Paradise Trinity Day

[Amigoo]

Re: PD Addendum design ("Say what?")

"The lower squared circle and its circle-squaring Pythagorean are used to create the larger squared circle above,
wherein the lower circle reflects its geometric similarity."

Say what?

"When a certain Pythagorean triangle squares a circle, the hypotenuse and long side may be exchanged,
creating a juxtapositioned Pythagorean with its own squared circle wherein the originating circle
is able to reflect its geometric similarity with complementary juxtaposition."

Say what?

"What."

Rod ... ...

[Amigoo]

Re: PD Addendum design (new & improved)
"Conceptual juxtaposition of Pythagorean triangles."
aka "Pivotal duality with juxtapositional attitude."

"Say what?"

Colorful Cartesian elaboration of the juxtaposition
of a squared circle having a diameter of 1 (lower left)
and a circle squared having a side length of 1 (upper right).

"One what?"

Yes ... for now.

Rod

[Amigoo]

Re: PD Addendum design (turn 90 degrees counterclockwise
and rename to Pivotal Duality Alignment Trial (PDAT))

A Pivotal Duality Alignment Trial is proffered for all who
would build bridges between the material and spiritual realms.

"Say what?"

A squared circles design may have enticing perspectives.


Rod

[Amigoo]

Re: PD Addendum design

A convenient link: http://aitnaru.org/psymmetry.html

Rod

[Amigoo]

Re: cPoP:52:53 (new design)
"convincing Patterns of Pi for Midwayer prompts x:52,x:53"

Answers the question: "What geometry precisely defines squared circles?"

This geometric matrix with its integrated patterns of Pi includes Pi-related line lengths,
also observed as Midwayer prompts x:52,x:53 (x:52 followed immediately by x:53):

From the chart of prompt associations (x:52 = two x:26):
x:26 = 0.8862 26 9254.. = sqrt Pi / 2
x:53 = 1.7724 53 8509.. = sqrt Pi

Rod

[Amigoo]

The numbers for the cPoP geometry.

Regarding the large red Pythagorean triangle:
hypotenuse = 2, long side = sqrt of Pi

Each circle represents a set of 3 dimensions:
d = circle's diameter, s = side of circle's square, i = side of inscribed square
<> = correspondence of numbers to 1, 2, and Pi
(values are then listed in vertical order)

The d,s,i (large circle to small) for all 4 circles:

<> 2, sqrt(Pi), sqrt(2)

2
1.7724538509055160272981674833411..
1.4142135623730950488016887242097..

<> sqrt(Pi), Pi/2, sqrt(Pi/2)

1.7724538509055160272981674833411..
1.5707963267948966192313216916398..
1.2533141373155002512078826424055..

<> 2(sqrt(1/Pi)), 1, sqrt((2(sqrt(1/Pi))sq)/2)

1.1283791670955125738961589031215..
1
0.79788456080286535587989211986876..

<> (sqrt(Pi))/2, Pi/4, (sqrt(Pi/2))/2

0.88622692545275801364908374167057..
0.78539816339744830961566084581988..
0.62665706865775012560394132120276..

Rod

[Amigoo]

Re: http://aitnaru.org/baktun14.html
(click on page to view PDF file of cPoP :52:53 design)

cPoP in squared circles with a new MoP (Model of Pi).

Rod

[Amigoo]

Re: cPoP :52:53 design (updated)

Now, the large blue arc (5th squared circle) elevates the symbolism of this esoteric geometry!
(effectively contrasting a "squared circle" with a "circle squared")

Say what? "It's revealed in this geometry that speaks for itself."

Rod

[Amigoo]

Re: cPoP :52:53 design
Cartesian contemplation ...

Proven formula: A = Pi times radius squared

Diameter of large blue circle = sqrt of Pi
= hypotenuse of red Pythagorean triangle.
Long side of triangle = Pi / 2.

Diameter of large green circle = 2
= hypotenuse of larger red Pythagorean triangle.
Long side of triangle = sqrt of Pi.

Therefore (long story short) ...

The Pythagorean triangle that defines the square of the blue circle
also defines the square of the green circle via geometric similarity.

Does "irrational" and "transcendental" exist in this geometry?

Rod

[Amigoo]

Re: cPoP :52:53 design

"squared circle" and "circle squared" refer to the larger circle of each of the two sets of circles,
especially the 2:1 diameter:side association. This (and the related Pythagorean triangles)
hints that the geometry is well-integrated and possibly Pi-quality precise.

The numbers in this focus:

<> 2, sqrt(Pi)
d = 2
s = 1.7724538509055160272981674833411..

<> 2(sqrt(1/Pi)), 1
d = 1.1283791670955125738961589031215..
s = 1

Rod ... ...

[Amigoo]

Re: cPoP :52:53 design

Further attempts to improve this foundational geometry of squared circles
keep leading back to: "What's the point?"

Apparently, a Pythagorean triangle having an hyptoneuse with length equal to 2
and a long side equal to the square root of Pi is the revelatory core geometry.

As I contemplate "long winter's nap", I'm intrigued by perceptual contrast:
"Squared circles exist." ~ "The universe began with a big bang."

Intuition (and thousands of hours in the creativty sandbox) persuade me
that only one of the two perceptions is true. But the first perception is
well-supported by the math formula: Area = Pi times radius squared;
I've observed this many times amongst the objects of cPop :52:53.

What's next? Zzzzz...

Rod




sup-port for the first
perception

[Amigoo]

sup-port for the first
perception

WTFinal words? (previous post) Who's editing these comments?
Those words are not residual from anything that I had typed!
... but they anticipated this subsequent post:

Re: Inquisitor's Epiphany (new design)

Selected meanings of the words in the title:
~ inquisitor - one who searches out or inquires.
~ epiphany - a moment in which something is seen or understood in a new way.

I was determined to "make sense" of the complex geometry in cPoP :52:53,
expecting to discover better lines and objects that add value to the quest.
So, what happens? Even more complex geometry, pairing 3 circles per set!

As squared-circles geometry reference, these 3-circle sets are superb!
But lesson learned: epiphany does not always correlate with simplicity;
"Squaring the circle" acquired its "impossible" reputation for good reason.

This design can take its time - shorter Zzzzzs are planned.

Rod

[Amigoo]

Re: Inquisitor's Epiphany (design completed)

Who knew? Squared circle geometry is octavian (re: octave equivalency)!

A quick look at the calendar cautioned that "Twas the night before Christmas ..."
would be every evening's theme song if this geometry didn't achieve closure soon.

Fortunately, a positive attitude and awareness that the design update
effort could rely on extensive pattern repetition (tedious mouse manipulation)
was complemented with the desired closure - a very impressive Cartesian display!

Rod ... ...

[Amigoo]

Re: cPop :52:53 - Inquisitor's Epiphany design

Since there are now six squared circles, the numbers were updated:
(also online: http://www.aitnaru.org/images/cPoP_5253.pdf )

<1> 2(2(sqrt(1/Pi))), 2, 2(sqrt((2(sqrt(1/Pi))sq)/2))
2.2567583341910251477923178062430..
2.0
1.5957691216057307117597842397365..

<2> 2, sqrt(Pi), sqrt(2)
2.0
1.7724538509055160272981674833411..
1.4142135623730950488016887242097..

<3> sqrt(Pi), Pi/2, sqrt(Pi/2)
1.7724538509055160272981674833411..
1.5707963267948966192313216916398..
1.2533141373155002512078826424055..

<4> 2(sqrt(1/Pi)), 1, sqrt((2(sqrt(1/Pi))sq)/2)
1.1283791670955125738961589031215..
1.0
0.79788456080286535587989211986876..

<5> 1, sqrt(Pi/2), sqrt(2)/2
1.0
0.88622692545275801364908374167057..
0.70710678118654752440084436210485..

<6> (sqrt(Pi))/2, Pi/4, (sqrt(Pi/2))/2

0.88622692545275801364908374167057..
0.78539816339744830961566084581988..
0.62665706865775012560394132120276..

Rod

[Amigoo]

Re: cPop :52:53 - Universal Pi
(new design in PDF)

2 cool for Christmas 2014!

"In the local universe, the anticipated alignment of
inhabited planets is a reflection of Universal Pi."

All three circles are squared;
diameter of yellow circle = 2;
side of large blue square = 2.

Within the smaller green circle,
hypotenuse of Pythagorean triangle = sqrt(Pi)
long side of Pythagorean = Pi/2.

Rod

[Amigoo]

Re: cPop :52:53 - Universal Pi
(new design in PDF)

Who knew? Squared circles prefer pairs (now 4 squared circles).
And such a powerful, circle-squaring Pythagorean triangle!

This entire Cartesian composition can be constructed from the Pythagorean
triangle in the center of the smallest yellow circle, including identification
of all 4 circles squared by the extended Pythagorean triangles!

Rod ... ...

[Amigoo]

Re: cPop :52:53 - Universal Pi design
Who knew? Imaginary geometry can be drawn!

WYSIWYG ... not!

I was puzzled by reading yesterday that a square cannot be constructed with sqrt(Pi).
[ sqrt(Pi) = 1.7724538509055160272981674833411.. ]

So, I added yet another set of squared circles to the Universal Pi geometry (now 6 circles)
to "compare and contrast": The largest set hosts a square with side length = 2; the smallest
set hosts a square with side length = 1.

So, the middle set [d = sqrt(Pi), square's side = Pi/2] must be imaginary!

Just 2 cool for Christmas!

Rod

[Amigoo]

Re: cPop :52:53 - Universal Pi design

PDF updated with numbers for Universal Pi geometry:
http://aitnaru.org/images/cPoP_5253.pdf

Rod

[Amigoo]

Re: Circular Continuum (new design)

I'm continually lured back to the creativity sandbox to discover more perspective on Pi and its related geometric objects. But keeping in mind that "Twas the night before Christmas" might yet lurk in the background of this geometry exploration, Circular Continuum (sans "numbers") should be the final 2014 contribution .

These recent patterns of Pi suggest that, when analyzed as integrated components of certain Cartesian compositions, Pi's "irrational" and "transcendental" essence reflects a puzzling disconnect between Pi's Cartesian reality and math theory.

Maybe more squared circle geometry "that speaks for itself" will inspire epiphanous, "out of the box" evaluations.

Rod

[Amigoo]

Re: Circular Continuum (design completed)

"Who knew? Squared circle geometry resonates with 777 symbolism!"

Diameter of largest magenta circle = 2; next magenta left = Pi/2.

Rod ... ...

[Amigoo]

Re: Points of Order (new design)

"In a well-ordered universe, Pythagorean points are well-taken."

A convincing spiral of 5 circles, all squared by similar Pythagorean triangles.

So ... "What's the point?"

Rod

[Amigoo]

Re: Points of Order design
A different set of numbers (for the blue right triangle).

Since this is the first geometry showing sqrt(Pi) as a straight line
segment of the side of a square of a circle (largest circle in design),
I was motivated to recall some basic trigonometry
(but internet samples helped):

Given: a right triangle with angles A,B,C
and sides a(djacent),o(pposite),h(ypotenuse).

If A = 27.597112635690604451732204752339.. degrees
and o = 1.772453850905516027298167483341.. (sqrt of Pi),
find length of side a (side of circle's square)

Since A = 27.597112635690604451732204752339.. degrees,
tan A = 0.52272320087706331513679711195253..

Formula:
tan A = opposite side / adjacent side
and opposite side / tan A = adjacent side

Calculate:
1.772453850905516027298167483341..
/ 0.52272320087706331513679711195253..
= 3.390807693118581671868072537264..
= side of largest circle's square.

"What's the point?"
The Pythagorean triangles in this geometry are similar.

"And that's important because ...?"
Integrated patterns of Pi are inherent in squared circles.

"Say what?"
What.

Rod

[Amigoo]

Re: Points of Order design
"In a well-ordered universe, Pythagorean points are well-taken."

The geometry was missing something that better associates the large circle
with the spiral of four circles. But what? "sqrt(Pi)" was the revelatory clue!

Because sqrt(Pi) squared = Pi, the missing geometry had to be a square!
The blue square in the lower right of this Cartesian composition is one
of three squares representing the Pythagorean formula: a2 + b2 = c2
(blue right triangle is the geometric object in focus)

Rod

[Amigoo]

Re: Points of Order design
"In a well-ordered universe, Pythagorean points are well-taken."

As I reviewed the blue Pythagorean triangle in the center of this geometric
composition and again noted the Pi square (lower right), I realized that this geometry
is the gold standard for a squared circle. If this geometry is not effectively present,
the circle is not squared.

Who knew? Pythagoras cast into the sea of mathematics the theorem that might now
be complemented with a Pythagorean Theorem of Squared Circles!

Rod ... ...

[Amigoo]

Re: Points of Order design (an observation)
Conceptually, the Pythagorean ABCs of squared circles.

The red Pythagorean triangle (vertical side = Pi/2, diagonal hypotenuse = sqrt(Pi))
in the middle of the squared circles spiral is sufficient geometry to construct this entire
composition. The short blue lines indicate the positions of the other two squares
of the Pythagorean Theorem (re: blue right triangle in center of design).

Rod