In the first billion digits of pi:

The longest ascending sequential sequence is 012345678.
The longest descending sequential sequence is 987654321 (or
2109876543 if you allow wrapping).
The longest self-similar sequence is 31415926

The longest constant sequence is 10 sixes. Sequences of ones,
sevens, eights, and nines max out at 9 digits. Sequences of zeros,
twos, threes, fours, and fives max out at 8 digits.

--
Remove del for email

On Wednesday, April 26, 2023 at 10:25:38 PM UTC+2, Dan joyce wrote:
> On Wednesday, April 26, 2023 at 9:56:54 AM UTC-4, sobriquet wrote:
> > On Tuesday, April 25, 2023 at 11:18:49 PM UTC+2, sobriquet wrote:
> > > On Tuesday, April 25, 2023 at 11:09:09 PM UTC+2, Dan joyce wrote:
> > > > On Tuesday, April 25, 2023 at 4:38:44 PM UTC-4, sobriquet wrote:
> > > > > On Tuesday, April 25, 2023 at 10:29:41 PM UTC+2, Dan joyce wrote:
> > > > > > On Tuesday, April 25, 2023 at 4:22:38 PM UTC-4, sobriquet wrote:
> > > > > > > On Tuesday, April 25, 2023 at 10:15:11 PM UTC+2, sobriquet wrote:
> > > > > > > > On Tuesday, April 25, 2023 at 7:28:04 PM UTC+2, Dan joyce wrote:
> > > > > > > > > On Tuesday, April 25, 2023 at 7:06:00 AM UTC-4, FromTheRafters wrote:
> > > > > > > > > > Dan joyce laid this down on his screen :
> > > > > > > > > > > On Monday, April 24, 2023 at 6:02:37 PM UTC-4, Barry Schwarz wrote:
> > > > > > > > > > >> On Mon, 24 Apr 2023 14:11:52 -0700 (PDT), Dan joyce
> > > > > > > > > > >> <danj...@gmail.com> wrote:
> > > > > > > > > > >>> On Monday, April 24, 2023 at 3:01:25?PM UTC-4, Barry Schwarz wrote:
> > > > > > > > > > >>>> On Mon, 24 Apr 2023 07:56:41 -0700 (PDT), Dan joyce
> > > > > > > > > > >>>> <danj...@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
> > > > > > > > > > >>>
> > > > > > > > > > >>> So in conclusion Barry, does this line increase in length without bounds
> > > > > > > > > > >>> as pi's decimal digits transposed into integers --->oo, or at this point,
> > > > > > > > > > >>> just a conjecture?
> > > > > > > > > > >>> Your thoughts?
> > > > > > > > > > >> It is indeed a conjecture and we have no idea if it is true or not.
> > > > > > > > > > >> And at the moment, I think we don't even have an idea how to prove it
> > > > > > > > > > >> one way of the other.
> > > > > > > > > > >> --
> > > > > > > > > > >> Remove del for email
> > > > > > > > > > >
> > > > > > > > > > > I believe this conjecture may never be proven true or false.
> > > > > > > > > > > And I will add, in my life time. I am 88 so the above statement is not too
> > > > > > > > > > > far fetched.
> > > > > > > > > > > Kind of a cool conjecture though!
> > > > > > > > > > It reminds me of Ulam's spiral of primes, though I don't know exactly
> > > > > > > > > > why. Maybe only because it is a visual representation of an interesting
> > > > > > > > > > set of numbers. How different do other interesting numbers' (I'll call
> > > > > > > > > > these mappings 'stamps') such as e or Phi look? Does your conjecture
> > > > > > > > > > seem to also apply to these?
> > > > > > > > > Phi, e and many other mathematical constants whos decimal expansion
> > > > > > > > > appears random would also apply.
> > > > > > > > > I haven't tested them but why not?
> > > > > > > > >
> > > > > > > > > The primes are a different breed ---
> > > > > > > > > The third column is the final number -x +x where the running totals
> > > > > > > > > of the second column is the abs line length starting with 2 ----- 3,5,7,11,13...
> > > > > > > > > 2+-3 =-1
> > > > > > > > > -1+ 5 = 4
> > > > > > > > > 4+-7 = -3
> > > > > > > > > -3+11= 8
> > > > > > > > > 8+-13=-5
> > > > > > > > > -5+17=12
> > > > > > > > > 12+-19=-7
> > > > > > > > > -7 + 23=16
> > > > > > > > > 16+-29=-13
> > > > > > > > > -13+31= 18
> > > > > > > > > 18+-37= -19
> > > > > > > > > -17+41= 24
> > > > > > > > > 24+-43=-19
> > > > > > > > > -19+47= 28
> > > > > > > > > 28+-53=-25
> > > > > > > > > A line +x\-x (third column above) ever extending in both directions on the x axis as the
> > > > > > > > > primes --->oo so does the length of this line.(abs second column)
> > > > > > > > You can have a single irrational number that encodes all the primes.. like this binary
> > > > > > > > expansion, sum for all primes P of (2^-P):
> > > > > > > >
> > > > > > > > 0.0110101000101000101000100000101...
> > > > > > > >
> > > > > > > > https://www.wolframalpha.com/input?i=Sum%5B1%2F2%5E%28Prime%5Bx%5D%29%2C+x%5D+
> > > > > > > >
> > > > > > > > https://www.wolframalpha.com/input?i=binary+0.414683
> > > > > > > I'm confused why wolfram alpha claims the sum diverges, since it's obviously just a particular
> > > > > > > number irrational number which has bits set for prime positions in the binary expansion.
> > > > > > Interesting how a (short) rational produces a binary irrational..
> > > > > It's an irrational number (regardless of representing it in decimal or binary expansion, but
> > > > > in binary expansion the prime number pattern is clearly visible).
> > > > I am not following about the prime number pattern, please explain?
> > > If we have a binary number between 0 and 1 with a binary expansion, we can label the
> > > positions of the digits that are set to 1 with a set of positive natural numbers.
> > >
> > > So for instance, a binary number with digits {2,3,5,7,11,13,17,19,23,29,31,37,..} set would look like:
> > >
> > > 0.0110101000101000101000100000101000001...
> > >
> > > So basically the number is the sum of the reciprocals of prime powers of 2.
> > > Which in binary representation visually clearly shows the prime positions of digits in
> > > the binary expansion.
> > https://en.wikipedia.org/wiki/Prime_constant
> Nice
> Where I got it wrong was the first post of yours where I stated a (short) rational number creating
> an irrational but that was wrong, thanks to Wolframs Alpha, when you corrected your post then things changed as did the binary sequence and the irrational connected to it.
> What is interesting is, drawing a horizontal line using the binary sequence where the first zero after
> the decimal point dose not draw the line but using only a right hand position then the next 1
> representing 2 draws a unit length left. Then the next 1 following the 1 representing 3 and draws
> a line to the right. The odd number of zeros between each 1 (prime) sets up a steady line going
> right for each 1 on the x axis. All it does is give a prime count in the length of the line -1.
> First 100 digits = 24 1 unit lengths. So 1 less in the count of 25 because of the 3(1) retracing over the 2(1). Simple, I know, but duplicating the alternating right\left pi's digits in the same way only with pi
> the line grows in 2 directions and this line actually starts @ x=-1 and then grows to the right --->oo.
> Counting the primes along the way less1 in unit line length.
> Setting up now to see how the irrational number that created the binary sequence on how that line
> goes using alternating directions for each digit.
> I am even looking at the Fibonacci sequence where the odd numbered gaps of zeros will dictate the same
> direction from the last term to the next term going in the same direction.. Even numbered gaps of zeros
> will reverse the direction from the previous Fibonacci number. Will this line be longer then the last term
> processed?
> Thanks for your input.

On Wednesday, April 26, 2023 at 8:30:30 PM UTC-4, sobriquet wrote:
> On Wednesday, April 26, 2023 at 10:25:38 PM UTC+2, Dan joyce wrote:
> > On Wednesday, April 26, 2023 at 9:56:54 AM UTC-4, sobriquet wrote:
> > > On Tuesday, April 25, 2023 at 11:18:49 PM UTC+2, sobriquet wrote:
> > > > On Tuesday, April 25, 2023 at 11:09:09 PM UTC+2, Dan joyce wrote:
> > > > > On Tuesday, April 25, 2023 at 4:38:44 PM UTC-4, sobriquet wrote:
> > > > > > On Tuesday, April 25, 2023 at 10:29:41 PM UTC+2, Dan joyce wrote:
> > > > > > > On Tuesday, April 25, 2023 at 4:22:38 PM UTC-4, sobriquet wrote:
> > > > > > > > On Tuesday, April 25, 2023 at 10:15:11 PM UTC+2, sobriquet wrote:
> > > > > > > > > On Tuesday, April 25, 2023 at 7:28:04 PM UTC+2, Dan joyce wrote:
> > > > > > > > > > On Tuesday, April 25, 2023 at 7:06:00 AM UTC-4, FromTheRafters wrote:
> > > > > > > > > > > Dan joyce laid this down on his screen :
> > > > > > > > > > > > On Monday, April 24, 2023 at 6:02:37 PM UTC-4, Barry Schwarz wrote:
> > > > > > > > > > > >> On Mon, 24 Apr 2023 14:11:52 -0700 (PDT), Dan joyce
> > > > > > > > > > > >> <danj...@gmail.com> wrote:
> > > > > > > > > > > >>> On Monday, April 24, 2023 at 3:01:25?PM UTC-4, Barry Schwarz wrote:
> > > > > > > > > > > >>>> On Mon, 24 Apr 2023 07:56:41 -0700 (PDT), Dan joyce
> > > > > > > > > > > >>>> <danj...@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
> > > > > > > > > > > >>>
> > > > > > > > > > > >>> So in conclusion Barry, does this line increase in length without bounds
> > > > > > > > > > > >>> as pi's decimal digits transposed into integers --->oo, or at this point,
> > > > > > > > > > > >>> just a conjecture?
> > > > > > > > > > > >>> Your thoughts?
> > > > > > > > > > > >> It is indeed a conjecture and we have no idea if it is true or not.
> > > > > > > > > > > >> And at the moment, I think we don't even have an idea how to prove it
> > > > > > > > > > > >> one way of the other.
> > > > > > > > > > > >> --
> > > > > > > > > > > >> Remove del for email
> > > > > > > > > > > >
> > > > > > > > > > > > I believe this conjecture may never be proven true or false.
> > > > > > > > > > > > And I will add, in my life time. I am 88 so the above statement is not too
> > > > > > > > > > > > far fetched.
> > > > > > > > > > > > Kind of a cool conjecture though!
> > > > > > > > > > > It reminds me of Ulam's spiral of primes, though I don't know exactly
> > > > > > > > > > > why. Maybe only because it is a visual representation of an interesting
> > > > > > > > > > > set of numbers. How different do other interesting numbers' (I'll call
> > > > > > > > > > > these mappings 'stamps') such as e or Phi look? Does your conjecture
> > > > > > > > > > > seem to also apply to these?
> > > > > > > > > > Phi, e and many other mathematical constants whos decimal expansion
> > > > > > > > > > appears random would also apply.
> > > > > > > > > > I haven't tested them but why not?
> > > > > > > > > >
> > > > > > > > > > The primes are a different breed ---
> > > > > > > > > > The third column is the final number -x +x where the running totals
> > > > > > > > > > of the second column is the abs line length starting with 2 ----- 3,5,7,11,13...
> > > > > > > > > > 2+-3 =-1
> > > > > > > > > > -1+ 5 = 4
> > > > > > > > > > 4+-7 = -3
> > > > > > > > > > -3+11= 8
> > > > > > > > > > 8+-13=-5
> > > > > > > > > > -5+17=12
> > > > > > > > > > 12+-19=-7
> > > > > > > > > > -7 + 23=16
> > > > > > > > > > 16+-29=-13
> > > > > > > > > > -13+31= 18
> > > > > > > > > > 18+-37= -19
> > > > > > > > > > -17+41= 24
> > > > > > > > > > 24+-43=-19
> > > > > > > > > > -19+47= 28
> > > > > > > > > > 28+-53=-25
> > > > > > > > > > A line +x\-x (third column above) ever extending in both directions on the x axis as the
> > > > > > > > > > primes --->oo so does the length of this line.(abs second column)
> > > > > > > > > You can have a single irrational number that encodes all the primes.. like this binary
> > > > > > > > > expansion, sum for all primes P of (2^-P):
> > > > > > > > >
> > > > > > > > > 0.0110101000101000101000100000101...
> > > > > > > > >
> > > > > > > > > https://www.wolframalpha.com/input?i=Sum%5B1%2F2%5E%28Prime%5Bx%5D%29%2C+x%5D+
> > > > > > > > >
> > > > > > > > > https://www.wolframalpha.com/input?i=binary+0.414683
> > > > > > > > I'm confused why wolfram alpha claims the sum diverges, since it's obviously just a particular
> > > > > > > > number irrational number which has bits set for prime positions in the binary expansion.
> > > > > > > Interesting how a (short) rational produces a binary irrational.
> > > > > > It's an irrational number (regardless of representing it in decimal or binary expansion, but
> > > > > > in binary expansion the prime number pattern is clearly visible).
> > > > > I am not following about the prime number pattern, please explain?
> > > > If we have a binary number between 0 and 1 with a binary expansion, we can label the
> > > > positions of the digits that are set to 1 with a set of positive natural numbers.
> > > >
> > > > So for instance, a binary number with digits {2,3,5,7,11,13,17,19,23,29,31,37,..} set would look like:
> > > >
> > > > 0.0110101000101000101000100000101000001...
> > > >
> > > > So basically the number is the sum of the reciprocals of prime powers of 2.
> > > > Which in binary representation visually clearly shows the prime positions of digits in
> > > > the binary expansion.
> > > https://en.wikipedia.org/wiki/Prime_constant
> > Nice
> > Where I got it wrong was the first post of yours where I stated a (short) rational number creating
> > an irrational but that was wrong, thanks to Wolframs Alpha, when you corrected your post then things changed as did the binary sequence and the irrational connected to it.
> > What is interesting is, drawing a horizontal line using the binary sequence where the first zero after
> > the decimal point dose not draw the line but using only a right hand position then the next 1
> > representing 2 draws a unit length left. Then the next 1 following the 1 representing 3 and draws
> > a line to the right. The odd number of zeros between each 1 (prime) sets up a steady line going
> > right for each 1 on the x axis. All it does is give a prime count in the length of the line -1.
> > First 100 digits = 24 1 unit lengths. So 1 less in the count of 25 because of the 3(1) retracing over the 2(1). Simple, I know, but duplicating the alternating right\left pi's digits in the same way only with pi
> > the line grows in 2 directions and this line actually starts @ x=-1 and then grows to the right --->oo.
> > Counting the primes along the way less1 in unit line length.
> > Setting up now to see how the irrational number that created the binary sequence on how that line
> > goes using alternating directions for each digit.
> > I am even looking at the Fibonacci sequence where the odd numbered gaps of zeros will dictate the same
> > direction from the last term to the next term going in the same direction. Even numbered gaps of zeros
> > will reverse the direction from the previous Fibonacci number. Will this line be longer then the last term
> > processed?
> > Thanks for your input.
>
> I dunno.. I'm kind of interested in a conceptual approach to randomness, like how randomness
> can be exploited to generate pleasing visual patterns (as one might argue that ordered patterns
> can often be somewhat boring in comparison).
>
> Like random color combinations:
>
> https://i.imgur.com/jUMw0JC.jpg
>
> Or random shapes:
>
> https://i.imgur.com/Mjl3prk.jpg
>
> So I'm wondering to what degree we can more or less extract the random aspect from
> such things like irrational numbers and recontextualize that, like in an aperiodic tiling.
>
> https://i.imgur.com/uDgSxcu.jpg
>
> https://i.imgur.com/1gQVRAO.png
>
> https://www.youtube.com/watch?v=nzsAIRontAA

On 4/20/2023 4:47 PM, Dan joyce wrote:
> 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?

Exactly. When dealing with pi, my first question is what are the
precision requirements for the task at hand?

On Thursday, April 27, 2023 at 4:19:18 PM UTC-4, Chris M. Thomasson wrote:
> On 4/20/2023 4:47 PM, Dan joyce wrote:
> > 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?
>
> Exactly. When dealing with pi, my first question is what are the
> precision requirements for the task at hand?

A good example given by NSSA
The most distant spacecraft from Earth is Voyager 1. It is about 12.5 billion miles away. [This answer was from 4 years back, and now Voyager 1 is over 13.8 billion miles away. ed.] Let’s say we have a circle with a radius of exactly that size (or 25 billion miles in diameter) and we want to calculate the circumference, which is pi times the radius times 2. Using pi rounded to the 15th decimal, as I gave above, that comes out to a little more than 78 billion miles. We don’t need to be concerned here with exactly what the value is (you can multiply it out if you like) but rather what the error in the value is by not using more digits of pi. In other words, by cutting pi off at the 15th decimal point, we would calculate a circumference for that circle that is very slightly off. It turns out that our calculated circumference of the 25 billion mile diameter circle would be wrong by 1.5 inches. Think about that. We have a circle more than 78 billion miles around, and our calculation of that distance would be off by perhaps less than the length of your little finger.

Chris M. Thomasson has brought this to us :
> On 4/20/2023 4:47 PM, Dan joyce wrote:
>> 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?
>
> Exactly. When dealing with pi, my first question is what are the precision
> requirements for the task at hand?

Pi equals three, that's probably good enough.

On 4/27/2023 1:58 PM, FromTheRafters wrote:
> Chris M. Thomasson has brought this to us :
>> On 4/20/2023 4:47 PM, Dan joyce wrote:
>>> 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?
>>
>> Exactly. When dealing with pi, my first question is what are the
>> precision requirements for the task at hand?
>
> Pi equals three, that's probably good enough.

What about using some convergent of continued fractions:

22/7

Good enough? ;^)

Chris M. Thomasson submitted this idea :
> On 4/27/2023 1:58 PM, FromTheRafters wrote:
>> Chris M. Thomasson has brought this to us :
>>> On 4/20/2023 4:47 PM, Dan joyce wrote:
>>>> 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?
>>>
>>> Exactly. When dealing with pi, my first question is what are the precision
>>> requirements for the task at hand?
>>
>> Pi equals three, that's probably good enough.
>
> What about using some convergent of continued fractions:
>
> 22/7
>
> Good enough? ;^)

Three is easier to store. :D

On Thursday, April 27, 2023 at 5:17:02 PM UTC-4, FromTheRafters wrote:
> Chris M. Thomasson submitted this idea :
> > On 4/27/2023 1:58 PM, FromTheRafters wrote:
> >> Chris M. Thomasson has brought this to us :
> >>> On 4/20/2023 4:47 PM, Dan joyce wrote:
> >>>> 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?
> >>>
> >>> Exactly. When dealing with pi, my first question is what are the precision
> >>> requirements for the task at hand?
> >>
> >> Pi equals three, that's probably good enough.
> >
> > What about using some convergent of continued fractions:
> >
> > 22/7
> >
> > Good enough? ;^)
> Three is easier to store. :D

I don't know what we are discussing about pi?
Some here believe it is not a legitimate number.
Are they really serious, when so many mathematical formulas are based on pi..
wikipedia.org/wiki/List_of_formulae_involving_π

Dan joyce formulated the question :
> On Thursday, April 27, 2023 at 5:17:02 PM UTC-4, FromTheRafters wrote:
>> Chris M. Thomasson submitted this idea :
>>> On 4/27/2023 1:58 PM, FromTheRafters wrote:
>>>> Chris M. Thomasson has brought this to us :
>>>>> On 4/20/2023 4:47 PM, Dan joyce wrote:
>>>>>> 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?
>>>>>
>>>>> Exactly. When dealing with pi, my first question is what are the
>>>>> precision requirements for the task at hand?
>>>>
>>>> Pi equals three, that's probably good enough.
>>>
>>> What about using some convergent of continued fractions:
>>>
>>> 22/7
>>>
>>> Good enough? ;^)
>> Three is easier to store. :D
>
> I don't know what we are discussing about pi?
> Some here believe it is not a legitimate number.

Yeah, we call them cranks. :)

> Are they really serious, when so many mathematical formulas are based on pi.
> wikipedia.org/wiki/List_of_formulae_involving_π

Unfortunately, some actually believe what they post, and others are
just playing.

On Thu, 27 Apr 2023 13:46:25 -0700 (PDT), Dan joyce
<danj4084@gmail.com> wrote:

>A good example given by NSSA
>The most distant spacecraft from Earth is Voyager 1. It is about 12.5 billion miles away. [This answer was from 4 years back,
>and now Voyager 1 is over 13.8 billion miles away. ed.] Let’s say we have a circle with a radius of exactly that size
>(or 25 billion miles in diameter) and we want to calculate the circumference, which is pi times the radius times 2. Using pi rounded
>to the 15th decimal, as I gave above, that comes out to a little more than 78 billion miles. We don’t need to be concerned here with
>exactly what the value is (you can multiply it out if you like) but rather what the error in the value is by not using more digits of pi.
>In other words, by cutting pi off at the 15th decimal point, we would calculate a circumference for that circle that is very slightly off.
>It turns out that our calculated circumference of the 25 billion mile diameter circle would be wrong by 1.5 inches.
>Think about that. We have a circle more than 78 billion miles around, and our calculation of
>that distance would be off by perhaps less than the length of your little finger.

Your arithmetic appears a little off.

To 50 decimal places,
pi_1 = 3.14159265358979323846264338327950288419716939937511
and
25E9 * pi_1 = 78539816339.74483096156608458198757210492923498437775

To 15 decimal places,
pi_2 = 3.141592653589793
and
25E9 * pi_2 = 78539816339.744825

The absolute value of the difference between the two products (in
miles) is 5.96156608458198757210492923498437775E-6

Multiplying by 5280 to get feet
produces 0.03147706892659289438071402636071751452

Multiplying by 12 to get inches
produces 0.37772482711911473256856831632861017424
which is roughly 4 times smaller than the value you got.

BUT ALL OF THAT IS IRRELEVANT.

When dealing with precision, the answer is only as precise as the
worst input. The fact that we have umpteen decimal places of pi
cannot compensate for the fact that the 25 billion mile diameter
is precise to only -9 decimal places. Therefore, the product cannot
be any more precise than -9 decimal places. In order to keep the
precision correct, we can only say the circumference is 78 billion.
(or perhaps 77 billion).

Now if NASA could tell us the craft was 25,000,000,010 miles away, you
could argue for better precision. This is very similar to one of your
first posts about races where you had position specified down to the
width of a hydrogen atom.

--
Remove del for email

On Thursday, April 27, 2023 at 8:00:37 PM UTC-4, Barry Schwarz wrote:
> On Thu, 27 Apr 2023 13:46:25 -0700 (PDT), Dan joyce
> <danj...@gmail.com> wrote:
>
> >A good example given by NSSA
> >The most distant spacecraft from Earth is Voyager 1. It is about 12.5 billion miles away. [This answer was from 4 years back,
> >and now Voyager 1 is over 13.8 billion miles away. ed.] Let’s say we have a circle with a radius of exactly that size
> >(or 25 billion miles in diameter) and we want to calculate the circumference, which is pi times the radius times 2. Using pi rounded
> >to the 15th decimal, as I gave above, that comes out to a little more than 78 billion miles. We don’t need to be concerned here with
> >exactly what the value is (you can multiply it out if you like) but rather what the error in the value is by not using more digits of pi.
> >In other words, by cutting pi off at the 15th decimal point, we would calculate a circumference for that circle that is very slightly off.
> >It turns out that our calculated circumference of the 25 billion mile diameter circle would be wrong by 1.5 inches.
> >Think about that. We have a circle more than 78 billion miles around, and our calculation of
> >that distance would be off by perhaps less than the length of your little finger.
> Your arithmetic appears a little off.
>
> To 50 decimal places,
> pi_1 = 3.14159265358979323846264338327950288419716939937511
> and
> 25E9 * pi_1 = 78539816339.74483096156608458198757210492923498437775
>
> To 15 decimal places,
> pi_2 = 3.141592653589793
> and
> 25E9 * pi_2 = 78539816339.744825
>
> The absolute value of the difference between the two products (in
> miles) is 5.96156608458198757210492923498437775E-6
>
> Multiplying by 5280 to get feet
> produces 0.03147706892659289438071402636071751452
>
> Multiplying by 12 to get inches
> produces 0.37772482711911473256856831632861017424
> which is roughly 4 times smaller than the value you got.
>
> BUT ALL OF THAT IS IRRELEVANT.
>
> When dealing with precision, the answer is only as precise as the
> worst input. The fact that we have umpteen decimal places of pi
> cannot compensate for the fact that the 25 billion mile diameter
> is precise to only -9 decimal places. Therefore, the product cannot
> be any more precise than -9 decimal places. In order to keep the
> precision correct, we can only say the circumference is 78 billion.
> (or perhaps 77 billion).
>
> Now if NASA could tell us the craft was 25,000,000,010 miles away, you
> could argue for better precision. This is very similar to one of your
> first posts about races where you had position specified down to the
> width of a hydrogen atom.
> --
> Remove del for email
I just quoted that info on pi form a web page about NASA calculations
for pi. Not sure who wrote it but apparently it is wrong.
Thanks for the correction.
I saw somewhere that they only used like 8 or 10 digits of pi for their moon
landing coordinates.

On Wednesday, April 26, 2023 at 4:25:38 PM UTC-4, Dan joyce wrote:
> On Wednesday, April 26, 2023 at 9:56:54 AM UTC-4, sobriquet wrote:
> > On Tuesday, April 25, 2023 at 11:18:49 PM UTC+2, sobriquet wrote:
> > > On Tuesday, April 25, 2023 at 11:09:09 PM UTC+2, Dan joyce wrote:
> > > > On Tuesday, April 25, 2023 at 4:38:44 PM UTC-4, sobriquet wrote:
> > > > > On Tuesday, April 25, 2023 at 10:29:41 PM UTC+2, Dan joyce wrote:
> > > > > > On Tuesday, April 25, 2023 at 4:22:38 PM UTC-4, sobriquet wrote:
> > > > > > > On Tuesday, April 25, 2023 at 10:15:11 PM UTC+2, sobriquet wrote:
> > > > > > > > On Tuesday, April 25, 2023 at 7:28:04 PM UTC+2, Dan joyce wrote:
> > > > > > > > > On Tuesday, April 25, 2023 at 7:06:00 AM UTC-4, FromTheRafters wrote:
> > > > > > > > > > Dan joyce laid this down on his screen :
> > > > > > > > > > > On Monday, April 24, 2023 at 6:02:37 PM UTC-4, Barry Schwarz wrote:
> > > > > > > > > > >> On Mon, 24 Apr 2023 14:11:52 -0700 (PDT), Dan joyce
> > > > > > > > > > >> <danj...@gmail.com> wrote:
> > > > > > > > > > >>> On Monday, April 24, 2023 at 3:01:25?PM UTC-4, Barry Schwarz wrote:
> > > > > > > > > > >>>> On Mon, 24 Apr 2023 07:56:41 -0700 (PDT), Dan joyce
> > > > > > > > > > >>>> <danj...@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
> > > > > > > > > > >>>
> > > > > > > > > > >>> So in conclusion Barry, does this line increase in length without bounds
> > > > > > > > > > >>> as pi's decimal digits transposed into integers --->oo, or at this point,
> > > > > > > > > > >>> just a conjecture?
> > > > > > > > > > >>> Your thoughts?
> > > > > > > > > > >> It is indeed a conjecture and we have no idea if it is true or not.
> > > > > > > > > > >> And at the moment, I think we don't even have an idea how to prove it
> > > > > > > > > > >> one way of the other.
> > > > > > > > > > >> --
> > > > > > > > > > >> Remove del for email
> > > > > > > > > > >
> > > > > > > > > > > I believe this conjecture may never be proven true or false.
> > > > > > > > > > > And I will add, in my life time. I am 88 so the above statement is not too
> > > > > > > > > > > far fetched.
> > > > > > > > > > > Kind of a cool conjecture though!
> > > > > > > > > > It reminds me of Ulam's spiral of primes, though I don't know exactly
> > > > > > > > > > why. Maybe only because it is a visual representation of an interesting
> > > > > > > > > > set of numbers. How different do other interesting numbers' (I'll call
> > > > > > > > > > these mappings 'stamps') such as e or Phi look? Does your conjecture
> > > > > > > > > > seem to also apply to these?
> > > > > > > > > Phi, e and many other mathematical constants whos decimal expansion
> > > > > > > > > appears random would also apply.
> > > > > > > > > I haven't tested them but why not?
> > > > > > > > >
> > > > > > > > > The primes are a different breed ---
> > > > > > > > > The third column is the final number -x +x where the running totals
> > > > > > > > > of the second column is the abs line length starting with 2 ----- 3,5,7,11,13...
> > > > > > > > > 2+-3 =-1
> > > > > > > > > -1+ 5 = 4
> > > > > > > > > 4+-7 = -3
> > > > > > > > > -3+11= 8
> > > > > > > > > 8+-13=-5
> > > > > > > > > -5+17=12
> > > > > > > > > 12+-19=-7
> > > > > > > > > -7 + 23=16
> > > > > > > > > 16+-29=-13
> > > > > > > > > -13+31= 18
> > > > > > > > > 18+-37= -19
> > > > > > > > > -17+41= 24
> > > > > > > > > 24+-43=-19
> > > > > > > > > -19+47= 28
> > > > > > > > > 28+-53=-25
> > > > > > > > > A line +x\-x (third column above) ever extending in both directions on the x axis as the
> > > > > > > > > primes --->oo so does the length of this line.(abs second column)
> > > > > > > > You can have a single irrational number that encodes all the primes.. like this binary
> > > > > > > > expansion, sum for all primes P of (2^-P):
> > > > > > > >
> > > > > > > > 0.0110101000101000101000100000101...
> > > > > > > >
> > > > > > > > https://www.wolframalpha.com/input?i=Sum%5B1%2F2%5E%28Prime%5Bx%5D%29%2C+x%5D+
> > > > > > > >
> > > > > > > > https://www.wolframalpha.com/input?i=binary+0.414683
> > > > > > > I'm confused why wolfram alpha claims the sum diverges, since it's obviously just a particular
> > > > > > > number irrational number which has bits set for prime positions in the binary expansion.
> > > > > > Interesting how a (short) rational produces a binary irrational..
> > > > > It's an irrational number (regardless of representing it in decimal or binary expansion, but
> > > > > in binary expansion the prime number pattern is clearly visible).
> > > > I am not following about the prime number pattern, please explain?
> > > If we have a binary number between 0 and 1 with a binary expansion, we can label the
> > > positions of the digits that are set to 1 with a set of positive natural numbers.
> > >
> > > So for instance, a binary number with digits {2,3,5,7,11,13,17,19,23,29,31,37,..} set would look like:
> > >
> > > 0.0110101000101000101000100000101000001...
> > >
> > > So basically the number is the sum of the reciprocals of prime powers of 2.
> > > Which in binary representation visually clearly shows the prime positions of digits in
> > > the binary expansion.
> > https://en.wikipedia.org/wiki/Prime_constant
> Nice
> Where I got it wrong was the first post of yours where I stated a (short) rational number creating
> an irrational but that was wrong, thanks to Wolframs Alpha, when you corrected your post then things changed as did the binary sequence and the irrational connected to it.
> What is interesting is, drawing a horizontal line using the binary sequence where the first zero after
> the decimal point dose not draw the line but using only a right hand position then the next 1
> representing 2 draws a unit length left. Then the next 1 following the 1 representing 3 and draws
> a line to the right. The odd number of zeros between each 1 (prime) sets up a steady line going
> right for each 1 on the x axis. All it does is give a prime count in the length of the line -1.
> First 100 digits = 24 1 unit lengths. So 1 less in the count of 25 because of the 3(1) retracing over the 2(1). Simple, I know, but duplicating the alternating right\left pi's digits in the same way only with pi
> the line grows in 2 directions and this line actually starts @ x=-1 and then grows to the right --->oo.
> Counting the primes along the way less1 in unit line length.
> Setting up now to see how the irrational number that created the binary sequence on how that line
> goes using alternating directions for each digit.
> I am even looking at the Fibonacci sequence where the odd numbered gaps of zeros will dictate the same
> direction from the last term to the next term going in the same direction.. Even numbered gaps of zeros
> will reverse the direction from the previous Fibonacci number. Will this line be longer then the last term
> processed?
> Thanks for your input.
I tried this with the Fibonacci numbers using the R,L,R,L,R... alternating direction and here is the simple results.
Starting on the x axis x=0 then 1> R,1<L,2>R,3<L,0>R,5<L,0>R,0<L,8>R,0<L,0>R,0<L,0>R,13<L,0>R,0<L,0>R,
0<L,0>R,0<L,0>R,21<L... Starting with the 3,5 both left the 8 is right then 13,21 both left and 34 is right.
What happens is the next Right fib number ends up x=2 and the sum of the 2 previous 2 fib numbers
are L,L where the sum of these previous 2 is a minus value +2 then adding the next fib number which is
a R direction the total sum x=2 and so on --->oo. The one R added equals x=+2 and the previous two
negative sums minus from the x=+2 too x=-n. So the line keeps growing longer moving only to the left negative and reaching only to x=2 when the next single R direction is added.

On Friday, April 28, 2023 at 12:44:47 AM UTC-4, Dan joyce wrote:
> On Wednesday, April 26, 2023 at 4:25:38 PM UTC-4, Dan joyce wrote:
> > On Wednesday, April 26, 2023 at 9:56:54 AM UTC-4, sobriquet wrote:
> > > On Tuesday, April 25, 2023 at 11:18:49 PM UTC+2, sobriquet wrote:
> > > > On Tuesday, April 25, 2023 at 11:09:09 PM UTC+2, Dan joyce wrote:
> > > > > On Tuesday, April 25, 2023 at 4:38:44 PM UTC-4, sobriquet wrote:
> > > > > > On Tuesday, April 25, 2023 at 10:29:41 PM UTC+2, Dan joyce wrote:
> > > > > > > On Tuesday, April 25, 2023 at 4:22:38 PM UTC-4, sobriquet wrote:
> > > > > > > > On Tuesday, April 25, 2023 at 10:15:11 PM UTC+2, sobriquet wrote:
> > > > > > > > > On Tuesday, April 25, 2023 at 7:28:04 PM UTC+2, Dan joyce wrote:
> > > > > > > > > > On Tuesday, April 25, 2023 at 7:06:00 AM UTC-4, FromTheRafters wrote:
> > > > > > > > > > > Dan joyce laid this down on his screen :
> > > > > > > > > > > > On Monday, April 24, 2023 at 6:02:37 PM UTC-4, Barry Schwarz wrote:
> > > > > > > > > > > >> On Mon, 24 Apr 2023 14:11:52 -0700 (PDT), Dan joyce
> > > > > > > > > > > >> <danj...@gmail.com> wrote:
> > > > > > > > > > > >>> On Monday, April 24, 2023 at 3:01:25?PM UTC-4, Barry Schwarz wrote:
> > > > > > > > > > > >>>> On Mon, 24 Apr 2023 07:56:41 -0700 (PDT), Dan joyce
> > > > > > > > > > > >>>> <danj...@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
> > > > > > > > > > > >>>
> > > > > > > > > > > >>> So in conclusion Barry, does this line increase in length without bounds
> > > > > > > > > > > >>> as pi's decimal digits transposed into integers --->oo, or at this point,
> > > > > > > > > > > >>> just a conjecture?
> > > > > > > > > > > >>> Your thoughts?
> > > > > > > > > > > >> It is indeed a conjecture and we have no idea if it is true or not.
> > > > > > > > > > > >> And at the moment, I think we don't even have an idea how to prove it
> > > > > > > > > > > >> one way of the other.
> > > > > > > > > > > >> --
> > > > > > > > > > > >> Remove del for email
> > > > > > > > > > > >
> > > > > > > > > > > > I believe this conjecture may never be proven true or false.
> > > > > > > > > > > > And I will add, in my life time. I am 88 so the above statement is not too
> > > > > > > > > > > > far fetched.
> > > > > > > > > > > > Kind of a cool conjecture though!
> > > > > > > > > > > It reminds me of Ulam's spiral of primes, though I don't know exactly
> > > > > > > > > > > why. Maybe only because it is a visual representation of an interesting
> > > > > > > > > > > set of numbers. How different do other interesting numbers' (I'll call
> > > > > > > > > > > these mappings 'stamps') such as e or Phi look? Does your conjecture
> > > > > > > > > > > seem to also apply to these?
> > > > > > > > > > Phi, e and many other mathematical constants whos decimal expansion
> > > > > > > > > > appears random would also apply.
> > > > > > > > > > I haven't tested them but why not?
> > > > > > > > > >
> > > > > > > > > > The primes are a different breed ---
> > > > > > > > > > The third column is the final number -x +x where the running totals
> > > > > > > > > > of the second column is the abs line length starting with 2 ----- 3,5,7,11,13...
> > > > > > > > > > 2+-3 =-1
> > > > > > > > > > -1+ 5 = 4
> > > > > > > > > > 4+-7 = -3
> > > > > > > > > > -3+11= 8
> > > > > > > > > > 8+-13=-5
> > > > > > > > > > -5+17=12
> > > > > > > > > > 12+-19=-7
> > > > > > > > > > -7 + 23=16
> > > > > > > > > > 16+-29=-13
> > > > > > > > > > -13+31= 18
> > > > > > > > > > 18+-37= -19
> > > > > > > > > > -17+41= 24
> > > > > > > > > > 24+-43=-19
> > > > > > > > > > -19+47= 28
> > > > > > > > > > 28+-53=-25
> > > > > > > > > > A line +x\-x (third column above) ever extending in both directions on the x axis as the
> > > > > > > > > > primes --->oo so does the length of this line.(abs second column)
> > > > > > > > > You can have a single irrational number that encodes all the primes.. like this binary
> > > > > > > > > expansion, sum for all primes P of (2^-P):
> > > > > > > > >
> > > > > > > > > 0.0110101000101000101000100000101...
> > > > > > > > >
> > > > > > > > > https://www.wolframalpha.com/input?i=Sum%5B1%2F2%5E%28Prime%5Bx%5D%29%2C+x%5D+
> > > > > > > > >
> > > > > > > > > https://www.wolframalpha.com/input?i=binary+0.414683
> > > > > > > > I'm confused why wolfram alpha claims the sum diverges, since it's obviously just a particular
> > > > > > > > number irrational number which has bits set for prime positions in the binary expansion.
> > > > > > > Interesting how a (short) rational produces a binary irrational.
> > > > > > It's an irrational number (regardless of representing it in decimal or binary expansion, but
> > > > > > in binary expansion the prime number pattern is clearly visible).
> > > > > I am not following about the prime number pattern, please explain?
> > > > If we have a binary number between 0 and 1 with a binary expansion, we can label the
> > > > positions of the digits that are set to 1 with a set of positive natural numbers.
> > > >
> > > > So for instance, a binary number with digits {2,3,5,7,11,13,17,19,23,29,31,37,..} set would look like:
> > > >
> > > > 0.0110101000101000101000100000101000001...
> > > >
> > > > So basically the number is the sum of the reciprocals of prime powers of 2.
> > > > Which in binary representation visually clearly shows the prime positions of digits in
> > > > the binary expansion.
> > > https://en.wikipedia.org/wiki/Prime_constant
> > Nice
> > Where I got it wrong was the first post of yours where I stated a (short) rational number creating
> > an irrational but that was wrong, thanks to Wolframs Alpha, when you corrected your post then things changed as did the binary sequence and the irrational connected to it.
> > What is interesting is, drawing a horizontal line using the binary sequence where the first zero after
> > the decimal point dose not draw the line but using only a right hand position then the next 1
> > representing 2 draws a unit length left. Then the next 1 following the 1 representing 3 and draws
> > a line to the right. The odd number of zeros between each 1 (prime) sets up a steady line going
> > right for each 1 on the x axis. All it does is give a prime count in the length of the line -1.
> > First 100 digits = 24 1 unit lengths. So 1 less in the count of 25 because of the 3(1) retracing over the 2(1). Simple, I know, but duplicating the alternating right\left pi's digits in the same way only with pi
> > the line grows in 2 directions and this line actually starts @ x=-1 and then grows to the right --->oo.
> > Counting the primes along the way less1 in unit line length.
> > Setting up now to see how the irrational number that created the binary sequence on how that line
> > goes using alternating directions for each digit.
> > I am even looking at the Fibonacci sequence where the odd numbered gaps of zeros will dictate the same
> > direction from the last term to the next term going in the same direction. Even numbered gaps of zeros
> > will reverse the direction from the previous Fibonacci number. Will this line be longer then the last term
> > processed?
> > Thanks for your input.
> I tried this with the Fibonacci numbers using the R,L,R,L,R... alternating direction and here is the simple results.
> Starting on the x axis x=0 then 1> R,1<L,2>R,3<L,0>R,5<L,0>R,0<L,8>R,0<L,0>R,0<L,0>R,13<L,0>R,0<L,0>R,
> 0<L,0>R,0<L,0>R,21<L... Starting with the 3,5 both left the 8 is right then 13,21 both left and 34 is right.
> What happens is the next Right fib number ends up x=2 and the sum of the 2 previous 2 fib numbers
> are L,L where the sum of these previous 2 is a minus value +2 then adding the next fib number which is
> a R direction the total sum x=2 and so on --->oo. The one R added equals x=+2 and the previous two
> negative sums minus from the x=+2 too x=-n. So the line keeps growing longer moving only to the left negative and reaching only to x=2 when the next single R direction is added.