Paradise Trinity Day

[Amigoo]

Re: "Aurora" design

An interesting discovery yesterday:

There is an Aurora, Texas, just north of DFW airport ... and it had a famous UFO incident in 1897:
http://en.wikipedia.org/wiki/Aurora,_Te ... O_incident
http://www.ufocasebook.com/Aurora.html

I recall reading about this incident over 30 years ago, but did not know the exact location of Aurora then (my internet experience began 20 years ago). Now, I see that Aurora is about an hour's drive from my home.

Of course, I began searching online yesterday for the location of the Pi~voT Ranch ... and found no evidence of it. So, I returned to the Aurora geometry and discovered more geometric lines which effectively square the inner circle (dark blue). And these lines complete the abstract representation of a longhorn steer!

This might be inspiration to search again for the Pi~voT ... and take a ride:

Re: http://en.wikipedia.org/wiki/Texas_Longhorn
"Due to their innate gentle disposition and intelligence, Texas Longhorns are increasingly being trained as riding steers."

1. "I did not know that!"
Re: http://wiki.answers.com/Q/How_did_the_T ... t_to_Texas

"Texas Longhorns, but only as a breed name, originated in Texas; they were named as such in the mid 1800's. The cattle from this breed have ancestry from the Spanish long-horned cattle that came with Christopher Columbus when he and many others landed in what was termed the 'New World.' These cattle, upon becoming feral from escaping from the Spanish settlements, adapted to the land in and around what is now known as Texas"

2. "I did not know that!"
http://www.wikihow.com/Tell-the-Differe ... nd-Heifers

"The best way to tell if a cow is a cow and a bull is a bull is to look between the hind legs for presence of an udder or a scrotum, respectively."


Rod ... ...

[Amigoo]

Re: "Back to Square One" design

I chased this geometry around the world for three years only to learn:
"The square root of Pi defines the square of a circle whose diameter is 2."

... but this closure is a sweet treat!

Rod

Postscript:

I was disappointed to discover that the local posting time of 12:36 could have been 12:37 (the x:37 prompt),
but then noticed the unusual prompt: 1 +2 +3 = 6 (good commentary on this experience).

[Amigoo]

Re: Back to Square One design
Improved colors for visual "proof" of a squared circle:

Imagine an isosceles right triangle (positioned as shown), whose two sides equal the square root of Pi, is centered above the vertical center of a circle whose diameter is 2.

Allow the right triangle to move downward until the two points of the bottom horizontal line touch the circle. At that precise moment, a chord of the circle representing one side on an inscribed square is effectively created, directly associating the square root of Pi with that circle of diameter 2.

Today's meditation:
"Even when one learns to walk on water, traveling by boat is faster."

Rod

[Amigoo]

Re: Back to Square One design

I've been exploring several versions of this geometry (different lines and colors but the same core geometry) and have finally settled on the current design.

I still conjecture that the circle cannot be squared but a squared circle can be proven geometrically (the "conundrum" mentioned previously):

In the design, the green scalene triangle (part of the larger isosceles right triangle) contains a 45-degree angle and, theoretically, a base length equal to the square root of Pi. In this concept, if the circle is squared, the left diagonal side (one end is attached to the Pi line) must equal the length of a side of a square inscribed in the primary circle. And for the square root of Pi, only one diameter provides such a side length: a circle having a diameter of 2.

That Pi is an irrational number is insignificant since the lines in the scalene triangle are complementary (perfectly balance the irrationality of Pi).

"To square the circle, one must circle the square."

Rod

[Amigoo]

Re: http://www.aitnaru.org/images/Pi_Corral.pdf (new "I-Square" design)

My "exit strategy" is limpid but the door is now wide open - maybe this will be the last design of this research plateau. The 2:22 PM prompt on my way to the computer to post this may indicate the sponsor of today's creativity and inspiration:

In those days long ago, a buried stone box was discovered by a minuscule group of reformed geometers. Having been popularly diagnosed with morbus cyclometricus, they vowed to maintain secrecy about their discovery (how their experience was revealed is still a mystery). The papyrus fragments in the box required translation but eventually revealed evidence of a remarkable device, once briefly used to square circles.

The device promised to be a commercial success but its nickname, "I-Square", ruined its reputation even before the first model hit the geometry store shelves. After all, why would geometers seriously consider purchasing an "Impossible Square", a nickname referring to the supposedly futile quest of squaring the circle.

Once the I-Square was reconstructed by the reformed geometers, its use became intuitive quickly. Fortunately, a tiny papyrus fragment contained the critical angle for the movable center line: 152.403 degrees. And this helped explain why another fragment contained the disclaimer "User calibration required". As shipped to the stores, the I-Square was calibrated to 152.403 degrees but required more delicate calibration, depending on the user's need for squared circle precision.

How did it work? "Child's play!" Align the length of the radius of a circle on the movable center line with one end of the radius at the adjusting dial (lower right of design). Draw the circle with its center at the other end of the radius - two points of the circle's square are immediately identified by the two sides of the 45-degree angle of the device.

Why did it fail to capture the circle-squaring market? Its nickname was the primary reason, but the early adopters kept yelling "I squared!" when first seeing the power of this fantastic device. These exclamations quickly found their way into anecdotal tales as "I sweared!". Unfortunately, swearing for any reason was taboo in those rural conservative communities.


Note: the I-Square design shows two circles and their squares, but the I-Square is the simplest of circle-squaring devices ever invented: a 45-degree angle with an adjustable center line that pivots at the vertex (location of the user calibration dial). Many of the early adopters reported that the factory-calibrated 152.403 degrees was perfect for most of their circle-squaring projects.

Rod ... ...

[Amigoo]

If not sponsored by 2:22 then 3:33 ?

After posting my previous comments, I wandered to the kitchen for snacks (by quantity consumed, "snacks" became "lunch"). While munching and contemplating a real model of the I-Square (maybe even laser-guided), I glanced at the clock: 3:33 PM.

Such a model would be easy to construct ... in various materials. But in this digital world, "entertainment" (gag gifts, educational mterials, wall hangings, etc.) would be the popular use. Maybe round paper dollars (the ones that seem to roll away faster and faster) could be converted to square dollars (the ones with a firm foundation).

Rod

[Amigoo]

After more reflection ...

The I-Square would have greatest commercial value as an electronic educational device:

As the Square One Conundrum design hints and the I-Square design suggests, "squaring the circle" is simply a question of how much precision is required. "Perfect" precision (atomic level and greater) is a bit difficult to achieve ... but not impossible.

An educational I-Square device, for example, could display the distance at which the requested precision (i.e., number of Pi decimal points) becomes significant for the inquiry. "Scientific applications generally require no more than 40 digits of Pi" (re: http://en.wikipedia.org/wiki/Pi ).

So far, the only practical use I can think of for a circle-to-square converter is if I wanted to dress like Sponge Bob. And that would be quite a conversion of a Santa mini-belly into square drawers!

Rod

[Amigoo]

New day - New inspiration:

The 152.403 degree angle of the I-Square design is all that's required for a clear plastic protractor to be used to square a circle (plus simple geometry). The edge of the protractor should allow the circle-squaring chord of the circle to be drawn.

A perfectly squared circle requires precision not permitted by paper and pencil, but such a protractor would be useful for many projects.

Rod

[Amigoo]

Important:
Advanced users calibrate I-Square to 152.40288736430939554826779524767 degrees
but wear a seat belt when forcing the device to this precision level.

Rod

[Amigoo]

Re: Square One Conundrum design (revised)

The addition of the large half circle to identify the circle's diameter provides entertaining symbolism:

If a geometer who keeps trying to "square the circle" has the disease of morbus cyclometricus, are they "off their rocker" (the half circle) when they stop believing that it's possible?

Rod ... ...

[Amigoo]

Explanation of the I-Square method:

An easy way to square a circle with a protractor and compass:

Draw a 45-degree angle with the sides at 135 and 180 degrees.
Draw a center line from the vertex of the angle at 152.403 degrees*.
Mark the radius of the circle along the center line from the vertex.
Draw the circle with the compass point at the left end of the radius.

The two sides of the 45-degree angle and the vertex identify three points of the circle's square and creates a chord that has length equal to the side length of the circle's square.

This is not a solution to "squaring the circle" but reveals the scalene triangle that might be useful in proving that the circle is squared (via the diagonal chord that has length equal to the side length of a square inscribed in the circle).

* Draw center line at 152.40288736430939554826779524767 degrees for best precision.

[Amigoo]

Re: Square One Conundrum design (revised again)

In every squared circle, there's another waiting to be revealed.
Improved design, more deserving of the appellation "Conundrum".

Rod

[Amigoo]

An 11:22 AM prompt when sending this email to friends convinced me to copy it here:

Re: http://www.amazon.com/School-Smart-Chal ... cr_acr_txt

A concise summary of this method and (apparently) good geometry tools for teaching it the old-fashioned way (without electronic devices).

Go Dallas, Texas! (note the location of the first two reviewers)

Rod

[Amigoo]

Re: "Draw center line at 152.4028873643.." written in the I-Square method

Maybe closure for the x:52 prompt ...

To this day, I had not found convincing correlation between the geometry and my recurring x:52 prompts. But a sense of closure (at least for this research plateau) is hinting that the I-Square method in general and the 152.4028873643.. degree angle in particular is the "message" of the x:52 prompt.

By the way, 12:21 PM appeared on a nearby clock minutes ago when I began associating this prompt (x:52) with this angle.

Also, I had seen many times x:52 followed immediately by x:53. Checking my chart of geometry/prompt associations, I see that x:53: relates to the square root of Pi (1.7724 53 8509..) and to half the square root of 2 / 2 (0.3535 53 3905..). Therefore, this closure might include confirmation of conjecture three years ago: a solution to "squaring the circle" requires geometric association of the square root of Pi with the square root of 2.

I'm still not convinced that a solution is possible, yet believe that it may be possible to prove that a circle is squared (the conundrum ). But it was important in these years of research that I believe that a solution was possible since proving a circle's square was not an enticing quest.

Rod ... ...

[Amigoo]

Speaking of Midwayer prompt 12:21 ...

I had just finished updating the Tips for the I-Square method, when I glanced at the clock: 12:21 AM.
This "Math reference" shows how closely Pi is complemented by the I-Square method:

Tips:

1). Try a circle with a diameter of 2 units (or 20, 200, 2000, etc.) to correlate results directly with half the square root of Pi.
2). The 62.4028873643.. degree radius also squares any circle with simple geometry!
3). Math reference:

Input half square root of Pi to calculate precise angle:
acos(0.88622692545275801364908374167057)
= 27.597112635690604451732204752339 degrees

Subtract from 90 degrees to calculate angle of radius:
90 - 27.597112635690604451732204752339
= 62.40288736430939554826779524767 degrees

Add 90 degrees to calculate angle of I-Square center line:
90 + 62.40288736430939554826779524767
= 152.40288736430939554826779524767 degrees

Rod

[Amigoo]

Still updating files with the research ...
and created one more design: "Point-to-Point Pi".

"Point-to-point Pi in a scalene overture of overlapping inscribed squares."

Geometric evidence that a circle of diameter 2 must be squared if the square root of Pi (horizontal blue line within the circle) thus separates two inscribed squares within the circle. Any other separation (length of line) is not the square root of Pi.

Point-to-Point Pi is in this smaller file:
http://www.aitnaru.org/images/Alright_Triangles.pdf

The I-Square Method is now described in the primary file:
http://www.aitnaru.org/images/Pi_Corral.pdf

11:11 AM said hello [and now 12:21 as I type] when I decided to post these comments: I had started on my way to the kitchen to make coffee first when I postponed that a few minutes to stop by a bathroom. That delay was just sufficient to view 11:11 on my way to the kitchen once more.

Rod

[Amigoo]

After one more attempt to improve the color and clarity of the Point-to-Point Pi design, I briefly wandered to a computer folder where this "recipe" (below) was saved. Its title seemed to correlate with other point-to-point subjects: distances of astronomic objects. A review of this "big whirl recipe" (notes condensed from UB Papers 15 and 42) was enlightening yet I planned to quickly return to the geometry.

Then 11:11 PM appeared, hinting that the notes might have relationship to this geometry research. Besides, this simple recipe for creating matter (and galaxies, etc.) might easily be found in any astronomy cookbook in a mansion world library:

The force-charge of space (space potency, circulating around Paradise in much of the Master Universe) is first transmuted by the Paradise force organizers into primordial energy (puissant energy). Paradise force organizers originate nebulae “when once started, can never be stopped or limited until the all pervading forces are mobilized for the eventual appearance of the ultimatonic units of universe matter.”

Then, Associate Master Force Organizers continue the transmutation of the force-charge, creating gravity-responding energy from the puissant energy. This gravity energy is “becoming directly responsive to the circular grasp of Paradise (absolute) gravity while disclosing a certain potential for sensitivity to the linear-gravity pull inherent in the soon appearing material mass of the electronic and the post-electronic stages of energy and matter.”

Next, Power Directors (and then, Power Centers) continue the transmutation: “the organization of evolved energy into matter entails the concentration of energy into discrete masses of definite dimensions and established weight - precise gravity reaction.” “Power Directors transmute energy into matter” as well as create power circuits and energy channels for the superuniverse; Power centers (and other forces) transmute ultimatons into electrons which later become detectable matter.

Serve hot and swirling.

[11light11]

Thanks Rod! I just love this thread . . . .

[Amigoo]

Michele,

Re: http://en.wikipedia.org/wiki/Universe

"The Universe is commonly defined as the totality of existence ... The observable universe is about 46 billion light years in radius."

This is a fascinating subtlety of the UB: The "recipe" (my previous comment) and this reference (below) to other other universes indicate science revelation. Current science hypothesizes that only one Big Bang has occurred - the UB describes the process as a big whirl, having occurred multiple times ... and still occurring:

"In the not-distant future, new telescopes will reveal to the wondering gaze of Urantian astronomers no less than 375 million new galaxies in the remote stretches of outer space. At the same time these more powerful telescopes will disclose that many island universes formerly believed to be in outer space are really a part of the galactic system of Orvonton. The seven superuniverses are still growing; the periphery of each is gradually expanding; new nebulae are constantly being stabilized and organized; and some of the nebulae which Urantian astronomers regard as extragalactic are actually on the fringe of Orvonton and are traveling along with us." (12:2.3)

An interesting prompt: I've mentioned 12:21 recently and noticed this prompt again just after submitting email to a UK mathematician about the I-Square Method. 12:21 might relate to progress, indicating "cycle completion" or "complementing action".

Rod

[Amigoo]

Regarding "parallel universes" (more science fiction than science theory) ...

If we believe that humans are indwelt by a unique "spark of our creator", parallel universes with human "clones" of another universe cannot exist.
But similarity of universes (at least, similarity of inhabited planets) seems possible with "375 million new galaxies" in the celestial plans.

Rod ... ...

[11light11]

Hi Rod!

Thanks for the link, and for sharing your new updates! 46 billion light years!!! It's wild to think about the wording, that this is the circumference of the "observable" universe. Boggles the mind just how far this truly extends!

Have you ever heard the theory that perhaps the universe isn't expanding -- perhaps it expands and contracts, as a heart beats?

With love! Michele

[Amigoo]

Re: http://aitnaru.org/threepoints.html

Updated to include the red isoceles right triangle whose lines (and extensions not displayed) identify more uniqueness of this Pi-squared circle.

Yes, Michele, I've read a little about the expanding/contracting theories. And the UB refers to space respiration!

Rod

[Amigoo]

Re: http://aitnaru.org/threepoints.html (design updated)

The geometry of this Three Points design (image name = "ACOS I") hints that, since Pi is "enclosed" within the scalene triangle, its transcendental nature dissipates. Maybe a transcendental number becomes algebraic when corraled by a triangle, inscribed within a circle.

A summary of the summary:

The actual geometry of a squared circle can be studied with construction based on the 62.4.. radius (simple geometry squares the circle; precision increases with more decimal digits):

Input half the square root of Pi to calculate the precise angle:
acos(0.88622692545275801364908374167057..)
= 27.597112635690604451732204752339.. degrees

Subtract from 90 degrees to calculate the angle of the radius:
90 – 27.597112635690604451732204752339..
= 62.40288736430939554826779524767.. degrees

Tips: Let circle's diameter = 2 (or 20, 200, 2000, etc.) to see numeric correlation with Pi. After the circle is squared, look for other geometric objects (especially parallelograms, triangles, inscribed squares) to appreciate possibilities for proving the square.

[Amigoo]

"I think I'm thinking" ... (reflection on the I-Square research)

Conjecture: A transcendental number may become algebraic when corraled by a triangle, inscribed within a circle.

Consider the scalene triangle:

Given: Circle’s diameter = 2 units; 45-degree angle includes Pi side (square root of Pi); one non-Pi side has length equal to a side of the circle’s inscribed square (effectively, square root of 2, irrational but not transcendental).

Observation: The length of the other non-Pi side must adjust as the Pi side becomes more precise (gains decimal digits). Does this non-Pi side balance, absorb, or dissipate Pi’s transcendentalism and/or the other side’s irrationality?

Rod

[Amigoo]

Re: http://www.aitnaru.org/images/Alright_Triangles.pdf

"Perhaps, a transcendental number becomes algebraic when corraled by a triangle, inscribed within a circle."

The new "Transcendental Transition" geometry is a good study of the transcendental nature of Pi (its square root). For once the green, horizontal square root line, 45-degree angle, enclosing circle, and side of the circle's inscribed square are drawn perfectly (in theory), the length of the Pi line cannot change ... even one-trillionth of a centimeter!

And this hints that Pi's transcendentalism ends! What math would describe such transition?

Rod