Paradise Trinity Day

[Amigoo]

Re: http://www.aitnaru.org/homepage/freewill.html

The research notes are now available as a downloadable PDF file.
Click on the design image to view the notes.



Rod

[LurkerAbyss]

Wow

Was just reading about squaring the circle not too long ago, and I didn't know there was a way around squaring the circle without Pi, to be honest.

How do you feel about the progress and results so far, Rod?

Love
Lucky

[Amigoo]

Lucky,

I'm not a mathematician, but the numbers may hint that Pi announced retirement on 6/5/11 as a constant relating to circle measurement. We'll know for sure in the days ahead when the formulas are verified ... especially by those who believe that "squaring the circle" is impossible.

But if the theory is proven correct, the unfolding of the discovery will be unlike previous mathematical discoveries - I perceive this more as symbolism relating to the dawning of a new spiritual era.

Rod

[Amigoo]

Commentary and some history of this research ...

A friend teaches high school math (including trigonometry), but does not have an interest in this research. I suspect that Pi has been so esoteric for centuries that many mathematicians need to know first how a solution is derived before they will believe it. And this hints that theoretical mathematicians are like a benevolent priesthood - if they claim that a math solution is impossible, many followers will not believe otherwise ... and they will not believe that a solution can be simple.

Months ago, an acquaintance (a computer engineer) tried to convince me with higher level math that the problem could not be reduced to "points on a circle" because the number of points could never be determined - they were infinite because of Pi. His argument actually helped my research because I learned not to focus on the total number of points - just on the points that were needed for a solution (8). And I next learned that those 8 reduce to 2 because of simple geometry!

But this is why I believed that a non-mathematician had a reasonable chance of discovering how to square a circle: brilliant mathematicians had used their greatest brain power over the centuries to attempt solutions, even to the point of "proving" that a solution was impossible. Therefore, I theorized that a solution would probably not be found by traditional math research ... even when using powerful computers. This perspective also relates to my comment about "belief and faith" with the online design.

This simple trigonometric solution derives from a non-mathematical approach: primarily pattern recognition and intuition - even the 0.886226 (the cosine) was often a pattern to me and not a value. The resulting math discovery seems enticingly simple: precisely define the cosine angle in a right triangle and the trigonometry (plus a good calculator) will do the necessary work.


Midwayer prompting continues to evolve ...

Interestingly, prompting now includes this series: x:07, 17, 27, 37, 47, 57. This series might represent a symbolic odometer, perhaps suggesting that the engine for the final project solution has been started. Maybe my x:53 prompts (still a mystery) will now suggest the final research direction. A strengthening hint: the 27-degree angle is present (or as potential) in other parts of the design.

Rod

[Amigoo]

I find the x:07 - x:57 prompting series intriguing and it seems to provide another sample of how the Midwayers communicate creatively with prompts.

In the revolving series:
x:07, 17, 27, 37, 47, 57, 67, 77, 87, 97, 107, 117, 127, 137, 147, 157, 167, 177, 187, 197, 207

There are 6 displayable clock increments (07 - 57), followed by 4 non-displayable increments (67 - 97) for a total of 10 increments. The incrementing value (10) plus these displayable and non-displayable increments seem to emphasize the decimal system (and Urantia is a decimal planet). Most other incrementing values (except 100, 1000, etc.) would not have allowed the base numbers to repeat during each cycle.

Also, the math associated with the geometry is very decimal oriented. In addition, the use of a 10/100/1000/etc. based line length or circle diameter (e.g., L - 100, D = 1000) has been very helpful in discovering relationships among the geometry components.

These related prompts began last October when I selected a square of 100 units (L = 10) for investigation. Soon, the ratio of a 45-degree diagonal line between the midpoints of two adjoining sides of the square became apparent: 7.07107... units. When I learned this, the x:07 prompts began ... and have continued to this day.

Although none of the prompts have been direct answers to problems, they have been entertaining, encouraging and motivating. In all, good and sufficient evidence that a worthy project is unfolding.

[Amigoo]

In fairness to Pi ...

The 1128379.16709552 diameter that was selected for the examples is biased toward the new formula (to cause the display of many zeros in the calculation, suggesting accuracy). A more neutral diameter (relative to the final calculations) will provide better examples. However, the calculations are very close: the differences in the calculations typically occur only after 6 or more decimal places.

In the following comparison (when testing a new diameter), the top horizontal side of the right triangle shows this difference (rPi = side length with the new formula; cPi = side length with the current Pi formula):

rPi = 522723.20087707008050712732250369
cPi = 522723.20087705749026211017854768

Summary ...

Nothing has been proven yet except the fact of differences in the very close calculations. But the trigonometry theory appears to be holding firm: calculate (or measure) the right triangle's cosine angle precisely and the square of the circle can be calculated precisely ... without using Pi. At least, the new formulas appear to be challenging Pi.

Rod

[Amigoo]

The current research and prompts seem to be leading in this direction: once the cosine angle is determined precisely, a method for "squaring the circle" will have been discovered. Considering how many hours it has taken to reach this point, that discovery might take months.

But my 9:07 prompt just now suggests "maintain confidence, but keep working on the research ... at a more reasonable pace". That the trigonometry theory still appears correct means that greater precision is the proper goal. I'll update the research notes attached to the online design if better numbers appear in the ongoing research.

A nice closing: 9:17 just appeared.

Rod

[Amigoo]

Re: "once the cosine angle is determined precisely, a method for 'squaring the circle' will have been discovered"

This latest number crunching (see online PDF file, June 8, 2011) shows that the trigonometry formulas easily keep up with Pi. But my mind still can't accept that an angle size cannot be known precisely. Yet, like Pi, the current numbers try to emphasize this. So, I'll consider this a temporary stalemate - Pi still rules!

If anything has been accomplished, it's that Pi can be related to this right triangle trigonometry ... and that a squared circle can be defined by 8 points (or just 2) on the circle's circumference. As far as removing Pi's ghostlike influence on the calculations, that perfect achievement awaits a precise cosine angle ... and a geometry map for getting there.

Rod

[Amigoo]

On reflection, a method to "square the circle" might be found in this updated information (see second diagram of the online PDF file). With the trigonometry now tracking Pi nearly digit for digit, this second diagram seems to offer significant clues. At least, this is where my focus will remain for a while.

These have been the long-term goals of this project (in this order):
1) Show how to prove geometrically that a circle is squared.
2) Show how to "square the circle" (draw the geometric lines by the classical Greek rules).

I now theorize that proving the exact cosine angle (a mathematical solution) will lead to completion of goal #1 and this might lead to completion of goal #2. In the latest number crunching, digits directly from Pi (multiplied by increments of 10) are used in the new Area formula to determine the necessary cosine angle. Several calculations firmly suggest that all known digits of Pi can be processed by the formula. Although this approach seems to verify the formulas (the trigonometry), it is not a geometric solution.

At first, the thought of an angle having hundreds of decimal places was overwhelming - theoretically such an angle could exist! But my experience with trying to relate to an infinite number of points on a circumference suggested a different perspective: the classical Greek challenge is not concerned with the size of the angle - just the discovery of geometric relationships amongst lines.

Because the rules of geometry (and trigonometry) have been confirmed for centuries, these are appropriate tools for crafting a solution ... with one minor problem: Pi is a "good guess" (not to mention its esoteric brilliance) about the relationship of a circle's diameter to its circumference.

So, the challenge of this new era is to measure the geometric objects of our universe with even more precision - "infinite" is not an accurate description of most of those measurements. At least, only when a circle is known to be of infinite size should the diameter/circumference ratio be a "good guess". Ask our First Source and Center, who lives at the very beginning of "infinite".

But this is just perspective from a non-mathematician who believes that many complicated solutions are unnecessary. The geometry of a squared circle now looks rather simple - the problem is how to describe that geometry with wiggly numbers. Or maybe they just wiggle in my mind as I try to relate them to circles and straight lines.

Rod

[LurkerAbyss]

Hey Rod,

I have to say, I am fascinated and I think you are pretty much right on the edge of something here.. I feel it. Always nice to hear from others learning about things from "non-" perspectives.. you could call me a non-mathematician, non-historian, non-scientist, non-religionist, etc. and yet I am interested in all of those things. Indeed, it seems to be some of us with a far broader and open experience who are not constrained by a focus or a bias; we may simply just be lovers and architects of the universe.

Love
Lucky

[Amigoo]

Lucky,

Thanks for the comment!

I knew better in high school, but over the many years of not paying attention to geometry I had forgotten the most basic terminology: compass and protractor. Until a few years ago, a compass was that sharp pointed thing in my desk drawer that I used sometimes to create a new hole in a belt (when the "most local universe" was expanding). And a protractor was that poorly designed windshield ice scraper (it was vulnerable to breakage).

So, when my interest in the Greek challenges from antiquity suddenly arose in 2008 (I was sitting at a desk, playing with three pennies on an arc drawn on paper (re: trisecting an angle)), I knew that I would need to rediscover the necessary tools. But which tool was the protractor and which was the compass? A few online pictures solved that terminology problem.

And speaking of the belt issue, I learned that a mathematical constant that sounded like a favorite dessert needed more respect from me. Respect was soon established, but my mind never lost the perspective that this constant is an extremely well-reasoned mathematical guess. And the continuing defense of Pi as proof of our inability to "square the circle" seemed remarkably intense.

But this was my deciding clue to bond with Pi one slice at a time. Naturally, a first creative concept was to solve the problem by recommending that only square pans be used for baking. But then I realized that most aficionados of this dessert preferred a wedge - not a square piece. And that meant that circular pans would continue to be used for many more centuries.

In those early days of this research, I used pan lids and rulers to get better perspective on the challenge. As recently as a week ago I had brought a large trash can into the house to wrap tape around it and measure the length of that tape. A good experiment: my measurement reflected Pi's influence and I was able to measure precisely ... to 1/10 of an inch! But new insight: I was never going to be able to measure trash cans that had less precision in their diameters ... unless I used a more detailed tape measure and had bionic eyes.

This real world appreciation sent me back to the computer to crunch numbers ... many wiggly numbers. As with the trash can experience, I quickly learned that many wiggly numbers cannot solve the problem. If a circle is going to be squared, the answer must be discovered amongst relationships of geometric lines. And neither should bionic eyes be required to observe those determining relationships.

At least, this is what I read in this morning's organic green tea leaves.

Rod

[Amigoo]

The online PDF file was updated once more - this one might be worth saving! The concept, formulas, cosine and other wiggly numbers seem reliable.

Also, new relationships of geometric lines are shown, including those which hint that "the light at the end of the tunnel" may soon be observed.

Rod

[Amigoo]

The light blue triangle in the second diagram of the PDF file was found to be imperfect when examined with magnification, so it was removed. I'll leave this PDF file alone now until I'm certain about any "circle squaring" geometry. The two diagrams should help to explain the focus of the trigonometry and to conceptualize possible solutions.

The Pi information and calculations are correct, but are very dependent on the Angle of Squaring Radii (half of which provides the cosine value). That there are 30 decimal places in the ASR is a strong clue that discovering "circle squaring" geometry will remain a significant challenge ... perhaps requiring more good fortune and inspiration than mathematical analysis.

And this perspective seems to explain why Midwayer prompting is involved: belief and faith will probably be the deciding factors in any discovery of "circle squaring" geometry - not mathematical brilliance (that approach has been tried for many centuries).

Of course, yielding to the historical persuasion of Pi is an option, but that same persuasion expressed in trigonometry would be more convincing. Is it possible to claim that no geometric relationships exist which would determine a precise ASR?

Rod

[LurkerAbyss]

wow

love you bro

Love
Lucky

[Amigoo]

Lucky,

When the light blue triangle was removed from the second diagram, the light blue square insisted on returning to the first diagram - symbolism of universe balance? Judging by patterns and line lengths (when other lines are drawn on the first diagram), it is not possible to claim that no geometric relationships exist which would determine a precise ASR.

In other words, the elusive solution seems to be lurking in this geometry. But unlike the infinite digits of PI, the abyss may not be immeasurable as currently believed by many mathematicians. A slower pace now in the research may allow inspiration to direct this intriguing path.

Rod

[Amigoo]

Apparently, "slower pace" was not going to be encouraged - another subtle hint about a mysterious agenda.

The 4th design in the online file reflects the greatest accuracy yet! The light green diagonal line was a clear signal that an important pattern would soon appear. This 45-degree diagonal line was a reminder of the .707107 relationship of a diagonal line which connected the midpoints of two adjacent sides of a square.

This clue soon led to the discovery of the magenta square whose side length equals the diameter of the magenta circle above the square. And this circle appears within an isosceles triangle whose left (blue-green) and right (golden) sides just happen to form "707" (when the left side is reversed horizontally).

This online PDF file may be worth downloading and saving for future reference.

Rod

[Amigoo]

An observation of the 4th diagram ...

The large golden right triangle, magenta circle and magenta square below are closely related geometrically (each includes a line length of 2,000,000 units) – an auspicious association of the basic geometric objects: straight line, circle, triangle, square.

Rod

[LurkerAbyss]

Thanks Rod; PDF downloaded and saved.

Might be a trivial thing but I liked the quote at the end and was just wondering what you get from it, "two points on a circle may reflect a triangular relationship".

The beauty of geometry to me is that, at the end of the day, it seems to reflect and teach us about how perfectly this universe really seems to be made.. I am astonished and moved by the very thought that humanity continues to be able to discover how God's universe works, and how perfectly it really is designed; it seems to me that there's nothing in this Universe that doesn't make sense.

EDIT:
Somehow Rod, I also have a hunch that any serious discoveries and milestones in this never ending mission will extend to third- and fourth- (etc.) dimensional geometry and open up new possibilities (or capabilities, rather) for discovering more about temporal-spatial existence and other mysteries.

Love
Lucky

[Amigoo]

Lucky,

"Two points on a circle" refers specifically to the geometry (two points on a circle in relationship to each other and to the center of the circle). But the expression may also relate to our separate physical and spiritual natures which have potential to unite in harmony with the increasing influence and presence of God.

At least, this is where my mind wandered just now when considering the question.

Rod

[Amigoo]

Diagram 4 was updated once more and now has better design balance.

While reflecting on what symbolism the magenta circle and square below might represent,
the lower case letter "i" had noticeable presence. And what might "i" symbolize? I AM

... which complements the trinity of colored circles having the same diameter (2,000,000 units) and also reflects on the design's message about the importance of belief and faith when attempting to "square a circle".

Rod

[Amigoo]

Very typical of progress: hours turn into days; days into weeks ... but the prompts continue, often with increasing frequency just when I've convinced myself to take a days-long break, believing that further progress might require months. This early evening event was especially entertaining:

Visually and by the numbers, you wouldn't expect these two line lengths to be related:
1352736.55127079 and 1913058.19096438

But on inspiration (and because of the 7:07 time prompt which effectively canceled a very late afternoon nap), the first number was divided by the second: 1352736.55127079 / 1913058.19096438 = 0.707107 (rounded)

This result immediately informed me that a special relationship exists between the large dark blue circle (see Diagram 4 of the online PDF file) and the small dark blue circle which have the same center. Since the large circle has twice the diameter of the small circle, this numeric relationship (0.707) between the lines is a good omen: the Angle of Squaring Radii (see first page of PDF file) might be geometrically provable.

At least, the numbers - even the "wiggly numbers" - are forecasting continuing progress. My x:53 prompt just now agrees (I may not learn the exact purpose of this prompt except by hindsight). And speaking of prompts, x:06 is a frequent visitor but doesn't seem to relate to the project prompts (x:07, x:53, x:27,) in the same way. X:06 seems to have the "signature" of an interested visitor.

Also, an interesting phenomenon: a forecast of good progress for the day is often indicated by a morning headache. I might describe this as a "thrashing of neurons" in anticipation of the day's exploration.

Rod

[Amigoo]

Diagram 4 in the PDF file was updated again.

These designs visually help to reinforce the concept that a squared circle is a finite entity. Pi may be transcendental, but it is just a mathematical tool used to estimate (but quite accurately!) some measurements of a circle.

This research is now focused on the discovery of a unique relationship of geometric objects that would determine the precise Angle of Squaring Radii (ASR - see first page of attached PDF file). The trigonometry concept is solid - all known digits of Pi can be used via the Area formula to create a more precise ASR. But the thought of an angle having a transcendental measurement seems illogical. At least, in the real world precision measurements usually stay above the sub-atomic perspective.

No discovery yet, but the possibility of such a relationship seems to increase with the ongoing research. For intentional symbolism of progress, I cut out a small piece of pie from the center.

Rod

[Amigoo]

Regarding "the trigonometry is solid" ...

If 0.88622692545275801364908374167057 (the cosine) seems familiar, it is half of the square root of Pi. This indicates how closely the trigonometry relates to a precisely squared circle ... and comments that "entering by the back door" may be a very long way to enter the house:

This cosine could have easily been converted into the Angle of Squaring Radii (ASR). But knowing how that value related to the trigonometry of a circle may have been the real lesson of the "long way".

Rod

[Amigoo]

Diagram 5 in the online PDF file was updated.

This needs to be verified again, but the small magenta circle appears to focus on the visual geometric proof that the circle is squared: the circumference of the circle and the diagonal line cross the top line of the square only when the circle is squared.

Rod

[LurkerAbyss]

Hey Rod! Sing us a song!

Have you ever pondered upon the three-dimensional implications of this research? In other words..CUBING the SPHERE? Think about it Rod.. think about it!

Love
Lucky