Paradise Trinity Day

[Amigoo]

Re: Three Pi Nice design (TPN)*
"... where C1 = 10, C2 = 5, D = C/Pi"

Three Pi Nice, Three Pi Nice!
See how they sum, see how they sum!
They all ran twixt the "impossible" fife
that tempered their tales with a rational right!
Did you ever see such a sight in your life as Three Pi Nice?

Morbus Cyclometricus in deed

* added to: http://aitnaru.org/images/Tasty_Pi.pdf
(TPN because both Pythagorean right triangle
and scalene triangle square the circle)

"So, what doesn't square the circle?"
"Disbelief" comes to mind.

Ro ... ...

[Amigoo]

Re: Corralation 2x2 design*

Geometric correlation of circle-squaring
Pythagorean right and scalene triangles
... in a Pi Corral.

They say the 7 heavens of Pi
are out of this world.

* added to: http://aitnaru.org/images/Tasty_Pi.pdf
(another version of Three Pi Nice, focused on
the circle-squaring Pythagorean right triangle
that downsteps by sqrt(2) in this design)

Ro ... ...

[Amigoo]

Re: Corralation 2x2 design,
updated in: http://aitnaru.org/images/Tasty_Pi.pdf

Another circle squared, creating two sets of paired circles, all related by sqrt(2).
Helps prove two inner squares (green) have precisely 1/4 and 1/16 the area
of the largest square, with complementary diamters for these circles.

Morbus Cyclometricus in deed!

"So, what's '2x2' ?" Just ask any geometer junior (or grasshopper)
Tip: "grasshopper" does not refer to weed tokin', jumpin', stumpin', or lumpin'.

"What's 'jumpin', stumpin', and lumpin' ?" Changin' brand, promotin', cookin'.
Tip: But I digress (or is it digrass?)

Rod

[Amigoo]

Re: iQ Corralation design*

"Anatomy of 'impossible' Quadrature,
reflecting Pythagorean sqrt(2) scalenity
and identifying a center of Quadrature"

* upscaled in: http://aitnaru.org/images/Tasty_Pi.pdf

Pythagorean scalenity with essence of sqrt(2) Feng Shui
in Cartesian composition of "impossible" Quadrature.
Geometry that proves Pi is precisely divisble by 2.

What doesn't square the circle in Cartesian space?!
(even Geometry Juniors know that the circle-squaring scalene
and Pythagorean right triangles form this isosceles trapezoid)

Ro ... ... (MC in deed!)

[Amigoo]

Re: iQ Starz design,
added to: http://aitnaru.org/images/Tasty_Pi.pdf

45 + 45 + 135 + 135 = 360,
where two sides reflect sqrt(2)
in Cartesian space where trapezoids rule!

Obviously, the isosceles trapezoid is the real iQ show
since it comprises the circle-squaring Pythagorean right
and the circle-squaring scalene triangles.


Next morning update ...

Breakfast Tweakies (better trapezoid identification) ...
including visible sqrt(2) choreography of this composition.

"impossible" Quadrature in Cartesian space
(unless there be a fault in our starz)


Quadrature's numbas ...

Given: Circle's Diameter = 2,
SoIS = sqrt(2), SoCS = sqrt(Pi)
where SoIS = Side of circle's Inscribed Square,
SoCS = Side of Circle's area Square

sqrt(2) = 1.4142135623730950488016887242097..
sqrt(Pi) = 1.7724538509055160272981674833411..

sqrt(Pi)/sqrt(2)
= 1.2533141373155002512078826424055..
2/sqrt(Pi)
= 1.1283791670955125738961589031215..

1.2533141373155002512078826424055..
x 1.1283791670955125738961589031215..
= 1.4142135623730950488016887242097.. sqrt(2)

Suggests sqrt(2) is host of the Pi Corral


Ro ... ... (MC in deed!)

[Amigoo]

Re: Center of Scalenity I, II designs,
added to: http://aitnaru.org/images/Tasty_Pi.pdf

When one side of an inscribed scalene triangle represents a side of that circle's area square
and another side represents a side of that circle's inscribed square, union of sqrt(Pi) and sqrt(2),
a center of scalenity appears that communicates probability that the universe itself has a center
... despite having an apparently non-existent boundary.


More about Quadrature's numbas ...

Sides of circle-squaring right triangle
a = 0.92650275035220848584275966758914.. sqrt(4-Pi)
b = 1.7724538509055160272981674833411.. sqrt(Pi)
c = 2.0 sqrt(4)

To confirm Pythagorean formula

(0.92650275035220848584275966758914..)^2
+ (1.7724538509055160272981674833411..)^2
= 4.0

Sides of circle-squaring scalene triangle
a = 1.4142135623730950488016887242097.. sqrt(2)
b = 1.7724538509055160272981674833411.. sqrt(Pi)
c = 1.9084505148775338043018185280302..

To calculate length of side c

sqrt(4-Pi) = 0.92650275035220848584275966758914..
/ sqrt(2) = 0.65513637756203355309393588562466..
sqrt(Pi) = 1.7724538509055160272981674833411..
/ sqrt(2) = 1.2533141373155002512078826424055..

0.65513637756203355309393588562466..
+ 1.2533141373155002512078826424055..
= 1.9084505148775338043018185280302.. side c


Final Pi notes ...

Center of Scalenity I design
a = 1.4142135623730950488016887242097..
b = 1.7724538509055160272981674833411..
c = 1.9084505148775338043018185280302..
= (sqrt(4-Pi)/sqrt(2)) + (sqrt(Pi)/(sqrt(2))

Center of Scalenity II design
a = (sqrt(Pi)/sqrt(2)) x (2/sqrt(Pi)) = sqrt(2)
b = (sqrt(2)/(2/sqrt(Pi))) x sqrt(2) = sqrt(Pi)


Ro ... ... (Morbus Cyclometricus in deed!)

[Amigoo]

Re: Center of Scalenity II design,
updated in: http://aitnaru.org/images/Tasty_Pi.pdf
(when Quadrature speaks Bible)

Quadrature, 2 by 2 (more Cartesian busyness)
Proves Pi is precisely divisible by 2 (by 2, by 2 ..) where long side of circle-squaring right triangle
that represents Side of Circle's area Square (SoCS) appears in the sqrt(2)-associated siblings.
Who knew?! Noah? (distant relative of Pythagoras?)

Wherever two or more Cartesian objects associate as "Pi corral",
there exists a center of scalenity wherein sqrt(Pi) (represents heaven)
and sqrt(2) (represents earth) unite in this Cartesian space.

Geometer's secret ...
Circle-squaring objects replicate similarly
across all sqrt(2) boundaries.

Rod

[Amigoo]

Re: Quadrasure design,
added to: http://aitnaru.org/images/Tasty_Pi.pdf
"Quadrature by 2 by sqrt(2)"

"Which came first, the circle or the square?" Yes!
"So what's your point?" Yes!

Ro ... ...

[Amigoo]

Re: Dial 7 design
added to: http://aitnaru.org/images/Tasty_Pi.pdf

Long distance charges* between heaven and earth
are astronomical but can be waived with good credit.


Geometer's secret ...
Circumference of large circle is 10, small circle 5.

The greater secret is that (with this juxtaposition of Cartesian objects)
sqrt(2) line length differences between large/small circle-squaring triangles
(overlapping scalene and Pythagorean right) is in-your-face obvious
and another line better identifies Pythagorean rights (a pun).

... not to mention that the area squares
of both circles have sqrt(2) relationship.


However, 8 is a better reference Circumference ...

D/SoCS =
2((2/sqrt(Pi))^2) / 2(2/sqrt(Pi))
= 1.1283791670955125738961589031215..
= 2/sqrt(Pi), sqrt(Pi)/(Pi/2), 2(sqrt(1/Pi))

C = 8
D = 2((2/sqrt(Pi))^2)
r = (2/sqrt(Pi))^2
SoCS = 2(2/sqrt(Pi))
A = Pi((2/sqrt(Pi))^2)

C = 8
D = 2.5464790894703253723021402139602..
r = 1.2732395447351626861510701069801..
SoCS = 2.2567583341910251477923178062431..
A = 5.0929581789406507446042804279205..

C/D = Pi
8/2.5464790894703253723021402139602..
= 3.1415926535897932384626433832795..


* https://www.ipublishing.co.in/angel-number-7-meaning
"Number 7 symbolizes the connection between heaven and earth."

Ro ... ...

[Amigoo]

Re: Key of Quadrature design*
(answers the question "What's the point?")

* added to: http://aitnaru.org/images/Tasty_Pi.pdf
(and followed by Master Key design).

Values increase/decrease by 2/sqrt(Pi)
except (2/sqrt(Pi))^2 for Area.

C = 8.0
D = 2.5464790894703253723021402139602..
A = 5.0929581789406507446042804279203..
r = 1.2732395447351626861510701069801..

C = 7.0898154036220641091926699333646..
D = 2.2567583341910251477923178062431..
A = 4.0
r = 1.1283791670955125738961589031215..

C = 6.2831853071795864769252867665591..
D = 2.0
A = 3.1415926535897932384626433832795..
r = 1.0


Re: Master Key of Quadrature design ...

E = MC^2 (Entropy = Morbus Cyclometricus ^2)
where 'Entropy' = 'inert uniformity' (end of Pi's decimals),
Morbus Cyclometricus = challenging path to entropy
and ^2 refers to (2/sqrt(Pi))^2.

Geometer's secret: C = 8.0, A = 4.0, D = 2.0, r = 1.0
are the only values needed to construct this geometry
that confirms the Key of Quadrature, 2/sqrt(Pi)
= 1.1283791670955125738961589031215..
^2 = 1.2732395447351626861510701069801..

Tip: Every circle has an enclosing square
that is area square of next larger circle

"Say what?!"
Ask a geometer of Quadraturial persuasion

(about the lookin' glass theme)
A spy whose pie is plump with Pi
oft settles a rump upon a stump,
samples the pie, plumps the rump,
circles the square, and ne'er deny.


Re: concentric circles of Quadrature ...

About large (outer) circle ...
D = 2((2/sqrt(Pi))^2)
= 2.5464790894703253723021402139602..
C = Pi(D)
= 3.1415926535897932384626433832795..
x 2.5464790894703253723021402139602..
= 8.0

About small (inner) circle ...
D = 2.0
C = Pi(D)
= 2(3.1415926535897932384626433832795..)
= 6.283185307179586476925286766559..

About Circumferences ratio ...
8.0 / 6.283185307179586476925286766559..
= 1.2732395447351626861510701069801..
= (2/sqrt(Pi))^2

About Diameters ratio ...
2.5464790894703253723021402139602..
/ 2.0 = 1.2732395447351626861510701069801..
= (2/sqrt(Pi))^2


Re: Goldilocks circle between outer/inner ...

About Circumference ...
C = 8.0 / (2/sqrt(Pi)) = 7.0898154036220641091926699333646..
or C = 2(Pi) x (2/sqrt(Pi)) = 7.0898154036220641091926699333646..

About Diameter ...
D = 2((2/sqrt(Pi))^2) / (2/sqrt(Pi)) = 2.2567583341910251477923178062431..
or D = 2.0 x (2/sqrt(Pi)) = 2.2567583341910251477923178062431..

Quadrature's Trinity of Trinities
(Circumferences, Diameters, Areas) ...

C = 8.0,
7.0898154036220641091926699333646..,
6.283185307179586476925286766559..

D = 2.5464790894703253723021402139602..,
2.2567583341910251477923178062431..,
2.0

A = 5.0929581789406507446042804279203..,
4.0,
3.1415926535897932384626433832795..


Ro ... ...

[Amigoo]

Re: Master Key of Quadrature design,
updated in: http://aitnaru.org/images/Tasty_Pi.pdf

Another Pythagorean triangle (red) added to highlight why
a trinity of lines crossing a point confirms Quadrature.

Also, these two triangles maintain similar circle-squaring
line length and angle relationships.

Rod

[Amigoo]

Re: Master Key of Quadrature design,
updated in: http://aitnaru.org/images/Tasty_Pi.pdf

Now with "Pi Fork" wherein paired lines have
2, sqrt(Pi), or 2/sqrt(Pi) length relationship.

The "lookin' glass" theme maintains that
squared circles' sqrt(2) siblings host Pi Forks.

Tip: "Forked Pi" is not polite expression.

Ro ... ...

[Amigoo]

Re: Master Key of Quadrature with Pi Forkz design,
updated in: http://aitnaru.org/images/Tasty_Pi.pdf
('z' plural refers to lines shape in Pythagorean Quadrature)

Just in time for Leftover Pi Day, Nov. 10, 2022 (Julian 314)
(just serve on a previous day to have leftover)

Note: Primary Pi Day is March 14, annually.

Ro ... ...

[Amigoo]

Re: MK Confirmation, MC Scalenity designs,
added to: http://aitnaru.org/images/Tasty_Pi.pdf
(appended with Golden Rectangle of Quadrature)

Re: Pythagorean circle-squaring right triangle
(also defines the two adjoined rectangles)

(c(a))/b = d(iagonal), (2(sqrt(4-Pi))/sqrt(Pi) = d
= 1.0454464017541266302735942239055..


Re: MC Scalenity design ...
Morbus Cyclometricus, in deed!
A Conundrum of Quadrature (CQ)
▄▄▄ ▄ ▄▄▄ ▄ . ▄▄▄ ▄▄▄ ▄ ▄▄▄

A good design for a T-shirt, mug, or wall poster,
especially with only CQ Morse Code sign below it.


More treats for Leftover Pi Day, Nov. 10, 2022 (Julian 314)
(just serve on a previous day to have leftover)
Note: Primary Pi Day is March 14, annually.

Ultimate foundation of Quadrature, 2/sqrt(Pi):
wherein Cartesian objects of child relate to parent by (2/sqrt(2))^2
and Pythagorean Quadrature absorbs the transcendence of Pi.


About areas of the two squared circles ...

4/Pi = 1.2732395447351626861510701069801.. radius^2
sqrt( = 1.1283791670955125738961589031215.. radius
x 2 = 2.2567583341910251477923178062431.. Diameter

1/Pi = 0.31830988618379067153776752674503.. radius^2
sqrt( = 0.56418958354775628694807945156077.. radius
x 2 = 1.1283791670955125738961589031215.. Diameter

Both Areas and Diameters relate by (2/sqrt(2))^2
... when sqrt(2) is host of the Pi Corral.


Re: https://www.imdb.com/title/tt8371130/
"Sacred geometry of the squared circle"

Who can tell? (this documentary?)
Does Wilcock's sacred geometry relate to 2/sqrt(Pi),
the ultimate foundation of Quadrature?

Notes: 2 refers to Diameter, sqrt(Pi) to SoCS
(Side of Circle's area Square). A circle's area square
rests upon 8 points of the circle and appears between
the circle's enclosing square and inscribed square.

OMG! How can the circle not be squared
but squared circles can exist?!


Ro ... ...

[Amigoo]

Re: Golden Rectangle of Quadrature design,
added to: http://aitnaru.org/images/Tasty_Pi.pdf

Re: Pythagorean circle-squaring right triangle
that defines the two adjoined rectangles, where
d also identifies next set of smaller rectangles
(90 degrees juxtaposition to beginning set).

(c(a))/b = d(iagonal), (2(sqrt(4-Pi))/sqrt(Pi) = d
= 1.0454464017541266302735942239055..

Fibonacci Golden Rectangle may be aesthetically pleasing,
but "impossible" Golden Rectangle of Morbus Cyclometricus
is ethereally illuminating ... and with new starz:

2/sqrt(Pi) = 1.1283791670955125738961589031215..
sqrt(Pi)/sqrt(4-Pi) = 1.9130583802711007947403078280205..


More geometric busyness (typical of Morbus Cyclometricus)

Pop Quiz (true or false?, regarding golden lines) ...
Every rectangle in this composition defines a squared circle.
Clue: Some line length pairs = sqrt(Pi)/sqrt(4-Pi)


Rod ... ...

[Amigoo]

Re: Stars of Q design,
added to: http://aitnaru.org/images/Tasty_Pi.pdf

MC in deed! (aka "How to visualize the 'impossible' " ...
where sqrt(2) is all over Quadrature like sticky on a bun!).

Circle-squaring lines (lengths) = sqrt(2), sqrt(Pi), 2,
(sqrt(Pi)/sqrt(2)) + (sqrt(4-Pi)/sqrt(2))

Angles created by red X = 45, 135, 45, 135 degrees.

Ro ... ...

[Amigoo]

Pop Quiz

Is transcendence of Pi absorbed in Pythagorean geometry
Consider these similar line length relationships:

1. Right triangle sides = 5,4,3 --> (c, b, a)
where (5/4) x (4/3) = 5/3 --> (c/b x b/a = c/a)

1.25 x 1.3333333333333333333333333333333..
= 1.6666666666666666666666666666667..

(a^2 + b^2 = c^2) --> 9 + 16 = 25

2. Circle-squaring right triangle sides = 2, sqrt(Pi), sqrt(4-Pi) --> (c, b ,a)
where (2/sqrt(Pi)) x (sqrt(Pi)/sqrt(4-Pi)) = (2/sqrt(4-Pi)) --> (c/b x b/a = c/a)

1.1283791670955125738961589031215..
x 1.9130583802711007947403078280205..
= 2.1586552217353950788554161024246..

(a^2 + b^2 = c^2) --> (4-Pi) + Pi = 4

Ro ... ...

[Amigoo]

Re: Starz of Q design,
updated and renamed in: http://aitnaru.org/images/Tasty_Pi.pdf

E = MC^2 (Entropy's randomness reflected by Morbus Cyclometricus ^2)

aka, "MC Busyness for the holidays and New Year".
A fractal generator, this "impossible" Quadrature ...
including the Golden Rectangle of Quadrature!

Rod

[Amigoo]

Re: Starz of Q design,
finalized in: http://aitnaru.org/images/Tasty_Pi.pdf

Every squared circle contains similar circle-squaring right triangle,
scalene triangle, isosceles trapezoid, another quadrilateral,
and a pentagon, all related by sqrt(2). Who knew

Thus, for Quadrature queasiness ...
take sqrt(2) to balance your composition:

1.7724538509055160272981674833411.. sqrt(Pi)
x 1.4142135623730950488016887242097.. sqrt(2)
= 2.506628274631000502415765284811..

1.7724538509055160272981674833411..
/ 1.4142135623730950488016887242097..
= 1.2533141373155002512078826424055..

2.506628274631000502415765284811..
/ 1.2533141373155002512078826424055..
= 2

Quadraturially speaking ...
If parents have the sqrt(2) genome,
their grandchildren will prefer square Pi.

Ro ... ...

[Amigoo]

Re: Starz of Scalenity,
added to: http://aitnaru.org/images/Tasty_Pi.pdf

"Pi Corral of Quadrature, hosted by sqrt(2),
with equidistant scalenifiability observable
in two overlapping isosceles trapezoids."
Morbus Cyclometricus in deed!


Geometer's secret to equidistant points of overlapping,
circle-squaring, scalene triangles that associate by sqrt(2):
The two longer sides of each triangle associate by sqrt(2),
giving equality of diagonals in isosceles trapezoids.

"Say what?!" Ask a Geometer Junior
of Quadraturial persuasion.


Geometer's secret to longer "equidistant points":
third diagonal was borrowed and lengthened to prove
sqrt(2) association of line to the three diagonals.

"Say what?!" Ask a Geometer Senior
of Quadraturial persuasion.


Ro ... ...

[Amigoo]

Re: Starz Light design,
added to: http://aitnaru.org/images/Tasty_Pi.pdf

"Nested circles, all squared, in a Pi Corral hosted by sqrt(2)"

A bit TMI, geometrically speaking ...
but convincing pattern of Quadraturial persuasion:
8 points that exist on every circle well-squared
... even when nested by sqrt(2)

Pattern also includes 8 similar, overlapping,
circle-squaring scalene triangles.


Overheard in a group of Geometry Juniors
(of Quadraturial persuasion) ...

"I knew that squaring the circle was "impossible",
but I didn't know that it would be so difficult!


Ro ... ...

[Amigoo]

Re: Starz Lite design,
added to: http://aitnaru.org/images/Tasty_Pi.pdf

In a Pi Corral hosted by sqrt(2),
Pi is precisely divisible by 2.

Who knew?!

Ro ... ...

[Amigoo]

Re: Concentricity D2C4 design,
added to: http://aitnaru.org/images/Tasty_Pi.pdf

"That's sum Pi!"

Suggests geometry in a Rosetta Stone of Quadrature
and new "Three Concentric Circles" for the millennium.

1.7724538509055160272981674833411.. sqrt(Pi
/ 1.5707963267948966192313216916398.. Pi/2
= 1.1283791670955125738961589031215..

2/1.7724538509055160272981674833411.. 2/sqrt(Pi)
= 1.1283791670955125738961589031215..

3.1415926535897932384626433832795.. Pi
+ 0.8584073464102067615373566167205.. (4-Pi)
= 4 (when RT sides = sqrt(Pi), sqrt(4-Pi), 2)
(aka "Pythagorean Theorem")


The Numbas of Concentricity C2D4

Code: Select all

            Diameter                             Circumference                             Area
5.6568542494923801952067548968388..  17.771531752633464988063523960243..   25.132741228718345907701147066236..

3.9374024864306049360726614986167..  12.369714725596574025114178793411..   12.176136379250304654555041586088..
3.1415926535897932384626433832795..   9.8696044010893586188344909998761..   7.7515691700749550438690787667753
2.506628274631000502415765284811..    7.8748049728612098721453229972335..   4.9348022005446793094172454999379..
2.0                                   6.283185307179586476925286766559..    3.1415926535897932384626433832795..
1.5957691216057307117597842397375..   5.013256549262001004831530569622..    2.0 
1.2732395447351626861510701069801..   4.0                                   1.2732395447351626861510701069801..            

1.7724538509055160272981674833411..     sqrt(Pi)
x 1.4142135623730950488016887242097..   sqrt(2)
= 2.506628274631000502415765284811..    sqrt(Pi) x sqrt(2)
/ 2 = 1.2533141373155002512078826424055.. common factor
^2 = 1.5707963267948966192313216916398..  Pi/2

sqrt(1.2732395447351626861510701069801..)
= 1.1283791670955125738961589031215..
= 2/(sqrt(Pi)) = sqrt(Pi)/(Pi/2)

Ro ... ...

[Amigoo]

Perfect Pi (according to the numbas)

Diameter: 1.2732395447351626861510701069801..
Area: 1.2732395447351626861510701069801..
Circumference: 4.0

sqrt(1.2732395447351626861510701069801..)
= 1.1283791670955125738961589031215..
= 2/(sqrt(Pi)) = sqrt(Pi)/(Pi/2)

A new millennium math conundrum?
"What is the circumference of a circle
whose diameter equals its area?"

Ro ... ...

[Amigoo]

Three Concentric Pi design,
added to: http://aitnaru.org/images/Tasty_Pi.pdf

The Temple of Square Pi is where those of Quadraturial persuasion
go to contemplate that which is claimed to be "impossible".

Popular t-shirt subtitle ...
"I've got a good mind to ... go within."

Note: Inner circle is the only circle where diameter = area
= (2/sqrt(Pi))^2 = 1.2732395447351626861510701069801..
= 4/Pi with circle's circumference = 4.

Geometer's secret riddle ...
Temple of Square Pi is open 12 hours a day,
with opening/closing time well integrated
for those fond of certain patterns.

"Morbus Cyclometricus in deed!"

Ro ... ...