think I should necessarily normalize the north_east vector.
> >>>>
> >>>> north_east = {-1, 1}
> >>>>
> >>>> Or, in this case I should define it as:
> >>>>
> >>>> left_up = {-1, 1}
> >>>>
> >>>> ;^)
> >>> You could also set a 10 sided n-gon drawing to the center of each line on the inner
> >>> circle that would give 10 directions from the end of the last known direction.
> >>> The naming convention for each direction could be, starting at the top line
> >>> going down the right side of the inner circle -- x, x1,x2 stopping at 1/4 circle
> >>> Then going down from x on the left top -- -x1, -x2 =5 directions for the top half.
> >>> On the bottom half right side-- y, y1, y2 left side bottom half-- -y1,-y2
> >>> Just another brainstorm.
> >> Humm... For some reason, these rules kind of remind me of playing the
> >> chaos game with an n-gon.
> >>
> >> https://en.wikipedia.org/wiki/Chaos_game
> >
> > There is no pictorial of 10 sided fractal?
> A ten sided fractal? Sure. One simple example is (z = z^11 + c) that
> obtains ten fold symmetry. You can create an n-gon and play the chaos
> game on it. You can plot a koch curve on the side of an n-gon. You can
> create 10 fold symmetry in a fractal many different ways. You can plot a
> vector field on 10-gons. Here is a two fold Julia set using one of my
> field line algorithms. This is not as chaotic as it might seem...
>
> https://i.ibb.co/r5P1sbL/image.png
weird!
> With regard to the chaos game, you could use digits of an irrational to
> choose the sides of a 10-gon to play the game with.
So why not pi?
The original graphics I sent you using each 1/pn (pn, = prime numbers) ,digits that formed a nice
repeated retrace after many digits of their decimal expansion. .
I am not sure how many angles (sides of a circle) are involved but I believe at least 10.
I will run pi in that algorithm and I am sure it will be Chadic but will it be a tighter image.
Meaning, after so many digits will it be closer to x=0 and y=0 the start of the plot then
the (4) 90 degree Brownian image pi plot I sent you?

On Thursday, April 20, 2023 at 2:06:07 PM UTC-4, Dan joyce wrote:
> On Wednesday, April 19, 2023 at 11:00:32 PM UTC-4, Chris M. Thomasson wrote:
> > On 4/19/2023 4:14 PM, Dan joyce wrote:
> > > On Wednesday, April 19, 2023 at 6:56:26 PM UTC-4, Chris M. Thomasson wrote:
> > >> On 4/19/2023 3:51 PM, Dan joyce wrote:
> > >>> On Wednesday, April 19, 2023 at 6:43:43 PM UTC-4, Chris M. Thomasson wrote:
> > >>>> On 4/19/2023 3:39 PM, Chris M. Thomasson wrote:
> > >>>>> On 4/19/2023 3:23 PM, Dan joyce wrote:
> > >>>>>> On Wednesday, April 19, 2023 at 6:07:35 PM UTC-4, Chris M. Thomasson
> > >>>>>> wrote:
> > >>>>>>> On 4/19/2023 2:34 PM, Dan joyce wrote:
> > >>>>>>>> On Wednesday, April 19, 2023 at 4:10:57 PM UTC-4, Chris M. Thomasson
> > >>>>>>>> wrote:
> > >>>>>>>>> On 4/18/2023 7:11 PM, Chris M. Thomasson wrote:
> > >>>>>>>>>> On 4/16/2023 2:00 PM, Dan joyce 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
> > >>>>>>>>>>> 9 right x=9 y=-4
> > >>>>>>>>>>> 2 up x=9 y=-2
> > >>>>>>>>>>> 6 left x=3 y=-2
> > >>>>>>>>>>> 5 D x=3 y=-7
> > >>>>>>>>>>> 3 L x=0 y=-7
> > >>>>>>>>>>> 5 U x=0 y=-2
> > >>>>>>>>>>> 8 R x=8 y=-2
> > >>>>>>>>>>> 9 D x=8 y=-11
> > >>>>>>>>>>> 7 R x=15 y=-11
> > >>>>>>>>>>> 9 U x=15 y=-2
> > >>>>>>>>>>> 3 L x=12 y=-2
> > >>>>>>>>>>> 2
> > >>>>>>>>>>>
> > >>>>>>>>>>> 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
> > >>>>>>>>>>>
> > >>>>>>>>>>> We know pi's digits --->oo but the Cartesian coordinate plain
> > >>>>>>>>>>> will not
> > >>>>>>>>>>> --->oo in any direction, in fact using the above method it will
> > >>>>>>>>>>> cross
> > >>>>>>>>>>> or land on the x=0 or y=0 --->oo.
> > >>>>>>>>>>> I used this repeated order D,L,U,R,D,R,U,L to accommodate all the
> > >>>>>>>>>>> single
> > >>>>>>>>>>> numbers of pi 0,1,2,3,4,5,6,7,8,9 that gives a small repeated
> > >>>>>>>>>>> pattern of joining lines but it takes 140 iterations to complete the
> > >>>>>>>>>>> pattern.
> > >>>>>>>>>>> Then it just retraces the lines in the next 140 iterations and so on
> > >>>>>>>>>>> --->oo.
> > >>>>>>>>>>>
> > >>>>>>>>>>> etc.
> > >>>>>>>>>>> The numbers above from 0-9 repeated for each direction to 140
> > >>>>>>>>>>> iterations produces a unique 140 x\y coordinates and then repeats
> > >>>>>>>>>>> that
> > >>>>>>>>>>> same unique x\y coordinates in the next 140 iterations and so-on.
> > >>>>>>>>>>> When a zero or zeros like in pi are encountered it does not draw a
> > >>>>>>>>>>> line but uses only a direction change.
> > >>>>>>>>> Wait a minute. I am missing where it "changes direction" when a zero
> > >>>>>>>>> digit is encountered. You say x=0, y=0 so are you resetting the
> > >>>>>>>>> origin,
> > >>>>>>>>> or just treat the zero as a non-movement like encounter? In other
> > >>>>>>>>> words
> > >>>>>>>>> the current point is left alone when a zero digit is observed?
> > >>>>>>>>
> > >>>>>>>> Say a zero lands on the up direction after a digit >0 draws a line
> > >>>>>>>> on the left direction but
> > >>>>>>>> now a line is NOT drawn on the up direction because of the zero. So
> > >>>>>>>> say the next integer
> > >>>>>>>> after the zero is >o it just retraces the left direction but now
> > >>>>>>>> going in the right direction over the left direction over laying
> > >>>>>>>> either partially or same length or longer (a larger digit that
> > >>>>>>>> traced over the left direction).
> > >>>>>>>> When you consider the order of directions --- D,L,U,R,D,R,U,L repeat
> > >>>>>>>> --->oo
> > >>>>>>> [...]
> > >>>>>>>
> > >>>>>>> Is this close? Using only a 5-ary symbol system for simplicity:
> > >>>>>>>
> > >>>>>>> Say we define in a 2d plane the following normalized directions:
> > >>>>>>>
> > >>>>>>> neutral = { 0, 0 }
> > >>>>>>> up = { 0, 1 }
> > >>>>>>> down = { 0, -1 }
> > >>>>>>> left = { -1, 0 }
> > >>>>>>> right = { 1, 0 }
> > >>>>>>>
> > >>>>>>> where:
> > >>>>>>>
> > >>>>>>> 0 = neutral
> > >>>>>>> 1 = up
> > >>>>>>> 2 = down
> > >>>>>>> 3 = left
> > >>>>>>> 4 = right
> > >>>>>>>
> > >>>>>>>
> > >>>>>>> So, a number
> > >>>>>>>
> > >>>>>>> 10133402
> > >>>>>>>
> > >>>>>>> If we started with a point z at point { 0, 0 },
> > >>>>>>>
> > >>>>>>> it would be:
> > >>>>>>>
> > >>>>>>> z = { 0, 0 }
> > >>>>>>>
> > >>>>>>> // iterate the digits
> > >>>>>>>
> > >>>>>>> 1: z = z + up = { 0, 1 }
> > >>>>>>> 0: z = z + neutral = { 0, 1 }
> > >>>>>>> 1: z = z + up = { 0, 2 }
> > >>>>>>> 3: z = z + left = { -1, 2 }
> > >>>>>>> 3: z = z + left = { -2, 2 }
> > >>>>>>> 4: z = z + right = { -1, 2 }
> > >>>>>>> 0: z = z + neutral = { -1, 2 }
> > >>>>>>> 2: z = z + down = { -1, 1 }
> > >>>>>>>
> > >>>>>>> I only used a 5-ary set of symbols, but it would work.
> > >>>>>>
> > >>>>>> You are using the digit length as a direction where I am using the
> > >>>>>> order of direction
> > >>>>>> to point that next integer to ---- D,LU,R,D,R,U,L
> > >>>>>
> > >>>>> What is the LU? I think left and up. So z = z + left + up
> > >>>>>
> > >>>>> LU seems to be treated as a single digit with the commas in your
> > >>>>> direction list. Or is LU a normal vector like:
> > >>>>>
> > >>>>
> > >>>>
> > >>>>> north_east = normalize({-1, 1}) ?
> > >>>> ^^^^^^^^^^^^^^^^^^^^^^^^^^
> > >>>> [..]
> > >>>>
> > >>>> Humm.. I don't think I should necessarily normalize the north_east vector.
> > >>>>
> > >>>> north_east = {-1, 1}
> > >>>>
> > >>>> Or, in this case I should define it as:
> > >>>>
> > >>>> left_up = {-1, 1}
> > >>>>
> > >>>> ;^)
> > >>> You could also set a 10 sided n-gon drawing to the center of each line on the inner
> > >>> circle that would give 10 directions from the end of the last known direction.
> > >>> The naming convention for each direction could be, starting at the top line
> > >>> going down the right side of the inner circle -- x, x1,x2 stopping at 1/4 circle
> > >>> Then going down from x on the left top -- -x1, -x2 =5 directions for the top half.
> > >>> On the bottom half right side-- y, y1, y2 left side bottom half-- -y1,-y2
> > >>> Just another brainstorm.
> > >> Humm... For some reason, these rules kind of remind me of playing the
> > >> chaos game with an n-gon.
> > >>
> > >> https://en.wikipedia.org/wiki/Chaos_game
> > >
> > > There is no pictorial of 10 sided fractal?
> > A ten sided fractal? Sure. One simple example is (z = z^11 + c) that
> > obtains ten fold symmetry. You can create an n-gon and play the chaos
> > game on it. You can plot a koch curve on the side of an n-gon. You can
> > create 10 fold symmetry in a fractal many different ways. You can plot a
> > vector field on 10-gons. Here is a two fold Julia set using one of my
> > field line algorithms. This is not as chaotic as it might seem...
> >
> > https://i.ibb.co/r5P1sbL/image.png
> weird!
> > With regard to the chaos game, you could use digits of an irrational to
> > choose the sides of a 10-gon to play the game with.
> So why not pi?
> The original graphics I sent you using each 1/pn (pn, = prime numbers) ,digits that formed a nice
> repeated retrace after many digits of their decimal expansion. .
> I am not sure how many angles (sides of a circle) are involved but I believe at least 10.
> I will run pi in that algorithm and I am sure it will be Chadic but will it be a tighter image.
> Meaning, after so many digits will it be closer to x=0 and y=0 the start of the plot then
> the (4) 90 degree Brownian image pi plot I sent you?
Where the hxll did I come up with that word ("Chadic") = chaotic

On Wednesday, April 19, 2023 at 8:04:07 PM UTC-7, Chris M. Thomasson wrote:
> On 4/19/2023 3:49 PM, mitchr...@gmail.com wrote:
> > On Wednesday, April 19, 2023 at 1:02:53 PM UTC-7, Chris M. Thomasson wrote:
> >> On 4/19/2023 10:36 AM, mitchr...@gmail.com wrote:
> >>> On Sunday, April 16, 2023 at 2:00:04 PM UTC-7, Dan joyce 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
> >>>> 9 right x=9 y=-4
> >>>> 2 up x=9 y=-2
> >>>> 6 left x=3 y=-2
> >>>> 5 D x=3 y=-7
> >>>> 3 L x=0 y=-7
> >>>> 5 U x=0 y=-2
> >>>> 8 R x=8 y=-2
> >>>> 9 D x=8 y=-11
> >>>> 7 R x=15 y=-11
> >>>> 9 U x=15 y=-2
> >>>> 3 L x=12 y=-2
> >>>> 2
> >>>>
> >>>> 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
> >>>>
> >>>> We know pi's digits --->oo but the Cartesian coordinate plain will not
> >>>> --->oo in any direction, in fact using the above method it will cross
> >>>> or land on the x=0 or y=0 --->oo.
> >>>>
> >>>> I used this repeated order D,L,U,R,D,R,U,L to accommodate all the single
> >>>> numbers of pi 0,1,2,3,4,5,6,7,8,9 that gives a small repeated
> >>>> pattern of joining lines but it takes 140 iterations to complete the pattern.
> >>>> Then it just retraces the lines in the next 140 iterations and so on --->oo.
> >>>>
> >>>> 0 D A zero so no change so x=0,y=0
> >>>> 1 L x=-1 y=0
> >>>> 2 U x=-1 y=2
> >>>> 3 R x=2 y=2
> >>>> 4 D x=2 y=-2
> >>>> 5 R x=7 y=-2
> >>>> 6 U x=7 y=4
> >>>> 7 L x=0 y=4
> >>>> 8 D x=0 y=-4
> >>>> 9 L x=-9 y=-4
> >>>> 0 U x=-9 y=-4
> >>>> 1 R x=-8 y=-4
> >>>> 2 D x=-8 y=-6
> >>>> etc.
> >>>> The numbers above from 0-9 repeated for each direction to 140 iterations produces a unique 140 x\y coordinates and then repeats that same unique x\y coordinates in the next 140 iterations and so-on.
> >>>> When a zero or zeros like in pi are encountered it does not draw a line but uses only a direction change.
> >>>>
> >>>>
> >>>> A simple concept, but interesting.
> >>>>
> >>>> Dan
> >>>
> >>> Digits are not very accurate.
> >> How many accurate base-10 symbols of pi can you generate?
> >
> > That is the right question. So How do you verify?
> > Can you prove you can verify more than a few?
> > where is your proof of PI accuracy?
> > There is no accurate PI formula.
> Well, there is atan(1) * 4 = pi
>
> ;^)
>
> Time to implement a cordic. We can get it accurate up to a large number
> of digits, but we cannot get all of them because they go on forever.

How do you verify your accuracy?
PI formulas do not have it either.

On Thursday, April 20, 2023 at 6:28:07 PM UTC-4, mitchr...@gmail.com wrote:
> On Wednesday, April 19, 2023 at 8:04:07 PM UTC-7, Chris M. Thomasson wrote:
> > On 4/19/2023 3:49 PM, mitchr...@gmail.com wrote:
> > > On Wednesday, April 19, 2023 at 1:02:53 PM UTC-7, Chris M. Thomasson wrote:
> > >> On 4/19/2023 10:36 AM, mitchr...@gmail.com wrote:
> > >>> On Sunday, April 16, 2023 at 2:00:04 PM UTC-7, Dan joyce 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
> > >>>> 9 right x=9 y=-4
> > >>>> 2 up x=9 y=-2
> > >>>> 6 left x=3 y=-2
> > >>>> 5 D x=3 y=-7
> > >>>> 3 L x=0 y=-7
> > >>>> 5 U x=0 y=-2
> > >>>> 8 R x=8 y=-2
> > >>>> 9 D x=8 y=-11
> > >>>> 7 R x=15 y=-11
> > >>>> 9 U x=15 y=-2
> > >>>> 3 L x=12 y=-2
> > >>>> 2
> > >>>>
> > >>>> 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
> > >>>>
> > >>>> We know pi's digits --->oo but the Cartesian coordinate plain will not
> > >>>> --->oo in any direction, in fact using the above method it will cross
> > >>>> or land on the x=0 or y=0 --->oo.
> > >>>>
> > >>>> I used this repeated order D,L,U,R,D,R,U,L to accommodate all the single
> > >>>> numbers of pi 0,1,2,3,4,5,6,7,8,9 that gives a small repeated
> > >>>> pattern of joining lines but it takes 140 iterations to complete the pattern.
> > >>>> Then it just retraces the lines in the next 140 iterations and so on --->oo.
> > >>>>
> > >>>> 0 D A zero so no change so x=0,y=0
> > >>>> 1 L x=-1 y=0
> > >>>> 2 U x=-1 y=2
> > >>>> 3 R x=2 y=2
> > >>>> 4 D x=2 y=-2
> > >>>> 5 R x=7 y=-2
> > >>>> 6 U x=7 y=4
> > >>>> 7 L x=0 y=4
> > >>>> 8 D x=0 y=-4
> > >>>> 9 L x=-9 y=-4
> > >>>> 0 U x=-9 y=-4
> > >>>> 1 R x=-8 y=-4
> > >>>> 2 D x=-8 y=-6
> > >>>> etc.
> > >>>> The numbers above from 0-9 repeated for each direction to 140 iterations produces a unique 140 x\y coordinates and then repeats that same unique x\y coordinates in the next 140 iterations and so-on.
> > >>>> When a zero or zeros like in pi are encountered it does not draw a line but uses only a direction change.
> > >>>>
> > >>>>
> > >>>> A simple concept, but interesting.
> > >>>>
> > >>>> Dan
> > >>>
> > >>> Digits are not very accurate.
> > >> How many accurate base-10 symbols of pi can you generate?
> > >
> > > That is the right question. So How do you verify?
> > > Can you prove you can verify more than a few?
> > > where is your proof of PI accuracy?
> > > There is no accurate PI formula.
> > Well, there is atan(1) * 4 = pi
> >
> > ;^)
> >
> > Time to implement a cordic. We can get it accurate up to a large number
> > of digits, but we cannot get all of them because they go on forever.
> How do you verify your accuracy?
> PI formulas do not have it either.

By the many different formulas for pi that produce the same outcome --->oo
What else could you possibly want?

On Sunday, April 16, 2023 at 11:00:04 PM UTC+2, Dan joyce 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
> 9 right x=9 y=-4
> 2 up x=9 y=-2
> 6 left x=3 y=-2
> 5 D x=3 y=-7
> 3 L x=0 y=-7
> 5 U x=0 y=-2
> 8 R x=8 y=-2
> 9 D x=8 y=-11
> 7 R x=15 y=-11
> 9 U x=15 y=-2
> 3 L x=12 y=-2
> 2
>
> 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
>
> We know pi's digits --->oo but the Cartesian coordinate plain will not
> --->oo in any direction, in fact using the above method it will cross
> or land on the x=0 or y=0 --->oo.
>
> I used this repeated order D,L,U,R,D,R,U,L to accommodate all the single
> numbers of pi 0,1,2,3,4,5,6,7,8,9 that gives a small repeated
> pattern of joining lines but it takes 140 iterations to complete the pattern.
> Then it just retraces the lines in the next 140 iterations and so on --->oo.
>
> 0 D A zero so no change so x=0,y=0
> 1 L x=-1 y=0
> 2 U x=-1 y=2
> 3 R x=2 y=2
> 4 D x=2 y=-2
> 5 R x=7 y=-2
> 6 U x=7 y=4
> 7 L x=0 y=4
> 8 D x=0 y=-4
> 9 L x=-9 y=-4
> 0 U x=-9 y=-4
> 1 R x=-8 y=-4
> 2 D x=-8 y=-6
> etc.
> The numbers above from 0-9 repeated for each direction to 140 iterations produces a unique 140 x\y coordinates and then repeats that same unique x\y coordinates in the next 140 iterations and so-on.
> When a zero or zeros like in pi are encountered it does not draw a line but uses only a direction change.
>
>
> A simple concept, but interesting.
>
> Dan

A bit like the horseshoe map.

https://i.imgur.com/LPUTO6m.png

https://www.quantamagazine.org/how-mathematicians-make-sense-of-chaos-20220302/

Dan joyce <danj4084@gmail.com> wrote:
> On Monday, April 17, 2023 at 8:58:20???PM UTC-4, Graham Cooper wrote:
>> On Tuesday, April 18, 2023 at 9:42:33???AM UTC+10, Dan joyce wrote:
>> > On Monday, April 17, 2023 at 2:09:51???AM UTC-4, James Waldby wrote:
>> > > Dan joyce <danj...@gmail.com> wrote:
>> > > > Each digit of pi treated as an integer,
>> > > > Starting with 3 and x=0 and y=0.
[snip about using pi's digits as step lengths on XY plane while
cycling thru directions D,L,U,R,D,R,U,L in turn] ...
>> > > > When a zero or zeros like in pi are encountered it does not draw a line but uses only a direction change.
>> > > More clearly, it draws a line of length 0 for digit 0.
>> > Right.

>> > > Here are some stats for the million-digit case:
>> > > Min Max Cross Zero Last @Orig
>> > > X: -2076 439 1074 563 -1648 1 #54073
>> > > Y: -752 1842 753 388 -574
....
>> > > Got pi to 100100009 digits, will process 100100000
>> > > Min Max Cross Zero Last @Orig
>> > > X: -14627 20240 7388 3841 -5522 1 #54073
>> > > Y: -33180 12484 6953 3590 -27829
....
>> > Pi's digits continue on --->oo but are bounded by x\y coordinates
>> > to never wander off in any direction --->oo on the Cartesians
>> > Coordinate plain. What's your thoughts on this?

Pi is thought to be simply normal ??? ie all digits and digit sequences
of equal likelihood, per [2] ??? and the process takes steps without
directional bias. So in the long run X and Y min and max excursions
should all be of the same order of magnitude. Generally, there's no
finite bound on how distant from the origin the trace may go as you
process more and more digits.

[Snip Graham Cooper remarks like "I dont believe in RANDOM WALK
HYPOTHESIS that he crosses 0,0 INF times."]

> I am just thinking of the average mean of the pi digits.
> My thoughts are it creates a bounded limit on the x\y coordinates.
> I could be dead wrong on my hypothesis.

Well, plain wrong, at least, unless you mean it's bounded if you bound
the number of digits, which is obviously true. An easy distance bound
on X and Y min or max for n digits is 10*n/2. But as noted in [1], in
an ordinary random walk the breadth of the walk grows like sqrt(n) for
n digits, which is not bounded unless n is bounded.

I put a program and a graph of X and Y min and max values up at
<https://gitlab.com/ghjwp7/piwalkstats>. To see the graph for X and Y
min and max values for a billion digits, click the qdata1000000000.png
link. To see the README with explanatory notes scroll down the
webpage. To see a program, click its link. - jiw

[1] <https://en.wikipedia.org/wiki/Random_walk>
[2] <https://en.wikipedia.org/wiki/Normal_number>

On Friday, April 21, 2023 at 4:04:26 PM UTC+10, James Waldby wrote:
> Dan joyce <danj...@gmail.com> wrote:
> > On Monday, April 17, 2023 at 8:58:20???PM UTC-4, Graham Cooper wrote:
> >> On Tuesday, April 18, 2023 at 9:42:33???AM UTC+10, Dan joyce wrote:
> >> > On Monday, April 17, 2023 at 2:09:51???AM UTC-4, James Waldby wrote:
> >> > > Dan joyce <danj...@gmail.com> wrote:
> >> > > > Each digit of pi treated as an integer,
> >> > > > Starting with 3 and x=0 and y=0.
> [snip about using pi's digits as step lengths on XY plane while
> cycling thru directions D,L,U,R,D,R,U,L in turn] ...
> >> > > > When a zero or zeros like in pi are encountered it does not draw a line but uses only a direction change.
> >> > > More clearly, it draws a line of length 0 for digit 0.
> >> > Right.
>
> >> > > Here are some stats for the million-digit case:
> >> > > Min Max Cross Zero Last @Orig
> >> > > X: -2076 439 1074 563 -1648 1 #54073
> >> > > Y: -752 1842 753 388 -574
> ...
> >> > > Got pi to 100100009 digits, will process 100100000
> >> > > Min Max Cross Zero Last @Orig
> >> > > X: -14627 20240 7388 3841 -5522 1 #54073
> >> > > Y: -33180 12484 6953 3590 -27829
> ...
> >> > Pi's digits continue on --->oo but are bounded by x\y coordinates
> >> > to never wander off in any direction --->oo on the Cartesians
> >> > Coordinate plain. What's your thoughts on this?
> Pi is thought to be simply normal ??? ie all digits and digit sequences
> of equal likelihood, per [2] ??? and the process takes steps without
> directional bias. So in the long run X and Y min and max excursions
> should all be of the same order of magnitude. Generally, there's no
> finite bound on how distant from the origin the trace may go as you
> process more and more digits.
>
> [Snip Graham Cooper remarks like "I dont believe in RANDOM WALK
> HYPOTHESIS that he crosses 0,0 INF times."]

an hypothesis that may be testable.

what % of million step random walks fail to cross 0,0 at all ?

AT Infinity the infinite cartesian plane is fully populated as its evenly distributed

but that doesn't prove it! there could be a discontinuity steps-->oo and steps=oo

On Friday, April 21, 2023 at 2:04:26 AM UTC-4, James Waldby wrote:
> Dan joyce <danj...@gmail.com> wrote:
> > On Monday, April 17, 2023 at 8:58:20???PM UTC-4, Graham Cooper wrote:
> >> On Tuesday, April 18, 2023 at 9:42:33???AM UTC+10, Dan joyce wrote:
> >> > On Monday, April 17, 2023 at 2:09:51???AM UTC-4, James Waldby wrote:
> >> > > Dan joyce <danj...@gmail.com> wrote:
> >> > > > Each digit of pi treated as an integer,
> >> > > > Starting with 3 and x=0 and y=0.
> [snip about using pi's digits as step lengths on XY plane while
> cycling thru directions D,L,U,R,D,R,U,L in turn] ...
> >> > > > When a zero or zeros like in pi are encountered it does not draw a line but uses only a direction change.
> >> > > More clearly, it draws a line of length 0 for digit 0.
> >> > Right.
>
> >> > > Here are some stats for the million-digit case:
> >> > > Min Max Cross Zero Last @Orig
> >> > > X: -2076 439 1074 563 -1648 1 #54073
> >> > > Y: -752 1842 753 388 -574
> ...
> >> > > Got pi to 100100009 digits, will process 100100000
> >> > > Min Max Cross Zero Last @Orig
> >> > > X: -14627 20240 7388 3841 -5522 1 #54073
> >> > > Y: -33180 12484 6953 3590 -27829
> ...
> >> > Pi's digits continue on --->oo but are bounded by x\y coordinates
> >> > to never wander off in any direction --->oo on the Cartesians
> >> > Coordinate plain. What's your thoughts on this?
> Pi is thought to be simply normal ??? ie all digits and digit sequences
> of equal likelihood, per [2] ??? and the process takes steps without
> directional bias. So in the long run X and Y min and max excursions
> should all be of the same order of magnitude. Generally, there's no
> finite bound on how distant from the origin the trace may go as you
> process more and more digits.
>
> [Snip Graham Cooper remarks like "I dont believe in RANDOM WALK
> HYPOTHESIS that he crosses 0,0 INF times."]
> > I am just thinking of the average mean of the pi digits.
> > My thoughts are it creates a bounded limit on the x\y coordinates.
> > I could be dead wrong on my hypothesis.
>
> Well, plain wrong, at least, unless you mean it's bounded if you bound
> the number of digits, which is obviously true. An easy distance bound
> on X and Y min or max for n digits is 10*n/2. But as noted in [1], in
> an ordinary random walk the breadth of the walk grows like sqrt(n) for
> n digits, which is not bounded unless n is bounded.
>
> I put a program and a graph of X and Y min and max values up at
> <https://gitlab.com/ghjwp7/piwalkstats>. To see the graph for X and Y
> min and max values for a billion digits, click the qdata1000000000.png
> link. To see the README with explanatory notes scroll down the
> webpage. To see a program, click its link. - jiw
>
> [1] <https://en.wikipedia.org/wiki/Random_walk>
> [2] <https://en.wikipedia.org/wiki/Normal_number>

In your random walk is the direction for the next digit predetermined?
Like in my walk ---
D,L,U,R,D,R,U,L... repeat that order.
3,1,4,1,5,9,2,6...
Zeros are treated only as a direction change for the next number >0 where the zero has no trace.
Multiple zeros in a row give multiple direction changes for the next number >0.
You give quite in-depth view of your 1,000,000,000 random walk.
What are your first few x\y coordinates of say the first 32 digits of pi starting with 3.
Just checking to see if we are on the same page of a random walk of pi.

Dan joyce <danj4084@gmail.com> wrote:
> On Friday, April 21, 2023 at 2:04:26???AM UTC-4, James Waldby wrote:
>> Dan joyce <danj...@gmail.com> wrote:
>> > On Monday, April 17, 2023 at 8:58:20???PM UTC-4, Graham Cooper wrote:
>> >> On Tuesday, April 18, 2023 at 9:42:33???AM UTC+10, Dan joyce wrote:
>> >> > On Monday, April 17, 2023 at 2:09:51???AM UTC-4, James Waldby wrote:
>> >> > > Dan joyce <danj...@gmail.com> wrote:
>> >> > > > Each digit of pi treated as an integer,
>> >> > > > Starting with 3 and x=0 and y=0.
>> [snip about using pi's digits as step lengths on XY plane while
>> cycling thru directions D,L,U,R,D,R,U,L in turn] ...
>> >> > > > When a zero or zeros like in pi are encountered it does not draw a line but uses only a direction change.
>> >> > > More clearly, it draws a line of length 0 for digit 0.
>> >> > Right.
>>
>> >> > > Here are some stats for the million-digit case:
>> >> > > Min Max Cross Zero Last @Orig
>> >> > > X: -2076 439 1074 563 -1648 1 #54073
>> >> > > Y: -752 1842 753 388 -574
>> ...
>> >> > > Got pi to 100100009 digits, will process 100100000
>> >> > > Min Max Cross Zero Last @Orig
>> >> > > X: -14627 20240 7388 3841 -5522 1 #54073
>> >> > > Y: -33180 12484 6953 3590 -27829
>> ...
>> >> > Pi's digits continue on --->oo but are bounded by x\y coordinates
>> >> > to never wander off in any direction --->oo on the Cartesians
>> >> > Coordinate plain. What's your thoughts on this?
>> Pi is thought to be simply normal ??? ie all digits and digit sequences
>> of equal likelihood, per [2] ??? and the process takes steps without
>> directional bias. So in the long run X and Y min and max excursions
>> should all be of the same order of magnitude. Generally, there's no
>> finite bound on how distant from the origin the trace may go as you
>> process more and more digits.
>>
>> [Snip Graham Cooper remarks like "I dont believe in RANDOM WALK
>> HYPOTHESIS that he crosses 0,0 INF times."]
>> > I am just thinking of the average mean of the pi digits.
>> > My thoughts are it creates a bounded limit on the x\y coordinates.
>> > I could be dead wrong on my hypothesis.
>>
>> Well, plain wrong, at least, unless you mean it's bounded if you bound
>> the number of digits, which is obviously true. An easy distance bound
>> on X and Y min or max for n digits is 10*n/2. But as noted in [1], in
>> an ordinary random walk the breadth of the walk grows like sqrt(n) for
>> n digits, which is not bounded unless n is bounded.
>>
>> I put a program and a graph of X and Y min and max values up at
>> <https://gitlab.com/ghjwp7/piwalkstats>. To see the graph for X and Y
>> min and max values for a billion digits, click the qdata1000000000.png
>> link. To see the README with explanatory notes scroll down the
>> webpage. To see a program, click its link. - jiw
>>
>> [1] <https://en.wikipedia.org/wiki/Random_walk>
>> [2] <https://en.wikipedia.org/wiki/Normal_number>
>
> In your random walk is the direction for the next digit predetermined?
> Like in my walk ---
> D,L,U,R,D,R,U,L... repeat that order.
> 3,1,4,1,5,9,2,6...

Right, just as stated in the programs and the README at <https://gitlab.com/ghjwp7/piwalkstats>.

> Zeros are treated only as a direction change for the next number >0 where the zero has no trace.

It seems to me that talking about "a direction change for the next
number >0 where the zero has no trace" is ambiguous and unclear. Just
say that directions are taken regularly in turn, in D,L,U,R,D,R,U,L
order, and that each digit controls the step length in that direction.

> Multiple zeros in a row give multiple direction changes for the next number >0.
> You give quite in-depth view of your 1,000,000,000 random walk.

> What are your first few x\y coordinates of say the first 32 digits
> of pi starting with 3. Just checking to see if we are on the same
> page of a random walk of pi.

Here is some output from "piWalk1.jl 35 35":

Got pi to 98 digits, will process 35
3.141592653589793238462643383279502...
1. 3 D x= 0 y= -3
2. 1 L x= -1 y= -3
3. 4 U x= -1 y= 1
4. 1 R x= 0 y= 1
5. 5 D x= 0 y= -4
6. 9 R x= 9 y= -4
7. 2 U x= 9 y= -2
8. 6 L x= 3 y= -2
9. 5 D x= 3 y= -7
10. 3 L x= 0 y= -7
11. 5 U x= 0 y= -2
12. 8 R x= 8 y= -2
13. 9 D x= 8 y=-11
14. 7 R x= 15 y=-11
15. 9 U x= 15 y= -2
16. 3 L x= 12 y= -2
17. 2 D x= 12 y= -4
18. 3 L x= 9 y= -4
19. 8 U x= 9 y= 4
20. 4 R x= 13 y= 4
21. 6 D x= 13 y= -2
22. 2 R x= 15 y= -2
23. 6 U x= 15 y= 4
24. 4 L x= 11 y= 4
25. 3 D x= 11 y= 1
26. 3 L x= 8 y= 1
27. 8 U x= 8 y= 9
28. 3 R x= 11 y= 9
29. 2 D x= 11 y= 7
30. 7 R x= 18 y= 7
31. 9 U x= 18 y= 16
32. 5 L x= 13 y= 16
33. 0 D x= 13 y= 16
34. 2 L x= 11 y= 16
35. 8 U x= 11 y= 24
Min Max Cross Zero Last @Orig
X: -1 18 1 5 11 0 #0
Y: -11 24 5 0 24

Graham Cooper <grahamcooper7@gmail.com> wrote:
> On Friday, April 21, 2023 at 4:04:26???PM UTC+10, James Waldby wrote:
>> Dan joyce <danj...@gmail.com> wrote:
>> > On Monday, April 17, 2023 at 8:58:20???PM UTC-4, Graham Cooper wrote:
>> >> On Tuesday, April 18, 2023 at 9:42:33???AM UTC+10, Dan joyce wrote:
>> >> > On Monday, April 17, 2023 at 2:09:51???AM UTC-4, James Waldby wrote:
>> >> > > Dan joyce <danj...@gmail.com> wrote:
....
>> >> > Pi's digits continue on --->oo but are bounded by x\y coordinates
>> >> > to never wander off in any direction --->oo on the Cartesians
>> >> > Coordinate plain. What's your thoughts on this?
>> Pi is thought to be simply normal - ie all digits and digit sequences
>> of equal likelihood, per [2] - and the process takes steps without
>> directional bias. So in the long run X and Y min and max excursions
>> should all be of the same order of magnitude. Generally, there's no
>> finite bound on how distant from the origin the trace may go as you
>> process more and more digits.
>>
>> [Snip Graham Cooper remarks like "I dont believe in RANDOM WALK
>> HYPOTHESIS that he crosses 0,0 INF times."]
>
> an hypothesis that may be testable.

Statistical analysis, as in [1], can address that question. I don't
know if there's a practical way to test its plausibility by
experiment. Based on how erratic the traces are in the billion digit
graph linked in [2], my programs would take days or weeks of
computation to give enough data for checking statistical prediction.

[1] <https://en.wikipedia.org/wiki/Random_walk>,
[2] <https://gitlab.com/ghjwp7/piwalkstats>

> what % of million step random walks fail to cross 0,0 at all ?

Don't know; if you are referring to "standard" random walks as in [1],
the percentage would be tiny (<<0.1%) but for the digit-step walks
considered in this thread, it could be huge (>1%).

> AT Infinity the infinite cartesian plane is fully populated as its
> evenly distributed but that doesn't prove it! there could be a
> discontinuity steps-->oo and steps=oo

Hazy notions about "at infinity" and "steps=oo" are harmful rather
than relevant. Can I assume you agree that it's practical to make
statistical predictions of how the walk pattern's bounds will grow as
n grows, and of the probability that any given (x,y) coordinate will
be encountered once, twice, ... during a walk of length n, and that if
you do the math, you will see that for any given value of k, for any
probability level p < 1, for any given (x,y), there exists a value of
n such that if you process n steps, the probability of hitting (x,y) k
or more times exceeds p ?

On Saturday, April 22, 2023 at 6:18:36 AM UTC+10, James Waldby wrote:
> Graham Cooper <graham...@gmail.com> wrote:
> > On Friday, April 21, 2023 at 4:04:26???PM UTC+10, James Waldby wrote:
> >> Dan joyce <danj...@gmail.com> wrote:
> >> > On Monday, April 17, 2023 at 8:58:20???PM UTC-4, Graham Cooper wrote:
> >> >> On Tuesday, April 18, 2023 at 9:42:33???AM UTC+10, Dan joyce wrote:
> >> >> > On Monday, April 17, 2023 at 2:09:51???AM UTC-4, James Waldby wrote:
> >> >> > > Dan joyce <danj...@gmail.com> wrote:
> ...
> >> >> > Pi's digits continue on --->oo but are bounded by x\y coordinates
> >> >> > to never wander off in any direction --->oo on the Cartesians
> >> >> > Coordinate plain. What's your thoughts on this?
> >> Pi is thought to be simply normal - ie all digits and digit sequences
> >> of equal likelihood, per [2] - and the process takes steps without
> >> directional bias. So in the long run X and Y min and max excursions
> >> should all be of the same order of magnitude. Generally, there's no
> >> finite bound on how distant from the origin the trace may go as you
> >> process more and more digits.
> >>
> >> [Snip Graham Cooper remarks like "I dont believe in RANDOM WALK
> >> HYPOTHESIS that he crosses 0,0 INF times."]
> >
> > an hypothesis that may be testable.
> Statistical analysis, as in [1], can address that question. I don't
> know if there's a practical way to test its plausibility by
> experiment. Based on how erratic the traces are in the billion digit
> graph linked in [2], my programs would take days or weeks of
> computation to give enough data for checking statistical prediction.
>
> [1] <https://en.wikipedia.org/wiki/Random_walk>,
> [2] <https://gitlab.com/ghjwp7/piwalkstats>
> > what % of million step random walks fail to cross 0,0 at all ?
> Don't know; if you are referring to "standard" random walks as in [1],
> the percentage would be tiny (<<0.1%) but for the digit-step walks
> considered in this thread, it could be huge (>1%).
> > AT Infinity the infinite cartesian plane is fully populated as its
> > evenly distributed but that doesn't prove it! there could be a
> > discontinuity steps-->oo and steps=oo
> Hazy notions about "at infinity" and "steps=oo" are harmful rather
> than relevant. Can I assume you agree that it's practical to make
> statistical predictions of how the walk pattern's bounds will grow as
> n grows, and of the probability that any given (x,y) coordinate will
> be encountered once, twice, ... during a walk of length n, and that if
> you do the math, you will see that for any given value of k, for any
> probability level p < 1, for any given (x,y), there exists a value of
> n such that if you process n steps, the probability of hitting (x,y) k
> or more times exceeds p ?

could be p=49% , 49.9%, 49.99% ..

its looks like 50% of the plane is being covered

and some walks there is 0 fill area about 0,0

HYPOTHESIS
prob( crossing 0,0 once ) = 50%
prob( crossing 0,0 twice ) = 25%
prob( crossing 0,0 thrice ) = 12.5%

On Friday, April 21, 2023 at 3:54:34 PM UTC-4, James Waldby wrote:
> Dan joyce <danj...@gmail.com> wrote:
> > On Friday, April 21, 2023 at 2:04:26???AM UTC-4, James Waldby wrote:
> >> Dan joyce <danj...@gmail.com> wrote:
> >> > On Monday, April 17, 2023 at 8:58:20???PM UTC-4, Graham Cooper wrote:
> >> >> On Tuesday, April 18, 2023 at 9:42:33???AM UTC+10, Dan joyce wrote:
> >> >> > On Monday, April 17, 2023 at 2:09:51???AM UTC-4, James Waldby wrote:
> >> >> > > Dan joyce <danj...@gmail.com> wrote:
> >> >> > > > Each digit of pi treated as an integer,
> >> >> > > > Starting with 3 and x=0 and y=0.
> >> [snip about using pi's digits as step lengths on XY plane while
> >> cycling thru directions D,L,U,R,D,R,U,L in turn] ...
> >> >> > > > When a zero or zeros like in pi are encountered it does not draw a line but uses only a direction change.
> >> >> > > More clearly, it draws a line of length 0 for digit 0.
> >> >> > Right.
> >>
> >> >> > > Here are some stats for the million-digit case:
> >> >> > > Min Max Cross Zero Last @Orig
> >> >> > > X: -2076 439 1074 563 -1648 1 #54073
> >> >> > > Y: -752 1842 753 388 -574
> >> ...
> >> >> > > Got pi to 100100009 digits, will process 100100000
> >> >> > > Min Max Cross Zero Last @Orig
> >> >> > > X: -14627 20240 7388 3841 -5522 1 #54073
> >> >> > > Y: -33180 12484 6953 3590 -27829
> >> ...
> >> >> > Pi's digits continue on --->oo but are bounded by x\y coordinates
> >> >> > to never wander off in any direction --->oo on the Cartesians
> >> >> > Coordinate plain. What's your thoughts on this?
> >> Pi is thought to be simply normal ??? ie all digits and digit sequences
> >> of equal likelihood, per [2] ??? and the process takes steps without
> >> directional bias. So in the long run X and Y min and max excursions
> >> should all be of the same order of magnitude. Generally, there's no
> >> finite bound on how distant from the origin the trace may go as you
> >> process more and more digits.
> >>
> >> [Snip Graham Cooper remarks like "I dont believe in RANDOM WALK
> >> HYPOTHESIS that he crosses 0,0 INF times."]
> >> > I am just thinking of the average mean of the pi digits.
> >> > My thoughts are it creates a bounded limit on the x\y coordinates.
> >> > I could be dead wrong on my hypothesis.
> >>
> >> Well, plain wrong, at least, unless you mean it's bounded if you bound
> >> the number of digits, which is obviously true. An easy distance bound
> >> on X and Y min or max for n digits is 10*n/2. But as noted in [1], in
> >> an ordinary random walk the breadth of the walk grows like sqrt(n) for
> >> n digits, which is not bounded unless n is bounded.
> >>
> >> I put a program and a graph of X and Y min and max values up at
> >> <https://gitlab.com/ghjwp7/piwalkstats>. To see the graph for X and Y
> >> min and max values for a billion digits, click the qdata1000000000.png
> >> link. To see the README with explanatory notes scroll down the
> >> webpage. To see a program, click its link. - jiw
> >>
> >> [1] <https://en.wikipedia.org/wiki/Random_walk>
> >> [2] <https://en.wikipedia.org/wiki/Normal_number>
> >
> > In your random walk is the direction for the next digit predetermined?
> > Like in my walk ---
> > D,L,U,R,D,R,U,L... repeat that order.
> > 3,1,4,1,5,9,2,6...
> Right, just as stated in the programs and the README at <https://gitlab.com/ghjwp7/piwalkstats>.
> > Zeros are treated only as a direction change for the next number >0 where the zero has no trace.
> It seems to me that talking about "a direction change for the next
> number >0 where the zero has no trace" is ambiguous and unclear. Just
> say that directions are taken regularly in turn, in D,L,U,R,D,R,U,L
> order, and that each digit controls the step length in that direction.
> > Multiple zeros in a row give multiple direction changes for the next number >0.
> > You give quite in-depth view of your 1,000,000,000 random walk.
>
> > What are your first few x\y coordinates of say the first 32 digits
> > of pi starting with 3. Just checking to see if we are on the same
> > page of a random walk of pi.
> Here is some output from "piWalk1.jl 35 35":
>
> Got pi to 98 digits, will process 35
> 3.141592653589793238462643383279502...
> 1. 3 D x= 0 y= -3
> 2. 1 L x= -1 y= -3
> 3. 4 U x= -1 y= 1
> 4. 1 R x= 0 y= 1
> 5. 5 D x= 0 y= -4
> 6. 9 R x= 9 y= -4
> 7. 2 U x= 9 y= -2
> 8. 6 L x= 3 y= -2
> 9. 5 D x= 3 y= -7
> 10. 3 L x= 0 y= -7
> 11. 5 U x= 0 y= -2
> 12. 8 R x= 8 y= -2
> 13. 9 D x= 8 y=-11
> 14. 7 R x= 15 y=-11
> 15. 9 U x= 15 y= -2
> 16. 3 L x= 12 y= -2
> 17. 2 D x= 12 y= -4
> 18. 3 L x= 9 y= -4
> 19. 8 U x= 9 y= 4
> 20. 4 R x= 13 y= 4
> 21. 6 D x= 13 y= -2
> 22. 2 R x= 15 y= -2
> 23. 6 U x= 15 y= 4
> 24. 4 L x= 11 y= 4
> 25. 3 D x= 11 y= 1
> 26. 3 L x= 8 y= 1
> 27. 8 U x= 8 y= 9
> 28. 3 R x= 11 y= 9
> 29. 2 D x= 11 y= 7
> 30. 7 R x= 18 y= 7
> 31. 9 U x= 18 y= 16
> 32. 5 L x= 13 y= 16
> 33. 0 D x= 13 y= 16
> 34. 2 L x= 11 y= 16
> 35. 8 U x= 11 y= 24
> Min Max Cross Zero Last @Orig
> X: -1 18 1 5 11 0 #0
> Y: -11 24 5 0 24
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.

Dan joyce <danj4084@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.

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
....

Graham Cooper <grahamcooper7@gmail.com> wrote:
> On Saturday, April 22, 2023 at 6:18:36???AM UTC+10, James Waldby wrote:
>> Graham Cooper <graham...@gmail.com> wrote:
>> > On Friday, April 21, 2023 at 4:04:26???PM UTC+10, James Waldby wrote:
>> >> Dan joyce <danj...@gmail.com> wrote:
>> >> > On Monday, April 17, 2023 at 8:58:20???PM UTC-4, Graham Cooper wrote:
>> >> >> On Tuesday, April 18, 2023 at 9:42:33???AM UTC+10, Dan joyce wrote:
>> >> >> > On Monday, April 17, 2023 at 2:09:51???AM UTC-4, James Waldby wrote:
>> >> >> > > Dan joyce <danj...@gmail.com> wrote:
>> ...
>> >> >> > Pi's digits continue on --->oo but are bounded by x\y coordinates
>> >> >> > to never wander off in any direction --->oo on the Cartesians
>> >> >> > Coordinate plain. What's your thoughts on this?
>> >> Pi is thought to be simply normal - ie all digits and digit sequences
>> >> of equal likelihood, per [2] - and the process takes steps without
>> >> directional bias. So in the long run X and Y min and max excursions
>> >> should all be of the same order of magnitude. Generally, there's no
>> >> finite bound on how distant from the origin the trace may go as you
>> >> process more and more digits.
>> >>
>> >> [Snip Graham Cooper remarks like "I dont believe in RANDOM WALK
>> >> HYPOTHESIS that he crosses 0,0 INF times."]
>> >
>> > an hypothesis that may be testable.
>> Statistical analysis, as in [1], can address that question. I don't
>> know if there's a practical way to test its plausibility by
>> experiment. Based on how erratic the traces are in the billion digit
>> graph linked in [2], my programs would take days or weeks of
>> computation to give enough data for checking statistical prediction.
>>
>> [1] <https://en.wikipedia.org/wiki/Random_walk>,
>> [2] <https://gitlab.com/ghjwp7/piwalkstats>
>> > what % of million step random walks fail to cross 0,0 at all ?
>> Don't know; if you are referring to "standard" random walks as in [1],
>> the percentage would be tiny (<<0.1%) but for the digit-step walks
>> considered in this thread, it could be huge (>1%).
>> > AT Infinity the infinite cartesian plane is fully populated as its
>> > evenly distributed but that doesn't prove it! there could be a
>> > discontinuity steps-->oo and steps=oo

>> Hazy notions about "at infinity" and "steps=oo" are harmful rather
>> than relevant. Can I assume you agree that it's practical to make
>> statistical predictions of how the walk pattern's bounds will grow as
>> n grows, and of the probability that any given (x,y) coordinate will
>> be encountered once, twice, ... during a walk of length n, and that if
>> you do the math, you will see that for any given value of k, for any
>> probability level p < 1, for any given (x,y), there exists a value of
>> n such that if you process n steps, the probability of hitting (x,y) k
>> or more times exceeds p ?

> could be p=49% , 49.9%, 49.99% ..
>
> its looks like 50% of the plane is being covered
>
> and some walks there is 0 fill area about 0,0
>
> HYPOTHESIS
> prob( crossing 0,0 once ) = 50%
> prob( crossing 0,0 twice ) = 25%
> prob( crossing 0,0 thrice ) = 12.5%

Here are counts (histogram data) for 1, 2, 3...50 crossings, for
unit-length steps in random directions -- that is, a more-or-less
"standard" random walk, vs one that cycles thru directions and uses
digits for step lengths. This is from 100000 each million-step walks,
of which 81.1% at some point returned to origin, and 18.9% did not.

Histo data: [15143, 12465, 10315, 8123, 6705, 5619, 4333, 3530, 2843,
2350, 1817, 1547, 1224, 971, 817, 636, 508, 452, 323, 279, 235, 201,
130, 120, 88, 75, 51, 39, 41, 28, 31, 17, 13, 14, 7, 5, 8, 5, 3, 1, 1,
4, 6, 1, 1, 1, 0, 0, 0, 0]

This data suggests (for million-step walks of this kind) probabilities
of about 18%, 15%, 12.5%, 10%, 8%,... for 0, 1, 2, 3, 4... returns to
origin. Program zeroMillions.jl is available on my gitlab site, along
with more data outputs in file results-piWalk1.
<https://gitlab.com/ghjwp7/piwalkstats>

My earlier predictions of the numbers of not-returning-to-origin walks
obviously were way too small... - jiw

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.

--
Ben.

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.

"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

--
Ben.

sim of 4 steps and 6 steps

DDDD
DDDL
DDDU
DDDR
DDLD
DDLL
DDLU
DDLR
DDUD
DDUL
DDUU ---- 1
DDUR
DDRD
DDRL
DDRU
DDRR
DLDD
DLDL
DLDU
DLDR
DLLD
DLLL
DLLU
DLLR
DLUD
DLUL
DLUU
DLUR ---- 2
DLRD
DLRL
DLRU ---- 3
DLRR
DUDD
DUDL
DUDU ---- 4
DUDR
DULD
DULL
DULU
DULR ---- 5
DUUD ---- 6
DUUL
DUUU
DUUR
DURD
DURL ---- 7
DURU
DURR
DRDD
DRDL
DRDU
DRDR
DRLD
DRLL
DRLU ---- 8
DRLR
DRUD
DRUL ---- 9
DRUU
DRUR
DRRD
DRRL
DRRU
DRRR
LDDD
LDDL
LDDU
LDDR
LDLD
LDLL
LDLU
LDLR
LDUD
LDUL
LDUU
LDUR ---- 10
LDRD
LDRL
LDRU ---- 11
LDRR
LLDD
LLDL
LLDU
LLDR
LLLD
LLLL
LLLU
LLLR
LLUD
LLUL
LLUU
LLUR
LLRD
LLRL
LLRU
LLRR ---- 12
LUDD
LUDL
LUDU
LUDR ---- 13
LULD
LULL
LULU
LULR
LUUD
LUUL
LUUU
LUUR
LURD ---- 14
LURL
LURU
LURR
LRDD
LRDL
LRDU ---- 15
LRDR
LRLD
LRLL
LRLU
LRLR ---- 16
LRUD ---- 17
LRUL
LRUU
LRUR
LRRD
LRRL ---- 18
LRRU
LRRR
UDDD
UDDL
UDDU ---- 19
UDDR
UDLD
UDLL
UDLU
UDLR ---- 20
UDUD ---- 21
UDUL
UDUU
UDUR
UDRD
UDRL ---- 22
UDRU
UDRR
ULDD
ULDL
ULDU
ULDR ---- 23
ULLD
ULLL
ULLU
ULLR
ULUD
ULUL
ULUU
ULUR
ULRD ---- 24
ULRL
ULRU
ULRR
UUDD ---- 25
UUDL
UUDU
UUDR
UULD
UULL
UULU
UULR
UUUD
UUUL
UUUU
UUUR
UURD
UURL
UURU
UURR
URDD
URDL ---- 26
URDU
URDR
URLD ---- 27
URLL
URLU
URLR
URUD
URUL
URUU
URUR
URRD
URRL
URRU
URRR
RDDD
RDDL
RDDU
RDDR
RDLD
RDLL
RDLU ---- 28
RDLR
RDUD
RDUL ---- 29
RDUU
RDUR
RDRD
RDRL
RDRU
RDRR
RLDD
RLDL
RLDU ---- 30
RLDR
RLLD
RLLL
RLLU
RLLR ---- 31
RLUD ---- 32
RLUL
RLUU
RLUR
RLRD
RLRL ---- 33
RLRU
RLRR
RUDD
RUDL ---- 34
RUDU
RUDR
RULD ---- 35
RULL
RULU
RULR
RUUD
RUUL
RUUU
RUUR
RURD
RURL
RURU
RURR
RRDD
RRDL
RRDU
RRDR
RRLD
RRLL ---- 36
RRLU
RRLR
RRUD
RRUL
RRUU
RRUR
RRRD
RRRL
RRRU
RRRR

4 steps
36/256

6 steps
400/4096

need RANDOM walk to plot RETURN TO 0,0 much further

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.

"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.

--
Ben.

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. Perhaps you can compare your 40th x,y
location to mine, which is (1.71, -1.45) for digit = 7. I show my
first 40 locations at the end of file README.rst and the plot is
walkPlot1000000S.png, both accessible via links on my gitlab page at
<https://gitlab.com/ghjwp7/piwalkstats> or as follows.
<https://gitlab.com/ghjwp7/piwalkstats/-/blob/master/README.rst>
<https://gitlab.com/ghjwp7/piwalkstats/-/blob/master/walkPlot1000000S.png>

I don't know how to control opacity in julia's Plots, or how to use a
spectrum in Plots, but approximated color-progression via red, brown,
orange, green, lime, cyan, blue for successive sevenths of the plot.
- jiw

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
>9 right x=9 y=-4
>2 up x=9 y=-2
>6 left x=3 y=-2
>5 D x=3 y=-7
>3 L x=0 y=-7
>5 U x=0 y=-2
>8 R x=8 y=-2
>9 D x=8 y=-11
>7 R x=15 y=-11
>9 U x=15 y=-2
>3 L x=12 y=-2
>2
>
>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
>
>We know pi's digits --->oo but the Cartesian coordinate plain will not
>--->oo in any direction, in fact using the above method it will cross
>or land on the x=0 or y=0 --->oo.

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.

--
Remove del for email

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.

--
Ben.

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?