You just need a piece of graph paper and a dark pencil or pen.
I am using James's sequence instead of mine because his is
the true lattice where mine is a rectangle.
Strange enough, our sequence is the same sequence but
my layout of directions are a little different starting out.
Do all these steps in order with your choice of writing instrument.
James's out going sequence of events for a square lattice is, starting with ---
(RU=1,LD=2,RD= 1)
After the initial square is drawn in the center of the lattice then the pattern begins for the path outward. Going counterclockwise ---
RU=3,LU=2,LD=4,RD=3
RU=5,LU=4,LD=6,RD=5
RU=7,LU=6,LD=8,RD=7
RU=9,LU=8,LD=9 then starts down the path created above on the left side in a reverse direction for the cell creation and a new pattern below going
clockwise---
-------------------------------------------------------------------------------------------------------------------------
RU=8,RD=8,LD=7,LU=7
RU=6,RD=6,LD=5,LU=5
RU=4,RD=4,LD=3,LU=3
RU=2,RD=2,LD=1,Lu=1
A simple pattern for a square lattice for both going out and coming back in.
My first order rectangular lattice will follow. (smallest ratio between between
the two sides)
Many more different rectangular lattices are possible.
Just take a few minutes and try the above instructions in that order.
Never lift the pen or pencil from the paper while drawing this out.

On Tuesday, June 13, 2023 at 8:47:05 AM UTC-4, Dan joyce wrote:
> You just need a piece of graph paper and a dark pencil or pen.
> I am using James's sequence instead of mine because his is
> the true lattice where mine is a rectangle.
> Strange enough, our sequence is the same sequence but
> my layout of directions are a little different starting out.
> Do all these steps in order with your choice of writing instrument.
> James's out going sequence of events for a square lattice is, starting with ---
> (RU=1,LD=2,RD= 1)
> After the initial square is drawn in the center of the lattice then the pattern begins for the path outward. Going counterclockwise ---
> RU=3,LU=2,LD=4,RD=3
> RU=5,LU=4,LD=6,RD=5
> RU=7,LU=6,LD=8,RD=7
> RU=9,LU=8,LD=9 then starts down the path created above on the left side in a reverse direction for the cell creation and a new pattern below going
> clockwise---
> -------------------------------------------------------------------------------------------------------------------------
> RU=8,RD=8,LD=7,LU=7
> RU=6,RD=6,LD=5,LU=5
> RU=4,RD=4,LD=3,LU=3
> RU=2,RD=2,LD=1,Lu=1
> A simple pattern for a square lattice for both going out and coming back in.
> My first order rectangular lattice will follow. (smallest ratio between between
> the two sides)
> Many more different rectangular lattices are possible.
> Just take a few minutes and try the above instructions in that order.
> Never lift the pen or pencil from the paper while drawing this out.
There are so many different ways these can be constructed, it is mind boggling.

Dan joyce <danj4084@gmail.com> writes:

> You just need a piece of graph paper and a dark pencil or pen.
> I am using James's sequence instead of mine because his is
> the true lattice where mine is a rectangle.
> Strange enough, our sequence is the same sequence but
> my layout of directions are a little different starting out.
> Do all these steps in order with your choice of writing instrument.
> James's out going sequence of events for a square lattice is, starting
> with --- (RU=1,LD=2,RD= 1)

I don't know exactly what you are doing here, but since it seems to be about
drawing edges on a square lattice I don't know why the sequence is not
just a list of R, L, U, D (or any other four symbols). Instead, I see
pairs of letters, and = sign and a number. What do these mean?

I know there have been lots of posts in other threads, but I stopped
reading those. This is a fresh top-level post starting a new thread, so
it should be self-contained with enough detail for anyone to be able to
follow.

--
Ben.

On Tuesday, June 13, 2023 at 9:44:46 AM UTC-4, Ben Bacarisse wrote:
> Dan joyce <danj...@gmail.com> writes:
>
> > You just need a piece of graph paper and a dark pencil or pen.
> > I am using James's sequence instead of mine because his is
> > the true lattice where mine is a rectangle.
> > Strange enough, our sequence is the same sequence but
> > my layout of directions are a little different starting out.
> > Do all these steps in order with your choice of writing instrument.
> > James's out going sequence of events for a square lattice is, starting
> > with --- (RU=1,LD=2,RD= 1)
> I don't know exactly what you are doing here, but since it seems to be about
> drawing edges on a square lattice I don't know why the sequence is not
> just a list of R, L, U, D (or any other four symbols). Instead, I see
> pairs of letters, and = sign and a number. What do these mean?
Ben,
As an example --- (RU=1,LD=2,RD= 1) RU=1 just means a right line of one unit long followed
by an up line of 1 unit long (always a 90 degree turn right or left in this case left) Then LD=2
just means left down twice, a line of 1unit long going left and then another line going down then repeat that for LD=2 always connecting each line and always at a right or left 90 degree turn depending
on the command. The third command RD= 1. is just two lines added where the last command
LD left off. Each connecting line is always a 90 degree connection to the last line drawn.
I hope this helps.
My ascii art sucks in order to explain it better.
Sorry, I should have explained it better..
..
> I know there have been lots of posts in other threads, but I stopped
> reading those. This is a fresh top-level post starting a new thread, so
> it should be self-contained with enough detail for anyone to be able to
> follow.
>
> --
> Ben.

Dan joyce <danj4084@gmail.com> writes:

> On Tuesday, June 13, 2023 at 9:44:46 AM UTC-4, Ben Bacarisse wrote:
>> Dan joyce <danj...@gmail.com> writes:
>>
>> > You just need a piece of graph paper and a dark pencil or pen.
>> > I am using James's sequence instead of mine because his is
>> > the true lattice where mine is a rectangle.
>> > Strange enough, our sequence is the same sequence but
>> > my layout of directions are a little different starting out.
>> > Do all these steps in order with your choice of writing instrument.
>> > James's out going sequence of events for a square lattice is, starting
>> > with --- (RU=1,LD=2,RD= 1)
>> I don't know exactly what you are doing here, but since it seems to be about
>> drawing edges on a square lattice I don't know why the sequence is not
>> just a list of R, L, U, D (or any other four symbols). Instead, I see
>> pairs of letters, and = sign and a number. What do these mean?
> Ben,
> As an example ---

In maths, it usually better to define notation rather than give
examples. But I think I have only one question left (see below).

> (RU=1,LD=2,RD= 1) RU=1 just means a right line of one unit long
> followed by an up line of 1 unit long (always a 90 degree turn right
> or left in this case left)

I'd write "ru".

> Then LD=2 just means left down twice, a
> line of 1unit long going left and then another line going down then
> repeat that for LD=2

So I'd write "ldld" for that.

> always connecting each line and always at a right
> or left 90 degree turn depending on the command.

Do the rules of the game really prohibit no turn, i.e. are rr, ll, dd
and uu excluded?

rl, lr, ud and du are excluded by the "only use each edge once" rule
which, if a turn /is/ required, will leave only eight valid pairs. This
might be why you prefer a "pair and count" notation.

> The third command RD=1. is just...

If I've understood, that is what I'd write with just "rd".

So your RU=1,LD=2,RD=1 is my "ruldldrd"?

Can you also state the claim about your sequence to which you want to
draw the attention of "all you doubters"? I.e. I don't know what people
have been doubting that you are now demonstrating.

--
Ben.

On Tuesday, June 13, 2023 at 10:56:43 AM UTC-4, Dan joyce wrote:
> On Tuesday, June 13, 2023 at 9:44:46 AM UTC-4, Ben Bacarisse wrote:
> > Dan joyce <danj...@gmail.com> writes:
> >
> > > You just need a piece of graph paper and a dark pencil or pen.
> > > I am using James's sequence instead of mine because his is
> > > the true lattice where mine is a rectangle.
> > > Strange enough, our sequence is the same sequence but
> > > my layout of directions are a little different starting out.
> > > Do all these steps in order with your choice of writing instrument.
> > > James's out going sequence of events for a square lattice is, starting
> > > with --- (RU=1,LD=2,RD= 1)
> > I don't know exactly what you are doing here, but since it seems to be about
> > drawing edges on a square lattice I don't know why the sequence is not
> > just a list of R, L, U, D (or any other four symbols). Instead, I see
> > pairs of letters, and = sign and a number. What do these mean?
> Ben,
> As an example --- (RU=1,LD=2,RD= 1) RU=1 just means a right line of one unit long followed
> by an up line of 1 unit long (always a 90 degree turn right or left in this case left) Then LD=2
> just means left down twice, a line of 1unit long going left and then another line going down then repeat that for LD=2 always connecting each line and always at a right or left 90 degree turn depending
> on the command. The third command RD= 1. is just two lines added where the last command
> LD left off. Each connecting line is always a 90 degree connection to the last line drawn.
> I hope this helps.
> My ascii art sucks in order to explain it better.
> Sorry, I should have explained it better..
> .
> > I know there have been lots of posts in other threads, but I stopped
> > reading those. This is a fresh top-level post starting a new thread, so
> > it should be self-contained with enough detail for anyone to be able to
> > follow.
> >
> > --
> > Ben.
I just noticed something, the only difference between my sequence and James's
is I come back in a 1/4 of a rotation sooner then his lattice. I come in on the top
and he comes back in on the left side a 1/4 of a rotation later. Making his a square
lattice and mine a rectangular lattice.
It starts the same way with a single completed cell like James's square lattice.
Like I said before many more different constructions of these lattices are possible.
We are just scratching the sueface.

On 13/06/2023 13:46, Dan joyce wrote:
> You just need a piece of graph paper and a dark pencil or pen.
> I am using James's sequence instead of mine because his is
> the true lattice where mine is a rectangle.
> Strange enough, our sequence is the same sequence but
> my layout of directions are a little different starting out.
> Do all these steps in order with your choice of writing instrument.
> James's out going sequence of events for a square lattice is, starting with ---
> (RU=1,LD=2,RD= 1)

[display with fixed width font]

--
| |
-- 0-
|
--
|

(0 is the start of the path...)

> After the initial square is drawn in the center of the lattice then the pattern begins for the path outward. Going counterclockwise ---
> RU=3,LU=2,LD=4,RD=3

--
| |
-- --
| |
-- -- --
| | | |
-- -- 0- --
| | |
-- -- --
| | |
-- --
|
--
|

ok, I get it - you're going to turn around at some point and wind back to finally end up at 0.

That's ok, but this is just a finite lattice. Your earlier thread had the title "Draw an infinite
square lattice without lifting the pencil", so you've failed at that! :)

Mike.

On Tuesday, June 13, 2023 at 11:21:15 AM UTC-4, Ben Bacarisse wrote:
> Dan joyce <danj...@gmail.com> writes:
>
> > On Tuesday, June 13, 2023 at 9:44:46 AM UTC-4, Ben Bacarisse wrote:
> >> Dan joyce <danj...@gmail.com> writes:
> >>
> >> > You just need a piece of graph paper and a dark pencil or pen.
> >> > I am using James's sequence instead of mine because his is
> >> > the true lattice where mine is a rectangle.
> >> > Strange enough, our sequence is the same sequence but
> >> > my layout of directions are a little different starting out.
> >> > Do all these steps in order with your choice of writing instrument.
> >> > James's out going sequence of events for a square lattice is, starting
> >> > with --- (RU=1,LD=2,RD= 1)
> >> I don't know exactly what you are doing here, but since it seems to be about
> >> drawing edges on a square lattice I don't know why the sequence is not
> >> just a list of R, L, U, D (or any other four symbols). Instead, I see
> >> pairs of letters, and = sign and a number. What do these mean?
> > Ben,
> > As an example ---
> In maths, it usually better to define notation rather than give
> examples. But I think I have only one question left (see below).
> > (RU=1,LD=2,RD= 1) RU=1 just means a right line of one unit long
> > followed by an up line of 1 unit long (always a 90 degree turn right
> > or left in this case left)
> I'd write "ru".
> > Then LD=2 just means left down twice, a
> > line of 1unit long going left and then another line going down then
> > repeat that for LD=2
> So I'd write "ldld" for that.
> > always connecting each line and always at a right
> > or left 90 degree turn depending on the command.
> Do the rules of the game really prohibit no turn, i.e. are rr, ll, dd
> and uu excluded?
>
> rl, lr, ud and du are excluded by the "only use each edge once" rule
> which, if a turn /is/ required, will leave only eight valid pairs. This
> might be why you prefer a "pair and count" notation.
>
> > The third command RD=1. is just...
>
> If I've understood, that is what I'd write with just "rd".
In this case 2 equal length lines at always 90 degrees at their connecting point.
Neve two connecting lines forming a longer straight line.
>
> So your RU=1,LD=2,RD=1 is my "ruldldrd"?
>
> Can you also state the claim about your sequence to which you want to
> draw the attention of "all you doubters"? I.e. I don't know what people
> have been doubting that you are now demonstrating.
Many reply's in my original post thought it was impossible but now I am not
sure if they are still doubters.
One even referred to a video on UTube where it showed a guy doing impossible things
such as walking on water and other impossible stunts.
Maybe I am not justified in reposting this but there is a lot to learn yet, including myself.
As you will gather if you read my last post.

Thanks for your input Ben.
> --
> Ben.

On Tuesday, June 13, 2023 at 11:41:08 AM UTC-4, Mike Terry wrote:
> On 13/06/2023 13:46, Dan joyce wrote:
> > You just need a piece of graph paper and a dark pencil or pen.
> > I am using James's sequence instead of mine because his is
> > the true lattice where mine is a rectangle.
> > Strange enough, our sequence is the same sequence but
> > my layout of directions are a little different starting out.
> > Do all these steps in order with your choice of writing instrument.
> > James's out going sequence of events for a square lattice is, starting with ---
> > (RU=1,LD=2,RD= 1)
> [display with fixed width font]
>
> --
> | |
> -- 0-
> |
> --
> |
>
> (0 is the start of the path...)
> > After the initial square is drawn in the center of the lattice then the pattern begins for the path outward. Going counterclockwise ---
> > RU=3,LU=2,LD=4,RD=3
> --
> | |
> -- --
> | |
> -- -- --
> | | | |
> -- -- 0- --
> | | |
> -- -- --
> | | |
> -- --
> |
> --
> |
>
>
>
> ok, I get it - you're going to turn around at some point and wind back to finally end up at 0.
>
> That's ok, but this is just a finite lattice. Your earlier thread had the title "Draw an infinite
> square lattice without lifting the pencil", so you've failed at that! :)
>
> Mike.
Mike,
I fail at a lot of things when it comes to math.
James pointed this out already and I agreed with him that it shouldn't
be called an infinite lattice but I used it because it sounded more sexy.
It really should be called a lattice --->oo. I knew this at the time
But thinking even about that one, it is not a lattice until it returns to the
origin created by the pathway --->oo. and then returning back to the origin
of the single cell first created at the center of this lattice.
I am still learning about this stuff. Many different ways to do it even a
random start and ending up eventually with an out going pathway for
the return trip back to complete the lattice.
It is fun using the old pixel technology.

I wonder what Georg Cantor or Euler's would say about it?
Maybe not so significant but I think still interesting.

On Tuesday, June 13, 2023 at 11:47:17 AM UTC-4, Dan joyce wrote:
> On Tuesday, June 13, 2023 at 11:21:15 AM UTC-4, Ben Bacarisse wrote:
> > Dan joyce <danj...@gmail.com> writes:
> >
> > > On Tuesday, June 13, 2023 at 9:44:46 AM UTC-4, Ben Bacarisse wrote:
> > >> Dan joyce <danj...@gmail.com> writes:
> > >>
> > >> > You just need a piece of graph paper and a dark pencil or pen.
> > >> > I am using James's sequence instead of mine because his is
> > >> > the true lattice where mine is a rectangle.
> > >> > Strange enough, our sequence is the same sequence but
> > >> > my layout of directions are a little different starting out.
> > >> > Do all these steps in order with your choice of writing instrument..
> > >> > James's out going sequence of events for a square lattice is, starting
> > >> > with --- (RU=1,LD=2,RD= 1)
> > >> I don't know exactly what you are doing here, but since it seems to be about
> > >> drawing edges on a square lattice I don't know why the sequence is not
> > >> just a list of R, L, U, D (or any other four symbols). Instead, I see
> > >> pairs of letters, and = sign and a number. What do these mean?
> > > Ben,
> > > As an example ---
> > In maths, it usually better to define notation rather than give
> > examples. But I think I have only one question left (see below).
> > > (RU=1,LD=2,RD= 1) RU=1 just means a right line of one unit long
> > > followed by an up line of 1 unit long (always a 90 degree turn right
> > > or left in this case left)
> > I'd write "ru".
> > > Then LD=2 just means left down twice, a
> > > line of 1unit long going left and then another line going down then
> > > repeat that for LD=2
> > So I'd write "ldld" for that.
> > > always connecting each line and always at a right
> > > or left 90 degree turn depending on the command.
> > Do the rules of the game really prohibit no turn, i.e. are rr, ll, dd
> > and uu excluded?
> >
> > rl, lr, ud and du are excluded by the "only use each edge once" rule
> > which, if a turn /is/ required, will leave only eight valid pairs. This
> > might be why you prefer a "pair and count" notation.
> >
> > > The third command RD=1. is just...
> >
> > If I've understood, that is what I'd write with just "rd".
> In this case 2 equal length lines at always 90 degrees at their connecting point.
> Neve two connecting lines forming a longer straight line.
> >
> > So your RU=1,LD=2,RD=1 is my "ruldldrd"?
> >
> > Can you also state the claim about your sequence to which you want to
> > draw the attention of "all you doubters"? I.e. I don't know what people
> > have been doubting that you are now demonstrating.
> Many reply's in my original post thought it was impossible but now I am not
> sure if they are still doubters.
> One even referred to a video on UTube where it showed a guy doing impossible things
> such as walking on water and other impossible stunts.
> Maybe I am not justified in reposting this but there is a lot to learn yet, including myself.
> As you will gather if you read my last post.
>
> Thanks for your input Ben.
> > --
> > Ben.
That should be (never) two connecting lines forming a longer line.
A proof reading idiot, that's me.

Dan joyce <danj4084@gmail.com> writes:

> On Tuesday, June 13, 2023 at 11:21:15 AM UTC-4, Ben Bacarisse wrote:
>> Dan joyce <danj...@gmail.com> writes:
>>
>> > On Tuesday, June 13, 2023 at 9:44:46 AM UTC-4, Ben Bacarisse wrote:
>> >> Dan joyce <danj...@gmail.com> writes:
>> >>
>> >> > You just need a piece of graph paper and a dark pencil or pen.
>> >> > I am using James's sequence instead of mine because his is
>> >> > the true lattice where mine is a rectangle.
>> >> > Strange enough, our sequence is the same sequence but
>> >> > my layout of directions are a little different starting out.
>> >> > Do all these steps in order with your choice of writing instrument.
>> >> > James's out going sequence of events for a square lattice is, starting
>> >> > with --- (RU=1,LD=2,RD= 1)
>> >> I don't know exactly what you are doing here, but since it seems to be about
>> >> drawing edges on a square lattice I don't know why the sequence is not
>> >> just a list of R, L, U, D (or any other four symbols). Instead, I see
>> >> pairs of letters, and = sign and a number. What do these mean?
>> > Ben,
>> > As an example ---
>> In maths, it usually better to define notation rather than give
>> examples. But I think I have only one question left (see below).
>> > (RU=1,LD=2,RD= 1) RU=1 just means a right line of one unit long
>> > followed by an up line of 1 unit long (always a 90 degree turn right
>> > or left in this case left)
>> I'd write "ru".
>> > Then LD=2 just means left down twice, a
>> > line of 1unit long going left and then another line going down then
>> > repeat that for LD=2
>> So I'd write "ldld" for that.
>> > always connecting each line and always at a right
>> > or left 90 degree turn depending on the command.
>> Do the rules of the game really prohibit no turn, i.e. are rr, ll, dd
>> and uu excluded?
>>
>> rl, lr, ud and du are excluded by the "only use each edge once" rule
>> which, if a turn /is/ required, will leave only eight valid pairs. This
>> might be why you prefer a "pair and count" notation.
>>
>> > The third command RD=1. is just...
>>
>> If I've understood, that is what I'd write with just "rd".

> In this case 2 equal length lines at always 90 degrees at their
> connecting point. Neve two connecting lines forming a longer straight
> line.

Yes, I can see that from your example, but the example does not answer
my question. Do the "rules of the game" prohibit a solution that has
rr, ll, uu or dd in it?

>> So your RU=1,LD=2,RD=1 is my "ruldldrd"?
>>
>> Can you also state the claim about your sequence to which you want to
>> draw the attention of "all you doubters"? I.e. I don't know what people
>> have been doubting that you are now demonstrating.
> Many reply's in my original post thought it was impossible but now I am not
> sure if they are still doubters.

But what are you claiming? I thought that you had accepted that the
original claim (an endless path that encloses, eventually, every square)
was not possible. So I am still wondering, what is it that people doubt
that you claim they should not doubt?

--
Ben.

On Tuesday, June 13, 2023 at 12:18:54 PM UTC-4, Ben Bacarisse wrote:
> Dan joyce <danj...@gmail.com> writes:
>
> > On Tuesday, June 13, 2023 at 11:21:15 AM UTC-4, Ben Bacarisse wrote:
> >> Dan joyce <danj...@gmail.com> writes:
> >>
> >> > On Tuesday, June 13, 2023 at 9:44:46 AM UTC-4, Ben Bacarisse wrote:
> >> >> Dan joyce <danj...@gmail.com> writes:
> >> >>
> >> >> > You just need a piece of graph paper and a dark pencil or pen.
> >> >> > I am using James's sequence instead of mine because his is
> >> >> > the true lattice where mine is a rectangle.
> >> >> > Strange enough, our sequence is the same sequence but
> >> >> > my layout of directions are a little different starting out.
> >> >> > Do all these steps in order with your choice of writing instrument.
> >> >> > James's out going sequence of events for a square lattice is, starting
> >> >> > with --- (RU=1,LD=2,RD= 1)
> >> >> I don't know exactly what you are doing here, but since it seems to be about
> >> >> drawing edges on a square lattice I don't know why the sequence is not
> >> >> just a list of R, L, U, D (or any other four symbols). Instead, I see
> >> >> pairs of letters, and = sign and a number. What do these mean?
> >> > Ben,
> >> > As an example ---
> >> In maths, it usually better to define notation rather than give
> >> examples. But I think I have only one question left (see below).
> >> > (RU=1,LD=2,RD= 1) RU=1 just means a right line of one unit long
> >> > followed by an up line of 1 unit long (always a 90 degree turn right
> >> > or left in this case left)
> >> I'd write "ru".
> >> > Then LD=2 just means left down twice, a
> >> > line of 1unit long going left and then another line going down then
> >> > repeat that for LD=2
> >> So I'd write "ldld" for that.
> >> > always connecting each line and always at a right
> >> > or left 90 degree turn depending on the command.
> >> Do the rules of the game really prohibit no turn, i.e. are rr, ll, dd
> >> and uu excluded?
> >>
> >> rl, lr, ud and du are excluded by the "only use each edge once" rule
> >> which, if a turn /is/ required, will leave only eight valid pairs. This
> >> might be why you prefer a "pair and count" notation.
> >>
> >> > The third command RD=1. is just...
> >>
> >> If I've understood, that is what I'd write with just "rd".
>
> > In this case 2 equal length lines at always 90 degrees at their
> > connecting point. Neve two connecting lines forming a longer straight
> > line.
> Yes, I can see that from your example, but the example does not answer
> my question. Do the "rules of the game" prohibit a solution that has
> rr, ll, uu or dd in it?
> >> So your RU=1,LD=2,RD=1 is my "ruldldrd"?
> >>
> >> Can you also state the claim about your sequence to which you want to
> >> draw the attention of "all you doubters"? I.e. I don't know what people
> >> have been doubting that you are now demonstrating.
> > Many reply's in my original post thought it was impossible but now I am not
> > sure if they are still doubters.
> But what are you claiming? I thought that you had accepted that the
> original claim (an endless path that encloses, eventually, every square)
> was not possible. So I am still wondering, what is it that people doubt
> that you claim they should not doubt?
>
> --
> Ben.
I never claimed it was impossible. I said just the opposite that was a proven
fact that it is possible,

On Tuesday, June 13, 2023 at 12:47:55 PM UTC-4, Dan joyce wrote:
> On Tuesday, June 13, 2023 at 12:18:54 PM UTC-4, Ben Bacarisse wrote:
> > Dan joyce <danj...@gmail.com> writes:
> >
> > > On Tuesday, June 13, 2023 at 11:21:15 AM UTC-4, Ben Bacarisse wrote:
> > >> Dan joyce <danj...@gmail.com> writes:
> > >>
> > >> > On Tuesday, June 13, 2023 at 9:44:46 AM UTC-4, Ben Bacarisse wrote:
> > >> >> Dan joyce <danj...@gmail.com> writes:
> > >> >>
> > >> >> > You just need a piece of graph paper and a dark pencil or pen.
> > >> >> > I am using James's sequence instead of mine because his is
> > >> >> > the true lattice where mine is a rectangle.
> > >> >> > Strange enough, our sequence is the same sequence but
> > >> >> > my layout of directions are a little different starting out.
> > >> >> > Do all these steps in order with your choice of writing instrument.
> > >> >> > James's out going sequence of events for a square lattice is, starting
> > >> >> > with --- (RU=1,LD=2,RD= 1)
> > >> >> I don't know exactly what you are doing here, but since it seems to be about
> > >> >> drawing edges on a square lattice I don't know why the sequence is not
> > >> >> just a list of R, L, U, D (or any other four symbols). Instead, I see
> > >> >> pairs of letters, and = sign and a number. What do these mean?
> > >> > Ben,
> > >> > As an example ---
> > >> In maths, it usually better to define notation rather than give
> > >> examples. But I think I have only one question left (see below).
> > >> > (RU=1,LD=2,RD= 1) RU=1 just means a right line of one unit long
> > >> > followed by an up line of 1 unit long (always a 90 degree turn right
> > >> > or left in this case left)
> > >> I'd write "ru".
> > >> > Then LD=2 just means left down twice, a
> > >> > line of 1unit long going left and then another line going down then
> > >> > repeat that for LD=2
> > >> So I'd write "ldld" for that.
> > >> > always connecting each line and always at a right
> > >> > or left 90 degree turn depending on the command.
> > >> Do the rules of the game really prohibit no turn, i.e. are rr, ll, dd
> > >> and uu excluded?
> > >>
> > >> rl, lr, ud and du are excluded by the "only use each edge once" rule
> > >> which, if a turn /is/ required, will leave only eight valid pairs. This
> > >> might be why you prefer a "pair and count" notation.
> > >>
> > >> > The third command RD=1. is just...
> > >>
> > >> If I've understood, that is what I'd write with just "rd".
> >
> > > In this case 2 equal length lines at always 90 degrees at their
> > > connecting point. Neve two connecting lines forming a longer straight
> > > line.
> > Yes, I can see that from your example, but the example does not answer
> > my question. Do the "rules of the game" prohibit a solution that has
> > rr, ll, uu or dd in it?
> > >> So your RU=1,LD=2,RD=1 is my "ruldldrd"?
> > >>
> > >> Can you also state the claim about your sequence to which you want to
> > >> draw the attention of "all you doubters"? I.e. I don't know what people
> > >> have been doubting that you are now demonstrating.
> > > Many reply's in my original post thought it was impossible but now I am not
> > > sure if they are still doubters.
> > But what are you claiming? I thought that you had accepted that the
> > original claim (an endless path that encloses, eventually, every square)
> > was not possible. So I am still wondering, what is it that people doubt
> > that you claim they should not doubt?
> >
> > --
> > Ben.
> I never claimed it was impossible. I said just the opposite that was a proven
> fact that it is possible,
Ben,
If you are referring to the naming convention of an infinite lattice, yes I was wrong there
but as far as a construct --->oo and back to the origin to complete the lattice then it is
already proven. Even then it is putting a value on --->oo which is wrong.
Just say you could put a value on --->oo then the way I explained it, it would only be a
--->oo lattice on completion back to the origin.
Please enlighten me.

Dan joyce <danj4084@gmail.com> writes:

> On Tuesday, June 13, 2023 at 12:18:54 PM UTC-4, Ben Bacarisse wrote:
>> Dan joyce <danj...@gmail.com> writes:

>> > Many reply's in my original post thought it was impossible but now
>> > I am not sure if they are still doubters.
>>
>> But what are you claiming? I thought that you had accepted that the
>> original claim (an endless path that encloses, eventually, every square)
>> was not possible. So I am still wondering, what is it that people doubt
>> that you claim they should not doubt?
>>
> I never claimed it was impossible. I said just the opposite that was a
> proven fact that it is possible,

Can you state the claim? What this proven fact?

--
Ben.

On Tuesday, June 13, 2023 at 9:02:39 PM UTC-4, Ben Bacarisse wrote:
> Dan joyce <danj...@gmail.com> writes:
>
> > On Tuesday, June 13, 2023 at 12:18:54 PM UTC-4, Ben Bacarisse wrote:
> >> Dan joyce <danj...@gmail.com> writes:
>
> >> > Many reply's in my original post thought it was impossible but now
> >> > I am not sure if they are still doubters.
> >>
> >> But what are you claiming? I thought that you had accepted that the
> >> original claim (an endless path that encloses, eventually, every square)
> >> was not possible. So I am still wondering, what is it that people doubt
> >> that you claim they should not doubt?
> >>
> > I never claimed it was impossible. I said just the opposite that was a
> > proven fact that it is possible,
> Can you state the claim? What this proven fact?
>
> --
> Ben.
Did you see James's finite lattice rendering that shows a path being built and then
getting filled back in with cells completing the lattice at the origin.
Good enough for a proof I'd say.
It's in one of James's post on the subject. There are 3 different post running on the subject
I am experimenting now with no pathways but building a rectangular lattice from
left to right 1,3,5,7,5,3,2,1 up and down cells. Giving 27 total connected cells drawn in a rectangular
lattice without raising the pencil. I believe these can be built larger, like 1,3,5,7,9,7,5,3,1. total cells and
larger. Maybe yes or may be no.

On 6/13/2023 6:27 PM, Dan joyce wrote:
> On Tuesday, June 13, 2023 at 9:02:39 PM UTC-4, Ben Bacarisse wrote:
>> Dan joyce <danj...@gmail.com> writes:
>>
>>> On Tuesday, June 13, 2023 at 12:18:54 PM UTC-4, Ben Bacarisse wrote:
>>>> Dan joyce <danj...@gmail.com> writes:
>>
>>>>> Many reply's in my original post thought it was impossible but now
>>>>> I am not sure if they are still doubters.
>>>>
>>>> But what are you claiming? I thought that you had accepted that the
>>>> original claim (an endless path that encloses, eventually, every square)
>>>> was not possible. So I am still wondering, what is it that people doubt
>>>> that you claim they should not doubt?
>>>>
>>> I never claimed it was impossible. I said just the opposite that was a
>>> proven fact that it is possible,
>> Can you state the claim? What this proven fact?
>>
>> --
>> Ben.
> Did you see James's finite lattice rendering that shows a path being built and then
> getting filled back in with cells completing the lattice at the origin.
> Good enough for a proof I'd say.
> It's in one of James's post on the subject. There are 3 different post running on the subject
> I am experimenting now with no pathways but building a rectangular lattice from
> left to right 1,3,5,7,5,3,2,1 up and down cells. Giving 27 total connected cells drawn in a rectangular
> lattice without raising the pencil. I believe these can be built larger, like 1,3,5,7,9,7,5,3,1. total cells and
> larger. Maybe yes or may be no.

Not sure what you mean. Are you ditching the other algorithms I and
James have created? You mean something like this?

https://i.ibb.co/gJb1HHL/image.png

On 6/13/2023 5:46 AM, Dan joyce wrote:
> You just need a piece of graph paper and a dark pencil or pen.
> I am using James's sequence instead of mine because his is
> the true lattice where mine is a rectangle.
> Strange enough, our sequence is the same sequence but
> my layout of directions are a little different starting out.
> Do all these steps in order with your choice of writing instrument.
> James's out going sequence of events for a square lattice is, starting with ---
> (RU=1,LD=2,RD= 1)
> After the initial square is drawn in the center of the lattice then the pattern begins for the path outward. Going counterclockwise ---
> RU=3,LU=2,LD=4,RD=3
> RU=5,LU=4,LD=6,RD=5
> RU=7,LU=6,LD=8,RD=7
> RU=9,LU=8,LD=9 then starts down the path created above on the left side in a reverse direction for the cell creation and a new pattern below going
> clockwise---
> -------------------------------------------------------------------------------------------------------------------------
> RU=8,RD=8,LD=7,LU=7
> RU=6,RD=6,LD=5,LU=5
> RU=4,RD=4,LD=3,LU=3
> RU=2,RD=2,LD=1,Lu=1
> A simple pattern for a square lattice for both going out and coming back in.
> My first order rectangular lattice will follow. (smallest ratio between between
> the two sides)
> Many more different rectangular lattices are possible.
> Just take a few minutes and try the above instructions in that order.
> Never lift the pen or pencil from the paper while drawing this out.

Check this out:

https://i.ibb.co/C14qksJ/image.png

Can be any size. It just keeps building it self.

On Tuesday, June 13, 2023 at 10:49:39 PM UTC-4, Chris M. Thomasson wrote:
> On 6/13/2023 5:46 AM, Dan joyce wrote:
> > You just need a piece of graph paper and a dark pencil or pen.
> > I am using James's sequence instead of mine because his is
> > the true lattice where mine is a rectangle.
> > Strange enough, our sequence is the same sequence but
> > my layout of directions are a little different starting out.
> > Do all these steps in order with your choice of writing instrument.
> > James's out going sequence of events for a square lattice is, starting with ---
> > (RU=1,LD=2,RD= 1)
> > After the initial square is drawn in the center of the lattice then the pattern begins for the path outward. Going counterclockwise ---
> > RU=3,LU=2,LD=4,RD=3
> > RU=5,LU=4,LD=6,RD=5
> > RU=7,LU=6,LD=8,RD=7
> > RU=9,LU=8,LD=9 then starts down the path created above on the left side in a reverse direction for the cell creation and a new pattern below going
> > clockwise---
> > -------------------------------------------------------------------------------------------------------------------------
> > RU=8,RD=8,LD=7,LU=7
> > RU=6,RD=6,LD=5,LU=5
> > RU=4,RD=4,LD=3,LU=3
> > RU=2,RD=2,LD=1,Lu=1
> > A simple pattern for a square lattice for both going out and coming back in.
> > My first order rectangular lattice will follow. (smallest ratio between between
> > the two sides)
> > Many more different rectangular lattices are possible.
> > Just take a few minutes and try the above instructions in that order.
> > Never lift the pen or pencil from the paper while drawing this out.
> Check this out:
>
> https://i.ibb.co/C14qksJ/image.png
>
> Can be any size. It just keeps building it self.
That is fantastic. It is weird that we are basically working on the same idea.

On 6/13/2023 7:59 PM, Dan joyce wrote:
> On Tuesday, June 13, 2023 at 10:49:39 PM UTC-4, Chris M. Thomasson wrote:
>> On 6/13/2023 5:46 AM, Dan joyce wrote:
>>> You just need a piece of graph paper and a dark pencil or pen.
>>> I am using James's sequence instead of mine because his is
>>> the true lattice where mine is a rectangle.
>>> Strange enough, our sequence is the same sequence but
>>> my layout of directions are a little different starting out.
>>> Do all these steps in order with your choice of writing instrument.
>>> James's out going sequence of events for a square lattice is, starting with ---
>>> (RU=1,LD=2,RD= 1)
>>> After the initial square is drawn in the center of the lattice then the pattern begins for the path outward. Going counterclockwise ---
>>> RU=3,LU=2,LD=4,RD=3
>>> RU=5,LU=4,LD=6,RD=5
>>> RU=7,LU=6,LD=8,RD=7
>>> RU=9,LU=8,LD=9 then starts down the path created above on the left side in a reverse direction for the cell creation and a new pattern below going
>>> clockwise---
>>> -------------------------------------------------------------------------------------------------------------------------
>>> RU=8,RD=8,LD=7,LU=7
>>> RU=6,RD=6,LD=5,LU=5
>>> RU=4,RD=4,LD=3,LU=3
>>> RU=2,RD=2,LD=1,Lu=1
>>> A simple pattern for a square lattice for both going out and coming back in.
>>> My first order rectangular lattice will follow. (smallest ratio between between
>>> the two sides)
>>> Many more different rectangular lattices are possible.
>>> Just take a few minutes and try the above instructions in that order.
>>> Never lift the pen or pencil from the paper while drawing this out.
>> Check this out:
>>
>> https://i.ibb.co/C14qksJ/image.png
>>
>> Can be any size. It just keeps building it self.
> That is fantastic.

Thanks.

> It is weird that we are basically working on the same idea.

Yeah. It's nice that we are doing it in different ways. The final
animation should show the various methods.

Dan joyce <danj4084@gmail.com> writes:

> On Tuesday, June 13, 2023 at 9:02:39 PM UTC-4, Ben Bacarisse wrote:
>> Dan joyce <danj...@gmail.com> writes:
>>
>> > On Tuesday, June 13, 2023 at 12:18:54 PM UTC-4, Ben Bacarisse wrote:
>> >> Dan joyce <danj...@gmail.com> writes:
>>
>> >> > Many reply's in my original post thought it was impossible but now
>> >> > I am not sure if they are still doubters.
>> >>
>> >> But what are you claiming? I thought that you had accepted that the
>> >> original claim (an endless path that encloses, eventually, every square)
>> >> was not possible. So I am still wondering, what is it that people doubt
>> >> that you claim they should not doubt?
>> >>
>> > I never claimed it was impossible. I said just the opposite that was a
>> > proven fact that it is possible,
>> Can you state the claim? What this proven fact?
>>
> Did you see James's finite lattice rendering that shows a path being
> built and then getting filled back in with cells completing the
> lattice at the origin.

Yes. Finite paths, enclosing a finite set of squares, have been obvious
right form the start. The spiral is prettier than the rectangle, I
grant you.

> Good enough for a proof I'd say.

So you won't state your claim -- the one you pretend is widely doubted?
I must conclude that you are simply claiming what this construction
demonstrates, must I? OK. So who doubts that for any n, there is a
path that encloses at least n squares? I thought that was clear form
the very start.

--
Ben.

On Wednesday, June 14, 2023 at 9:01:04 AM UTC-4, Ben Bacarisse wrote:
> Dan joyce <danj...@gmail.com> writes:
>
> > On Tuesday, June 13, 2023 at 9:02:39 PM UTC-4, Ben Bacarisse wrote:
> >> Dan joyce <danj...@gmail.com> writes:
> >>
> >> > On Tuesday, June 13, 2023 at 12:18:54 PM UTC-4, Ben Bacarisse wrote:
> >> >> Dan joyce <danj...@gmail.com> writes:
> >>
> >> >> > Many reply's in my original post thought it was impossible but now
> >> >> > I am not sure if they are still doubters.
> >> >>
> >> >> But what are you claiming? I thought that you had accepted that the
> >> >> original claim (an endless path that encloses, eventually, every square)
> >> >> was not possible. So I am still wondering, what is it that people doubt
> >> >> that you claim they should not doubt?
> >> >>
> >> > I never claimed it was impossible. I said just the opposite that was a
> >> > proven fact that it is possible,
> >> Can you state the claim? What this proven fact?
> >>
> > Did you see James's finite lattice rendering that shows a path being
> > built and then getting filled back in with cells completing the
> > lattice at the origin.
> Yes. Finite paths, enclosing a finite set of squares, have been obvious
> right form the start. The spiral is prettier than the rectangle, I
> grant you.
> > Good enough for a proof I'd say.
> So you won't state your claim -- the one you pretend is widely doubted?
> I must conclude that you are simply claiming what this construction
> demonstrates, must I? OK. So who doubts that for any n, there is a
> path that encloses at least n squares? I thought that was clear form
> the very start.
>
> --
> Ben.
I don't get what you are saying. I had doubters, that's all that I am saying.
Nothing more. I didn't make any accusations on who were the doubters.

Dan joyce <danj4084@gmail.com> writes:

> I don't get what you are saying. I had doubters, that's all that I am saying.
> Nothing more. I didn't make any accusations on who were the doubters.

Strange. All I was asking is what was being doubted? I still don't
know what you claiming that could be doubted. If you can't state your
claim, at least say what was being doubted by the doubters.

--
Ben.

On Thursday, June 15, 2023 at 12:29:22 PM UTC-4, Ben Bacarisse wrote:
> Dan joyce <danj...@gmail.com> writes:
>
> > I don't get what you are saying. I had doubters, that's all that I am saying.
> > Nothing more. I didn't make any accusations on who were the doubters.
> Strange. All I was asking is what was being doubted? I still don't
> know what you claiming that could be doubted. If you can't state your
> claim, at least say what was being doubted by the doubters.
>
> --
> Ben.
The Lattice that I proposed that could be drawn --->oo and back without lifting the pencil.
In other words without retracing or crossing a line. Simple as that.
That would be putting a value on --->oo which I said was wrong but making the point that
giving a very large value for it like the number of cells >10^600 or greater for the size of
this square lattice then after it reverses and comes back down the path producing cells
on the way the >10^600 cells when reaching the origin are realized.

Dan joyce <danj4084@gmail.com> writes:

> On Thursday, June 15, 2023 at 12:29:22 PM UTC-4, Ben Bacarisse wrote:
>> Dan joyce <danj...@gmail.com> writes:
>>
>> > I don't get what you are saying. I had doubters, that's all that I am saying.
>> > Nothing more. I didn't make any accusations on who were the doubters.
>> Strange. All I was asking is what was being doubted? I still don't
>> know what you claiming that could be doubted. If you can't state your
>> claim, at least say what was being doubted by the doubters.
>>
> The Lattice that I proposed that could be drawn --->oo and back
> without lifting the pencil.
> In other words without retracing or crossing a line. Simple as that.
> That would be putting a value on --->oo which I said was wrong but
> making the point that giving a very large value for it like the number
> of cells >10^600 or greater for the size of this square lattice then
> after it reverses and comes back down the path producing cells on the
> way the >10^600 cells when reaching the origin are realized.

I don't think anyone ever doubted this fact. Can you point to a
doubting post? Every finite number of cells can be fully enclosed by a
suitable path (one that follows the "continuous with no retracing"
rules). This can be done in many ways.

--
Ben.

On Thursday, June 15, 2023 at 6:51:36 PM UTC-4, Ben Bacarisse wrote:
> Dan joyce <danj...@gmail.com> writes:
>
> > On Thursday, June 15, 2023 at 12:29:22 PM UTC-4, Ben Bacarisse wrote:
> >> Dan joyce <danj...@gmail.com> writes:
> >>
> >> > I don't get what you are saying. I had doubters, that's all that I am saying.
> >> > Nothing more. I didn't make any accusations on who were the doubters..
> >> Strange. All I was asking is what was being doubted? I still don't
> >> know what you claiming that could be doubted. If you can't state your
> >> claim, at least say what was being doubted by the doubters.
> >>
> > The Lattice that I proposed that could be drawn --->oo and back
> > without lifting the pencil.
> > In other words without retracing or crossing a line. Simple as that.
> > That would be putting a value on --->oo which I said was wrong but
> > making the point that giving a very large value for it like the number
> > of cells >10^600 or greater for the size of this square lattice then
> > after it reverses and comes back down the path producing cells on the
> > way the >10^600 cells when reaching the origin are realized.
> I don't think anyone ever doubted this fact. Can you point to a
> doubting post? Every finite number of cells can be fully enclosed by a
> suitable path (one that follows the "continuous with no retracing"
> rules). This can be done in many ways.
>
> --
> Ben.
There were people that doubted it was possible.'
Just look at the one that sent the video for one.
There were others also if you read all the posts.
Where is it documented that what I found is not unique?
That I copied it somewhere?
Please enlighten me.
..