Paradise Trinity Day

[Amigoo]

Re: The Right Chord design

Geometrically proves why the circle-squaring right triangle exists
for diameters = 2.0 and sqrt(Pi) where SoCS = sqrt(Pi).

The chord created between D = 2.0 and D = sqrt(Pi)
effectively defines the circle-squaring right triangle.
But is this chord a dyad or triad (how many notes)?

Rod

[Amigoo]

Re: The Right Chord design
Updated in: http://aitnaru.org/images/The_Right_Triangle.pdf

This simple geometry proves why the circle-squaring right triangle exists in Cartesian space for two nested diameters 2.0 and sqrt(Pi), where sqrt(Pi) is the length of a Side of Circle's Square (SoCS); the chord created between D = 2.0 and D = sqrt(Pi) effectively defines this circle-squaring right triangle for D = 2.0.

Theoretically, this triangle exists! ... albeit the circle cannot be squared (it's "impossible") by the rules of this Greek geometry challenge from antiquity. If such an "impossible" circle-squaring triangle truly exists, a musical chord(s) should be created to represent it (probably a triad) ... and would be a unique gift to the world! Preferably, the chord(s) should evoke thoughts and feelings of the Paradise Trinity.

Rod

[Amigoo]

Re: Phi of Pi design
(the 3 ratios, aka "constants")

For a circle-squaring right triangle,
ratio of hypotenuse to long side = 1.1283791670955125738961589031215..
ratio of long side to short side = 1.9130583802711007947403078280203..
ratio of hypotenuse to short side =
1.1283791670955125738961589031215..
x 1.9130583802711007947403078280203..
= 2.1586552217353950788554161024243..

Rod

[Amigoo]

Re: Pythagorean Squares design,
The "impossible" game of Quadrature

(gameboard pieces and instructions also "impossible" ...
but will be found in box marked "a^2 + b^2 = c^2")

Rod

[Amigoo]

Re: Pythagorean Squares design,
The "impossible" game of Quadrature

Re: "The Paranormal Equation", 2013, by James D. Stein, PhD, p. 110

"One of the most innovative mathematicians of the 19th century, Georg Cantor,
developed the first legitimate mathematical treatment of infinite quantities."

A one-minute scan of this page highlighted "infinite quantities", helping explain
why a unique Pythagorean right triangle can square the circle:

The triangle's ratio of hypotenuse to long side*, with hypotenuse having length
equal to circle's diameter (e.g., 2.0), forces the other sides ("infinite quantities")
to balance and mathematically causing the circle to be squared

* 2(sqrt(1/Pi)) = sqrt(Pi)/(Pi/2) = 2.0/sqrt(Pi) = 1.1283791670955125738961589031215..

A golden opportunity for a new millennium mathematician to prove this
infinite quantity balance that permits "impossible" squaring of the circle!
... methinks.

Rod ... ...

[Amigoo]

Re: Pythagorean Squares design (updated)

Now, sqrt(Pi) is represented with this "two-tine Pi fork" along with ...
ratio of hypotenuse to long side = 1.1283791670955125738961589031215..
ratio of long side to short side = 1.9130583802711007947403078280203..
ratio of hypotenuse to short side =
1.1283791670955125738961589031215..
x 1.9130583802711007947403078280203..
= 2.1586552217353950788554161024243..

"Get transcendental Pi, then fork it!"
... if you would comprehend squared circles.
Bon Appétit ! geometrically speaking

Rod

[Amigoo]

Re: Pythagorean Squares design
(and "infinite quantity balance")

Highlights Pythagorean magic in right-triangle-squared circles:
Length ratio of the two red lines = sqrt(Pi), perhaps the perfect
indicator of "locked" squared circle geometry!

"Such a life on such a planet!"

Rod

[Amigoo]

Re: Sqrt(Pi) design

Who knew
"Infinite quantity balance", for a circle-squaring right triangle,
means three integrated and balanced quantities - not two!

A squared-circle trifecta of 3 sets of colored lines (red, yellow, green)
where each two-line set has a length ratio of sqrt(Pi) .

"Get transcendental Pi, then fork it!"
... if you would comprehend squared circles.

Rod

[Amigoo]

Re: Sqrt(Pi) design

Triangular balance of infinite quantities (all sqrt(Pi)) confirms the greater
Tri-Phi Pi (primary ratios of the circle-squaring right triangle).

~ ratio of hypotenuse to long side = 1.1283791670955125738961589031215..
~ ratio of long side to short side = 1.9130583802711007947403078280203..
~ ratio of hypotenuse to short side =
1.1283791670955125738961589031215..
x 1.9130583802711007947403078280203..
= 2.1586552217353950788554161024243..

Rod ... ...

[Amigoo]

Re: Sqrt(Pi) design

"final, final" (re: geometry that speaks for itself)
"textbook ready" (re: color coding for easier discussion)

Tri-Phi Pi ratios of the triangular lines and similar angles
all confirm that there's nothing more to document, since ...
squared circles exist but the circle cannot be squared
... apparently. "Go figure!"

Rod

[Amigoo]

Re: Sqrt(Pi) design
("final, final" ^2)

After multiple iterations of color coding, the current design is stable
... enough to start perceiving Tri-Phi Pi as the geometric "blockchain"
of integrated squared circles.

Rod ... ... (cruisin' around the blocks)

[Amigoo]

Re: Sqrt(Pi) Ratios design
w/ focus on circle-squaring right triangle

Squared circle geometry 101, color-coded for easier discussion.

Duplication of line letters might cause confusion ...
but geometers know how to identify the correct ratios.

Rod

[Amigoo]

Re: Sqrt(Pi) Ratios design

Long story short: http://aitnaru.org/images/Sqrt_Pi_Ratios.pdf
aka "The 'impossible' ABCs of squared circles"

Rod

[Amigoo]

Re: Sqrt(Pi) RIP design
(Sqrt(Pi) Ratios In Pattern)

Obviously, squared circles are best comprehended with geometric pattern.

Geometers' humor: "Which circle is squared?" Yes.

Rod ... ...

[Amigoo]

Re: Sqrt(Pi) RIP design, expanding into the Cartesian universe

Exploration of squared circle geometry (aka "quadrature") is not unlike climbing the highest mountains: "Why climb the mountain?" adventurers are asked. "Because it's there!" they respond enthusiastically. The same is true with quadrature: "Because it's there!" But not the Cartesian space to explore - the geometric points to simply connect (and the lines to color to make convincing designs).

Once geometric designs convince us to fathom squared circles, we are ready to fathom first contact with alien civilizations. However, our first contact with aliens (perhaps, even now exploring our planet) would likely parallel the reactions of the tribal Sentinelese of the Andaman Islands to modern man (their existence first recorded in "early Arab and Persian documents": https://www.atlasobscura.com/places/sentinelese ).

Considering the age of the universe, advanced civilizations would likely be much older than Earth's, suggesting that we would experience first contact as did the Sentinelese with modern man. This might explain why modern governments (if they know such things ) are reluctant to reveal the existence of aliens. How do you reveal what you do not fully comprehend, what appears more powerful, and what should not exist (according to some religions)?

Rod

[Amigoo]

Re: Sqrt(Pi) RIP design
(the supporting "wiggly numbers")

Hypotenuse and long side ratios of the
two integrated circle-squaring right triangles
where SoCS = Side of Circle's Square:

4 / 2(sqrt(Pi)) = 2(sqrt(1/Pi)) = circle-squaring ratio of RT

= 4.0 / 3.5449077018110320545963349666823..
= 1.1283791670955125738961589031215..

4.0 x 2(sqrt(1/Pi)) = 4(sqrt(1/Pi)) = large triangle's hypotenuse

= 4.0 x 1.1283791670955125738961589031215..
= 4.5135166683820502955846356124862..

4(sqrt(1/Pi)) / 2(sqrt(Pi)) = 2(sqrt(1/Pi))^2

= 4.5135166683820502955846356124862..
/ 3.5449077018110320545963349666823..
= 1.2732395447351626861510701069801..
= 2(sqrt(1/Pi))^2 = cross-triangles' increment

Circle's radius x sqrt(Pi) = SoCS

= 2(sqrt(1/Pi)) x sqrt(Pi)
= 1.1283791670955125738961589031215..
x 1.7724538509055160272981674833411..
= 2.0 = SoCS

Thus, 4.0 is the square of a circle
having a diameter of 4(sqrt(1/Pi))
... and the geometry agrees!

Rod ... ("The stars at night ...")

[Amigoo]

Re: Sqrt(Pi) RIP design

This early morning's Midwayer prompts: 1:28, 2:28, 3:28 AM
Maybe related to the circle-squaring ratio: 2(sqrt(1/Pi))
= 1.1283791670955125738961589031215..

Rod

[Amigoo]

Re: Sqrt(Pi) RIP design, "Best of Show"

Who knew

It's not possible to "square the circle" but squared circles exist!
since Ratios and Increments of Pi construct the right triangle
that squares the circle.

Rod ... ...

[Amigoo]

Re: Sqrt(Pi) RIP design
Geometers' secret:

Circle-squaring right triangle (C-SRT) is formed when the isosceles right triangle,
representing one-fourth of the circle's inscribed square, pivots upward and morphs
into the C-SRT ... at which point, line length ratios equal sqrt(Pi).

However, the circle is not squared (theoretically) ...
if Pi is not the ratio of circle's circumference to its diameter.

Rod

[Amigoo]

"Impossible" squared circle math
(SoCS = Side of Circle's Square)

Diameter / 2(sqrt(1/Pi)) = SoCS
SoCS x 2(sqrt(1/Pi)) = Diameter

Perhaps, a blockchain of squared circles where
SqCoins are numbered from 1 to infinity.

2(sqrt(1/Pi))
= 1.1283791670955125738961589031215..

Rod ... ... (still cruisin' the block)

[Amigoo]

Re: Sqrt(Pi) RIP design
Geometers' secret:

Ratio of diameter of inner circle (dark blue)
to diagonal of largest square (light blue) = Pi/2
= 1.5707963267948966192313216916398..

More proof that squared circles exist!
... when Pi itself defines those circles.

Now methinks ...
Advanced mathematics proved that a circle cannot be squared,
but it can also be used to prove that squared circles exist!
"Not have your cake and eat it too" comes to mind.

Rod ... ... (cruisin' the Neighborhood, waiting for the movie)

[Amigoo]

Re: Sqrt(Pi) RIP design
("final, final")

Geometry that proves that ...
... the circle cannot be squared
... or squared circles exist
... or none of the above
... or all of the above.

Rod (RIP indeed!)

[Amigoo]

Re: Sqrt(Pi) RIP design
(final ^3: http://aitnaru.org/images/Sqrt_Pi_Ratios.pdf )

Despite the weeks-long effort to "perfect" the geometry of Sqrt(Pi) RIP,
the first design of this file ("Sqrt(Pi)'s Tri-Phi Pi") remains the defining
"geometric lock" of a circle squared: sqrt(Pi) thrice-integrated.

Rod ... ...
90-degree turn is tough, but 62.403 and 27.597 are "impossible"

[Amigoo]

Re: Sqrt(Pi)'s Tri-Phi Pi design and the intriguing story,
https://www.kcet.org/redefine/how-did-t ... t-its-name

When squared-circle geometry tells this story, the Joshua Tree got its name
from the Obtuse Revelation of Pi and the Two-Tine Fork, as told by Uncle Joshua
when he lunched with fellow travelers in the shade of a giant Yucca brevifolia.

Uncle Joshua had sketched the two "tines" (similar and obtuse overlapping triangles)
in the warm sand as he explained the line length ratios of the major tine: long side
has length equal to Pi and the two short sides each have length equal to sqrt(Pi).

The division of Pi by 2 creates two similar, circle-squaring right triangles
where the ratio of hypotenuse-to-long-side = 2(sqrt(1/Pi)) = sqrt(Pi)/(Pi/2)
= 1.1283791670955125738961589031215..

Thereafter, a celebratory pie was first shared by two companions, each dining on their
ceremonial portion with a two-tine fork, preferably under the shade of a Joshua Tree
... if perchance encountered during one of life's "impossible" journeys.

Rod

[Amigoo]

Re: 1point913 design

Disappointed that I couldn't commit to a New Year's Resolution
(No more geometry designs in 2018), I finally rationalized that
the research for this one started in 2017.

Significance? 1.913.. is ratio of the circle-squaring right triangle's
long side to short side, an "impossible" constant ... apparently.

Given: ls = 1.0
ss = 0.52272320087706331513679711195284..
hyp = 1.1283791670955125738961589031216.. 2(sqrt(1/Pi))
ls / ss = 1.9130583802711007947403078280203..

Rod ... ... (but still cruisin' on the hypotenuse)