Paradise Trinity Day

[Amigoo]

Lucky,

No song tonight - the proving concept was not reliable. I'll take a longer break this time and update the PDF file when anything significant appears in the research. Yes, 3-dimensional objects would present mind-stretching challenges (but 2-dimensional objects are challenging enough when movement of the objects has to be imagined).

Rod

[Amigoo]

The previous 4:44 am posting (not as early in Dallas) was a nice surprise later. I'm usually aware of the posting time immediately after verifying that the posting occurred. This time, I just turned off the computer and went to bed.

However, "4:44" may just be a "Hello" since it doesn't seem to relate to this geometry project. Or it might reflect on the comments about 3-dimensional geometry, perhaps indicating the expertise of this particular Midwayer.

Rod

[LurkerAbyss]

I dunno roderigo amigo, been thinking about the geometry myself lately and I noticed that the 4s "444" etc. have been huge lately. Also just got 1:11 while typing this. I have a feeling something good is on the way.



Rod this is how you might feel sometimes about that there math --->
But I think pretty soon you're gonna be all like --->

Love
Lucky

[Amigoo]

Lucky,

I was just going to enjoy your comments without posting
until I noticed 3:33 appearing on my computer clock: 4:44 - 3:33 = 1:11

Also, I missed this "error" when editing Diagram 5 in the PDF file. I was going to correct it but noticed that the file had been saved at 11:11 am. The "error"? The two periods at the end of the sentence:

"... the points on the circle where its square should be placed. ."

The symbolism is subtle: I was thinking about the fact that only two points on a circle are actually required to square the circle when I typed that sentence.

Rod

[Amigoo]

The online PDF file has been edited again and now appears ready for use in classroom environments for the study of geometry, in general, and for contemplation of "squaring the circle". The web site might be controversial, but the PDF file does not refer to the web site and may be downloaded.

This material is a brief but visually effective presentation of conjecture that "squaring the circle" is indeed possible. And the timing seems perfect for the informal inauguration of an international challenge (especially involving older children) to discover circle squaring geometry. The thousands of intriguing geometric patterns viewed during this research suggest that multiple solutions might be discovered ... some day.

Rod

[LurkerAbyss]

Dear Sir Rod

Do you have any plans from here on out as to how you are going to start presenting this idea and "getting it out there"?

Maybe one day I'll drive an ice cream truck

Love
Lucky

[Amigoo]

Lucky,

In today's electronic world, once an idea is shared online, it is already "out there". The acceptance of this new concept will take a while, but there's no hurry. A few thousand years have already passed since the ancient Greeks proposed this geometry challenge, so this seed of a new perspective will need time to germinate.

Despite having shared a lot of information (for several years) about this research with family, friends and others, the momentum for belief in this new perspective is still negligible. But this is easily explained: "squaring the circle" was "proven" impossible in 1882: http://en.wikipedia.org/wiki/Squaring_the_circle

So, changing the overwhelming momentum of that proof by 27.5971126... degrees might take a while. Interestingly, a friend and former computer engineer encouraged continuing research by asking, "What's the point?" And that question provided insight to focus on just a few points on the circle ... and not on the billions of other points to which Pi would direct our attention.

So, to answer your question, I won't be "out there" much in the current heat of another Texas summer. However, cold ice cream should be good for occasionally stressed neurons.

Rod

[Amigoo]

This continuing geometry exploration keeps discovering the importance of the .707 ratio. I was uncertain about the importance of posting this comment and had finally decided to ignore the thought ... until I returned to my computer and noticed the time: 10:07 am.

Apparently, this explanation should be recorded with the unfolding story, especially since the x:07 time prompts began when this ratio was first understood last October. Also, this topic of "Paradise Trinity Day" was first posted (as inspiration relating to the date of 10/10/10) during the beginning days of this geometry research ... and just prior to the .707 discovery.

The .707 ratio:

For a square having a side length of 10 units, the length of a diagonal line drawn from midpoint to midpoint of two adjacent sides is calculated by the Pythagorean Theorem (A squared + B squared = C squared).
In this example, C squared = square root of (25 + 25) = 7.0710678...

Conversely, side length x 0.70710678... = length of diagonal line.

Despite my lifetime reluctance to deal with "wiggly numbers" (large numbers, those with many decimal places, irrational numbers, transcendental numbers, etc.), I've been repeatedly introduced to two in this research:

1. The cosine for half of the Angle of Squaring Radii: 0.8862269254... (also known as half of the square root of Pi).
2. The .707 ratio: 0.70710678118...


Rod

[Amigoo]

Regarding the "wiggly numbers", more comment for number 2 ("in more ways than one"):

1. The cosine for half of the Angle of Squaring Radii: 0.8862269254... (also known as half of the square root of Pi).
2. The .707 ratio: 0.70710678118... (also known as half of the square root of 2).


Rod

[Amigoo]

Today's symbolism of progress complements this country's national Independence Day celebration. The updated Diagram 6 in the online PDF file may be the timely indication that a "good and sufficient" plateau has been reached - it's now time to reflect on the journey before advancing once more to the destination (which is still unknown).

Another sign of "plateau" seems to be the resolution of the "x:53" Midwayer prompt: I had long associated this prompt with a geometry design, created almost a year ago as a stained glass pattern - it was named "Squared Circle No. 53". The design was updated a few weeks later, but the later version was not displayed online.

These appear to be the consistent geometric relationships of the prompts which have continued to appear since the .707 discovery in October, 2010:

x:07 - The .707 ratio: 0.70710678118... (also known as half of the square root of 2).

x:26, x:27 - The cosine for half of the Angle of Squaring Radii: 0.8862269254...
(also known as half of the square root of Pi).

x:53 - Half of the .707 ratio (0.3535533905)
(in Diagram 6, half of the .707 ratio is reflected by the large golden circle, partially displayed.

As prompts associated with "wiggly numbers" (they're now becoming more stable in my mind), the Midwayer's choice of digits is intriguing. Of course, they're constrained by the digits available for time display, but they also needed to choose digits that I would recognize as meaningful patterns ... and digits that I would associate with a unique large number. "x:27" is even more interesting since it is a rounding of 269 in 0.8862269254...

As if to say "Don't forget about us!", "x:06" just appeared as I'm typing ... and then "x:07".
And this prompting just now was "good and sufficient" sufficient mental distraction:

A riddle:

In all of the designs, the starting point is colored white.

Meditation:

If a circle and a square of the same area ran a race
which geometric object would win?

A Joke:

Why was the race delayed?
Answer: the racers could not find their starting points.

Same question - different answer:

Why was the race delayed?
Answer: the circle just rolled around and the square kept turning the corner.


Rod

[Amigoo]

Regarding the online PDF file ...

This geometry material was updated and re-organized for use as a classroom handout, perhaps as an extra credit project.
The question everyone has been asking is listed below Diagram 4 (but no answer is given).

And now, there's better doodle space (Diagram 7) ... so, doodle on another creative concept:

Diagram 7 appears to be a new ball court design or board game ... or both (either independent or integrated):
Imagine, a board game that can be continued on a court or vice versa.

Rod

[Amigoo]

Diagram 8 was added to the online PDF file:

The previous night, I had a dream that I received an envelope with a return address of "Square Circles Club 707". Later during the day, remembering only this part of the dream, I assumed that "707"referred to the .7070107 number so prevalent in this geometry research.

Maybe it did, but the next exploratory design suggests otherwise: 707 appears in the upper center portion of this new design. Considering the total geometry, this may be that elusive 707 visual component that I expect to appear when the research is close to completion.

I hope to open that envelope from Square Circles Club 707 in another dream.

Rod

[Amigoo]

I've been confident for months that the Midwayers had selected specific 2-digit numbers from the "wiggly numbers" to use as prompts (each 2-digit number associates with a specific wiggly number). But I became curious today how they had made their selection of the 2-digit numbers.

x:07 is the original prompt and first appeared as 7:07 pm on my van's dashboard clock one evening last October.

Being quite fond of patterns, I discovered an interesting pattern today relating to the Midwayers' 2-digit number selection for the other prompts. Notice the digit positions of the selected numbers (decimal places 5 and 6):

x:06 - 0.7071 06 7811... half of the square root of 2
x:53 - 0.3535 53 3905... (half of the square root of 2) / 2
x:26 - 0.8862 26 9254... half of the square root of Pi
x:13 - 0.4431 13 4627... (half of the square root of Pi) / 2

"x:13" is most recently fascinating because, as I typed this information, I noticed that I had never seen this 2-digit number as a prompt. Yet, notice digit positions 5 and 6 in its wiggly number: "13" is a pattern number and should have appeared as a prompt, but I didn't recall ever seeing this 2-digit number ... until minutes ago! 11:11 pm had recently appeared on my computer clock - I didn't look at the clock again until this appeared: 11:13 pm. So, x:13 is now confirmed as a member of this family of prompts.

One more comment for tonight's tale of wiggly number prompts: this unmistakable prompt occurred recently: 3:14 (Pi).

Despite many observations, I have not yet detected that any prompt - any wiggly number - is a direct clue for analysis of the geometry ... even when the prompts occur during the research. As a group, these prompts still remain a simple "Hello, we're observing the adventure. We cannot tell you if the goal is achievable, nor how to reach that goal."

So, my only clue that the goal is achievable is that this family of prompts keep appearing. Weak evidence for sure, but they (plus the other common prompts) tend to appear when I begin to doubt the goal ... like 11:33 pm just now ... and then 11:44 just after I reviewed these notes.

Rod

[Amigoo]

Postscript:

Last night was the first time that I had seen x:13 (11:13 pm) as a prompt (eventually you just know when a time display is a prompt and when it is not). For instance, when arising from bed this morning, I walked to the bathroom, passing a clock: 9:09 am (not a prompt, but an interesting pattern to me). Then, I went outside to pick up the newspaper and turn on a water sprinkler. Next, I went to the kitchen to make coffee, passing the next time display for the day: 9:13 am (this is the brand new member of my geometry family of prompts).

This x:13 experience is an excellent indication of just how near Midwayers can be. The arrival of x:13 as a new prompt occurred in real time ... as I was typing. But the "communication" from them (in my experience) is one-way and only as prompts. George Barnard and others around the world are able to receive transmissions (AKA, "channeling").

Just coincidence? What are the odds that this can occur with these specific "wiggly numbers"? This particular prompting had occurred for months (except for x:13) before I had noticed this pattern yesterday.

Could I have created such a pattern? Unlikely. Until a few months ago, I was barely conscious of Pi and the square root of 2. When found in print, they only elicted this response from me: zZzzzzz (and many more).

Or consider these related patterns:

1. Each 2-digit number (a prompt) is preceded by the same 2-digit number in reverse:

0.3535 53 3905... (half of the square root of 2) / 2
0.8862 26 9254... half of the square root of Pi
0.4431 13 4627... (half of the square root of Pi) / 2

2. Each of the first two decimal digits is the sum of the next two groups of 2-digit numbers:

0.88 6226 9254... half of the square root of Pi
0.44 3113 4627... (half of the square root of Pi) / 2

Of course, the Midwayers didn’t invent Pi or the square root of 2 (or did they?). But this type of prompting gives strong evidence of their pattern awareness … and therefore of their interaction with humans who have natural (evolutionary) pattern awareness.

Rod (AKA, Ro - an email typing mistake last year)

[Amigoo]

Additional patterns in the Midwayer's focus group of digits for prompting:

1a. On the left side of "06", the sum of each set of 2 digits
is the same as the 2-digit number on the right side of "06" (78):
0.7071 06 7811... half of the square root of 2

1b. 78 minus 6 = 72 (the next sequential number after 70, 71):
0.7071 06 7811... half of the square root of 2

2. The sum of each of the first three sets of 2-digit numbers is 8:
0.3535 53 3905... (half of the square root of 2) / 2

There could be more patterns within this chosen focus group!

Rod

[Amigoo]

Today, the 11th day since I commented about a dream wherein I received an envelope from Square Circles Club 707, was the day for the arrival of an envelope in the geometry (see Diagram 3 in the online PDF file). Since this geometry is closer to the goal, the news in the envelope may be encouraging!

Rod

[Amigoo]

Documenting this adventure online has multiple purposes:

- to acknowledge Midwayer prompting as being integral to the adventure.
- to share a lot of evidence of a unique prompting experience.
- to help me remember the facts of the adventure.
- to keep me from changing "the story" later.

And speaking of facts ...

While rereading my earlier notes, I discovered a connection to the more recent x:06 prompts
(see the post of Thu Dec 16, 2010 7:43 am):

This short experience with a sound phenomenon seems to prove how Midwayers are accustomed to projects with long timelines. Almost a year had passed before x:06 became a frequent prompt. That earlier experience might have been testing to see if I would respond to other patterns of prompts ... or to see if I had sufficient curiosity to investigate unusual phenomena ... or was symbolism for a later project (this geometry research).

Rod

[Amigoo]

My 5:55 pm prompt just now was the encouragement that I needed to mention the recent discovery that the geometry is close - not yet perfect. But I've been able to reduce the problem to this "simple" challenge: a solution will require a geometric relationship of half of the square root of 2 (0.7071067811...) and half of the square root of Pi (0.8862269254...).

Considering the many geometric relationships that I've observed during this project, the "simplicity" equates to mixing oil and water with a toothpick. But I'm ahead of this game: I'm using filtered water and extra virgin olive oil. Maybe my toothpicks need to be organic.

Rod

[Amigoo]

An interesting revelation today about the "envelope" in the geometry:

What I found is not the actual geometry but an interesting pattern - the envelope may need to be a different size and, perhaps, positioned differently. The small circles confirmed that the current geometry is inaccurate.

But these adventures stress the reason why "squaring the circle" seems impossible ... and will stress the fact some day (maybe) that finding a solution could only be achieved by belief, faith ... and persistence. Expertise and skill are relatively unimportant for this kind of search.

Rod

[Amigoo]

More numbers associations which show the sophistication of the Midwayers' selection of these 2-digit numbers from these "wiggly numbers" for use as time prompts (x:06, 07, 13, 26, 53):

The isolated 2-digit numbers refer to the time promts which have been occurring for months:
Examples: 53 = 5:53 pm, 13 = 9:13 pm, 06 = 6:06 am

Today's observation: associations of the square root of 2 with the square root of PI:

1.4142 13 5623... (square root of 2)
0.4431 13 4627... (half of the square root of Pi) / 2

0.8862 26 9254... (half of the square root of Pi)

1.7724 53 8509... (square root of Pi)
0.3535 53 3905... (half of the square root of 2) / 2

0.7071 06 7811... (half of the square root of 2)

Also, the sum of the two-digit sets is the prompt just below them: 13 + 13 = 26
and 53 + 53 = 06 (high order digit dropped in the summation)

Other noticeable prompt sequences that occur: x:52, 53, 54
also, x:07 in 10-min. increments: x:07, 17, 27, 37, 47, 57

Methinks that the Midwayers are quite proud of their pattern awareness!
As for me, I just try to observe and record the impressive displays.

Rod

[Amigoo]

Two PDF files now reflect this continuing geometry adventure.
Click on the image of the web page displayed to view the PDF file.

The original file is still focused on the geometry:
http://www.aitnaru.org/homepage/freewill.html

This recently added file is a recap of the comments that I've posted.
The story is from my experience with prompting for the past year:
http://www.aitnaru.org/homepage/harmonyofchords.html

Rod (AKA "Ro" - a typing mistake last year)

"Row, row, row your boat ... in the stream of time"
... or risk drifting to points unknown.

[Amigoo]

I was curious why I had received a prompt a few days ago of my lockbox number (with a lot of prompting experience one just knows when a time display is a prompt). This was the first occurrence of this prompt and seemed to suggest that it was time to archive the project materials. Yet, I had no sense of project completion ... and believed that completion - if ever - was months away.

But this early morning's arrival of a pattern in the geometry at 1:11 am might explain the lockbox prompt. The new geometry was obviously a pattern and not the line that I had been focused on for months. Yet, the pattern provided easy confirmation of how the other line should be drawn. This was also a humorous reminder that I had just written how important patterns were in the "On Midwayer Prompting" document (in the other PDF file).

Diagram 8 of this PDF file (click on the site page image) shows the "pattern" geometry: the circled diagonal dark green line in the upper left portion of this design is the pattern; the diagonal dark green line above it is the one that I had been concentrating on: http://www.aitnaru.org/homepage/freewill.html

As for the green "envelope" in Diagram 3, if the current geometry is accurate, that envelope was just symbolism for this discovery today (the envelope related to my dream days ago about receiving mail from Square Circles Club 707). And as symbolism, the message might have been the announcement for today's arrival.

However, experience cautions me to evaluate the geometry again before searching for the lockbox key.

Rod

[Amigoo]

Another clue about the quality of this geometry?

Notice the 45-degree magenta line in the design? That's the "half of the square root of 2" line relationship (0.707106...) whose discovery was followed soon after by my first 7:07 prompt one evening last October. I've been searching for 9 months to discover how this line might relate to a solution.

Rod

[Amigoo]

This project has become excellent disappointment training - the most recent discovery is yet closer to the goal, but not close enough. Perhaps, the pattern discovered at 1:11 am is an important clue.

As I considered the many excitement/disappointment cycles of the past and how the goal keeps getting closer, I thought of ((square root of 2) / 2) ... especially the "/ 2". The pattern of these cycles - patterns seem to be everywhere - suggest this caution:

If you're traveling to a distant place and each day you can only travel half the distance remaining, when will you arrive?
Never - you might be atomically close in the final days but, technically, you will never arrive. Maybe the universe allows for "atomically close" and one day declares that your accumulated effort is sufficient.

Being far from atomically close, I searched for other meaning in the cycles - this may be the important lesson: we learn to recover from disappointment, but we can always work on our recovery rate. If during each significant E/D cycle (maybe "EDucational"), our recovery rate improves by half again the previous rate, that new rate will provide astounding benefits for all of life's remaining trials.

So, from this morning's lesson about the square root of 2, what might be extrapolated about the square root of Pi? One thing I know for certain: if you're going to walk in circles, for better speed and efficiency, don't walk in squared circles.

Rod

[Sandy]

Hi Rod,

I smiled when I read your post and it really spoke to me even though I am not doing anything nearly as complicated as attempting to square a circle. I get frustrated rather quickly and so this thread and your continued exercise in patience is good for me.
This is very profound...

Being far from atomically close, I searched for other meaning in the cycles - this may be the important lesson: we learn to recover from disappointment, but we can always work on our recovery rate. If during each significant E/D cycle (maybe "EDucational"), our recovery rate improves by half again the previous rate, that new rate will provide astounding benefits for all of life's remaining trials.

Thank you for that! It is exactly what I needed to hear today! Funny how we can apply these lessons in so many different ways...each one in a way personal for someone else.
Love,
Sandy