Paradise Trinity Day

[Amigoo]

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

"According to sqrt(2), where Area = 4/Pi,
Diameter = 4/Pi, and Circumference = 4.

Morbus Cyclometricus in deed!

Ro ... ...

[Amigoo]

Pi Fourked I design,
added to: http://aitnaru.org/images/Tasty_Pi.pdf

According to sqrt(2), where largest Diameter = 4/Pi,
circle's Area = 4/Pi, and its Circumference = 4.

Now with more Quadrature-defining line length ratios.
"But wait! There's more!" (they say)

They kept insisting ...
so I layered on a circle-squaring scalene triangle.

"But wait! There's more!" (they say)
"Get real!" I said, then they responded:
"We're more Real than you!"

"Morbus Cyclometricus in deed!"

Ro ... ...

[Amigoo]

Pi Fourked II design,
added to: http://aitnaru.org/images/Tasty_Pi.pdf

"When sqrt(2) tells the story
... from its Pi Corral."

All the objects in the smallest squared circle
have sqrt(2) association to respective objects
in the largest circle ("Like Father, Like Son").


Two weeks later ...

Quadrature AAA (aka "Pi are sum square!")

For a circle where Diameter = 2,
Area of circle-squaring scalene triangle
= Area of large isosceles right triangle
+ Area of small right triangle:

= Pi/4 + (((sqrt(4-Pi)/sqrt(2))(sqrt(Pi)/sqrt(2)))/2)

= 0.78539816339744830961566084581988..
+ (((0.65513637756203355309393588562466..)
x (1.2533141373155002512078826424055..))/2)

= 0.78539816339744830961566084581988..
+ 0.41054584193408096912860688490007..

= 1.1959440053315292787442677307199..


Two days later ...

OMG! Morbus Cyclometricus redux

Area of this right triangle is precisely half
of the Area of this circle's "Pi Fork":

Area = 0.41054584193408096912860688490007.. right triangle
Area = 0.82109168386816193825721376980014.. Pi Fork (a right triangle)

"That's sum Pi !" (they say)
sqrt(2)^2 = 2 Who knew?!


Ro ... ...

[Amigoo]

Pi in the Skeye design,
added to: http://aitnaru.org/images/Tasty_Pi.pdf
Suggests plant-like growth of a new concept.

At maximum skeye loftiness, blue cross shows line length ratios
suggesting formula to square a circle: SoCS = D/(sqrt(2D))
where D = Diameter, SoCS = Side of Circle's Square,
where this square is inscribed in the circle's square.

Large circle's Diameter = (2/sqrt(Pi))^2
= 1.2732395447351626861510701069801..

SoCS = D/(sqrt(2D))
= 1.2732395447351626861510701069801.. (2/sqrt(Pi))^2
/ 1.5957691216057307117597842397375.. sqrt(2D)
= 0.79788456080286535587989211986882..

x 1.4142135623730950488016887242097.. sqrt(2)
= 1.1283791670955125738961589031216.. 2/sqrt(Pi)
^2 = 1.2732395447351626861510701069801.. (2/sqrt(Pi))^2

Area of circle's square = Diameter
when Diameter = (2/sqrt(Pi))^2


A wandering mind wanders to metaphysics ...

Geometric symbolism suggesting the square and the circle have gender!
... that circle is squared by the geometric marriage of sqrt(Pi) and sqrt(2)
... that the universe was created, reflecting a necessary dichotomy.

Why not the marriage of Pi and 2
Because Pi and 2 remain unique in their mathematical domains.
Only their respective essence of sqrt(Pi) and sqrt(2) can unite,
mathematically speaking, in Cartesian space. (they say)


Regarding the circle-squaring Pi Fork
where D = Diameter, SS = Short Side,
SoCS = Side of Circle's Square

D = 1.2732395447351626861510701069801..
SoCS = 1.1283791670955125738961589031215..
SS = 0.58982997002716101129132484048001..

Conversion to D = 2, SoCS = sqrt(Pi), SS = sqrt(4-Pi)

2 / 1.2732395447351626861510701069801..
= 1.5707963267948966192313216916398.. Pi/2
x 1.1283791670955125738961589031215.. 2/sqrt(Pi)
= 1.7724538509055160272981674833411.. sqrt(Pi)

0.58982997002716101129132484048001.. SS
x 1.5707963267948966192313216916398.. Pi/2
= 0.92650275035220848584275966758917.. sqrt(4-Pi)

2/sqrt(Pi) = 1.1283791670955125738961589031215..
sqrt(Pi)/sqrt(4-Pi) = 1.9130583802711007947403078280204..
(defines hypotenuse/long side and long side/short side
of every circle-squaring right triangle)

aka "Pythagorean Theorem" (a^2 + b^2 = c^2)
(4-Pi) + Pi = 4 (square each side of right triangle)
aka "Pythagorean Quadrature"


Postscript ...
It's a myth that a cave man chipped a stone squared circle
before real mathematicians chipped the stone wheel ...
and discarded the secret of Quadrature for millennia.

Ro ... ...

[Amigoo]

ABC of Pi design,
added to: http://aitnaru.org/images/Tasty_Pi.pdf
aka "Pi are Square"

Proves the "impossible" circle-squaring right triangle
divides into 2 similar triangles of Pythagorean persuasion
(area is precisely divisible by 2; each side by sqrt(2).

Whether divided by Noah (2 equal parts)
or by Goldilocks (2 sets of 3 parts),
a Pi divided is soon consumed.

Only geometers of Quadraturial persuasion
know how to select 1 part from one side
and 2 parts from the other side to divide
the Pi into two equal areas.

Tasty clue: Ask Pythagoras.

"There's something fishy about this Quadrature!"
Good catch! There's a Red Herring and similar stuff.

Ro ... ...

[Amigoo]

Spirit of Quadrature design, Big Whirl in pervaded space,
ongoing repercussion of the "mandate of the Ancients of Days
calling for organization of a new material creation".

(renamed from Three Concentric Squared),
added to: http://aitnaru.org/images/Tasty_Pi.pdf

The simple city of Quadrature - Areas = 4, 2, 1
Diameters = 2(2/sqrt(Pi)), sqrt(2)(2/sqrt(Pi)), 2/sqrt(Pi)

Ro ... ...

[Amigoo]

Pizzas 4 Pi design,
added to: http://aitnaru.org/images/Tasty_Pi.pdf
When Quadrature gets symmetrically mandalicious!

As the squares increase (or decrease) by sqrt(2),
circles of the squares increase (or decrease) by sqrt(2),
giving three concentric circles "to write home about".

Total area of 4 pizzas = area of the large pizza!
(since the areas of inscribed squares are equal,
proving that Pizza Pi is precisely divisible by 4).

aka "Basic Sherlock Pizza"


Re: https://www.dailymail.co.uk/news/articl ... izzas.html
"Can YOU figure out which one is bigger - A 20-inch pizza or two 10-inch pizzas?"

The visual is so obvious to mathematicians that they discover the Pizza Pi Revelation
when answering this question: "Which is bigger - A 20-inch pizza or four 10-inch pizzas?"

Using the same math where Area = Pi x r^2 ...

For 20-inch pizza ...
10^2 x Pi = 314.15926535897932384626433832795..

For four 10-inch pizzas ...
4(5^2) x Pi = 314.15926535897932384626433832795..

What's the tasty Pizza Pi Revelation ?
A Pizza Pi is precisely divisible by 4!
An "impossible" YUM


For precise servings of Pizza Pi,
follow the 4,2,1 pattern:

For 40-inch pizza ...
20^2 x Pi = 1256.6370614359172953850573533118..

For sixteen 10-inch pizzas ...
16(5^2) x Pi = 1256.6370614359172953850573533118..

This Pizza Pi is precisely divisible by 16.

For 20(sqrt(2))-inch pizza ...
(10(sqrt(2))^2 x Pi = 628.3185307179586476925286766559..

For eight 10-inch pizzas ...
8(5^2) x Pi = 628.3185307179586476925286766559..

This Pizza Pi is precisely divisible by 8.

For 20-inch pizza ...
10^2 x Pi = 314.15926535897932384626433832795..

For four 10-inch pizzas ...
4(5^2) x Pi = 314.15926535897932384626433832795..

This Pizza Pi is precisely divisible by 4.


Ro ... ... (delivers precise Pizza Pi)

[Amigoo]

Pizzas 4 Pi design,
updated in: http://aitnaru.org/images/Tasty_Pi.pdf

A Beautiful Day in the (Cartesian) Neighborhood!
with proof that sqrt(2) hosts both circle and square
in a Pi Corral in this Cartesian neighborhood.

A: "Breaker! Breaker! Pythonater!"
Q: "What happened in the Pi Corral?"

Ro ... ...

[Amigoo]

Square of Squares design*
"The Big Whirl of Quadrature"

* added to: http://aitnaru.org/images/Tasty_Pi.pdf


... Lotsa number crunchin'

Code: Select all

Side of Circles' Squares (SoCS) ...

= 2.2567583341910251477923178062431..    4/sqrt(Pi)
x 1.1283791670955125738961589031215..    2/sqrt(Pi)
2.0                                      2.0 
/ 1.2533141373155002512078826424055..    (sqrt(Pi)(sqrt(2))/2
= 1.5957691216057307117597842397375..    (4/sqrt(Pi))/sqrt(2)

2.2567583341910251477923178062431..      4/sqrt(Pi)
/ 1.5957691216057307117597842397375..    (4/sqrt(Pi))/sqrt(2)
= 1.4142135623730950488016887242097..    sqrt(2)

Circles' Squares ...

= 5.0929581789406507446042804279205..    (4/sqrt(Pi))^2
x 1.2732395447351626861510701069801..    (2/sqrt(Pi))^2
4.0                                      (2.0^2) 
/ 1.5707963267948966192313216916398..    Pi/2
= 2.5464790894703253723021402139602..    4/(Pi/2)

5.0929581789406507446042804279205..      (4/sqrt(Pi))^2)
/ 2.5464790894703253723021402139602..    4/(Pi/2)
= 2.0                                    sqrt(2)^2

Circles' Diameters ...

= 2.5464790894703253723021402139601..    8/Pi    
x 1.1283791670955125738961589031215..    2/sqrt(Pi)
2.2567583341910251477923178062431..      4/sqrt(Pi)
/ 1.2533141373155002512078826424055..    (sqrt(Pi)(sqrt(2))/2
= 1.8006326323142121391103983820135..    (8/Pi)/sqrt(2)   

2.5464790894703253723021402139601..      8/Pi
/ 1.8006326323142121391103983820135..    (8/Pi)/sqrt(2) 
= 1.4142135623730950488016887242097..    sqrt(2) 

... Proof Pi is inherent in the two squares
that enclose and are inscribed in the circle
where Diameter = 4/sqrt(Pi), Area = 4

Code: Select all

2.2567583341910251477923178062431..     4/sqrt(Pi)      
x 1.5957691216057307117597842397375..   (4/sqrt(Pi))/sqrt(2)   
= 3.6012652646284242782207967640269..
/ 2 = 1.8006326323142121391103983820135.

2 / 1.8006326323142121391103983820135..
= 1.1107207345395915617539702475152..      
x 2(1.4142135623730950488016887242097..)  2(sqrt(2))
= 3.1415926535897932384626433832796..     Pi 

Then there's Quadrature where
Area = Diameter, Circumference = 4

Code: Select all

A = 1.2732395447351626861510701069801..  (2/sqrt(Pi))^2
D = 1.2732395447351626861510701069801..  (2/sqrt(Pi))^2
C = 4.0 

The new sleight of Pi

Circumference / Diameter = Pi
4 / 1.2732395447351626861510701069801..
= 3.1415926535897932384626433832795.. Pi

Circumference / Area = Pi
4 / 1.2732395447351626861510701069801..
= 3.1415926535897932384626433832795.. Pi

Therefore, Area - Diameter = 0
((2/sqrt(Pi))^2) - ((2/sqrt(Pi))^2) = 0
or (4/Pi) - (4/Pi) = 0

When Circumference = 4.0, Diameter = Area
D = 1.2732395447351626861510701069801.. (2/sqrt(Pi))^2
A = 1.2732395447351626861510701069801.. (2/sqrt(Pi))^2
4.0 / 1.2732395447351626861510701069801..
= Pi = 3.1415926535897932384626433832795..


Enclosing vs inscribed squares when Circumference = 4.0

Code: Select all

= 1.6211389382774043431020714113556..    enclosing area
x 1.2732395447351626861510701069801..    (2/sqrt(Pi))^2
1.2732395447351626861510701069801..      diameter, area
/ 1.5707963267948966192313216916398..    Pi/2
= 0.81056946913870217155103570567782..   inscribed area

1.6211389382774043431020714113556..      enclosing area
/ 0.81056946913870217155103570567782..   inscribed area
= 2.0

Long story short: In Cartesian Quadrature,
either sqrt(2) is transcendental or Pi is not!
Tasty clue: Pi/2 = (sqrt(Pi)sqrt(2))/2

Now that circumference is finite, the onus is on the diameter, but
a diameter divided precisely by 2 then squared is transcendental?

Methinks Pi is transcendental because its formula is inherently transcendental.
A different formula would not be transcendental and would complement sqrt(2),
the obvious host of nested circles squared in a Cartesian Neighborhood.

Ro ... ...

[Amigoo]

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

Better representation of the two spirals (sqrt(Pi), sqrt(2)),
with obvious sqrt(2) curvature. The transcendental marriage
of sqrt(Pi) and sqrt(2) in tandem geometric spirals.

Note: Obviously, sqrt(2) is both curvature of the spirals
as well as the separation between the two spirals.


Quadrature simplified for Circumference = 4,
where SoS = Side of Square, Diameter = 4/Pi:

Code: Select all

^2 = 1.6211389382774043431020714113556..   enclosing Area
1.2732395447351626861510701069801..        Diameter, Area, SoS
x 0.70710678118654752440084436210485..     sqrt(2)/2
= 0.90031631615710606955519919100674..     inscribed SoS
^2 = 0.81056946913870217155103570567782..  inscribed Area

sqrt(1.2732395447351626861510701069801..)  Side of Circle's Square  

1.6211389382774043431020714113556..        enclosing Area
/ 0.81056946913870217155103570567782..     inscribed Area
= 2

Long story short (re: C = 4) ...
Since Circumference has a finite number of points,
Pi must be inherently transcendental (by design?).

When Circumference = 4, Area = Diameter.
Thus, C = Pi(D) becomes 4/Pi = D, effectively
identifying the Pythagorean right triangle
that squares the circle:

Ratio of hypotenuse (Diameter) to long side (SoCS)
= 2/sqrt(Pi) = 1.1283791670955125738961589031215..
where SoCS = Side of Circle's Square.

When C = 4, hypotenuse / long side =
1.2732395447351626861510701069801..
/ 1.1283791670955125738961589031215..
= 1.1283791670955125738961589031215..
Who knew?


Ro ... ...

[Amigoo]

Sum Pi design,
updated in: http://aitnaru.org/images/Tasty_Pi.pdf

"Best of Show" when geometry itself is the focus

Sub-title: "In this new millennium,
finger out the real geometry of Pi".

Ro ... ...

[Amigoo]

Sum Pi Too design,
added to: http://aitnaru.org/images/Tasty_Pi.pdf

"Long story short" ...
but immediately got longer!

With the square of the smallest circle having 1/4 the area
of the square of the largest circle, we're effectively teased ...
"Is this transcendental Quadrature or Cartesian legerdemain?"

Also, both diameter and circumference of smallest circle
are 1/2 the respective dimensions of the largest circle!

Who knew
sqrt(Pi)/sqrt(2) = (sqrt(Pi)(sqrt(2)))/2
= 1.2533141373155002512078826424055..
^2 = 1.5707963267948966192313216916398..
= Pi/2

When you're ready to order Sum Pi, order Sum Pi Too!
... or just ask for the Sqrt(2) Tandem.

Ro ... ...

[Amigoo]

Sum Pi 2EZ design,
added to: http://aitnaru.org/images/Tasty_Pi.pdf

"Long story short" ... and too easy

Visual proof (to geometers of Quadraturial persuasion)
that area of small circle is 1/2 area of the large circle
... gracias a sqrt(2)

Let large diameter = 2 and calculate the rest.


Who knew
Sqrt(2) is inherent in Pi ...

3.1415926535897932384626433832795.. Pi
/ 2 = 1.5707963267948966192313216916398.. Pi/2
sqrt( = 1.2533141373155002512078826424055..
= (sqrt(Pi)sqrt(2))/2

1.2533141373155002512078826424055..
/ sqrt(Pi) = 0.70710678118654752440084436210487.. sqrt(2)/2
1.2533141373155002512078826424055..
/ sqrt(2) = 0.88622692545275801364908374167057.. sqrt(Pi)/2


Ro ... ...

[Amigoo]

Sum Pi Starz design, at the E-R Bridge
added to: http://aitnaru.org/images/Tasty_Pi.pdf

"So, what's that geometry?" Just Sum Pi Starz.
(If you see one squared circle, you see one squared circle.)

Ro ... ...

[Amigoo]

Sum Pi Cube design, at the E-R Bridge
added to: http://aitnaru.org/images/Tasty_Pi.pdf

Color changes to highlight that this cube includes
two overlapping, circle-squaring scalene triangles
that define "impossible" Pythagorean Quadrature.

Once the crop circle makers transit the E-R Bridge
they should be able to duplicate Quadrature's cube,
the geometric union of sqrt(Pi) and sqrt(2).

Note: This is foundational Pythagorean Quadrature
where Diameter = Area when Circumference = 4.

About the Pi Corral and its Math Munchies ...

When Diameter = 2, Circumference = 2(Pi)
= 6.283185307179586476925286766559..
2 / 2(Pi) = 0.31830988618379067153776752674503..
Area = 3.1415926535897932384626433832795..
Circumference / Area = 2

When Circumference = 4, Diameter = 4/Pi
= 1.2732395447351626861510701069801..
(4/Pi) / 4 = 0.31830988618379067153776752674503..
Area = 1.2732395447351626861510701069801..
Diameter / Area = 1


Ro ... ...

[Amigoo]

Long story short ...
"Pi Divided Against Itself" (PDAI)

Pi/2 = ((sqrt(Pi)sqrt(2))/2)^2
= 1.5707963267948966192313216916398..
sqrt( = 1.2533141373155002512078826424055..

More story, aka "The Short Tail of Tres Pi" ...

"Pdaitry, or Pdaitric Geometry, is a branch of mathematics devoted to the study, analysis, and extrapolation
of the discrete unity of Pi in three nested Cartesian circles of 'impossible' Pythagorean Quadrature.
The mathcare professional is known as a Pdaitrist."

Abstract: The outer and inner circles, presenting in this unique Cartesian Neighborhood as three sets
(each presenting as one circle and two squares, Area square and Inscribed square), have discrete Diameters:
outer = 2, inner = 4/Pi (thus Circumference = 4). Because these two object sets have finite Cartesian position,
the distance between them is also finite (not infinitely decimal-pointy).

This geometric distance = 2/(4/Pi) = Pi/2. The square root of Pi/2 = 1.2533141373155002512078826424055..,
and defines the geometric distance of every object set (thus distance between similar objects).

"But wait! There's More!"
(in the Cartesian Neighborhood)

Ro ... ...

[Amigoo]

The Short Tails of Tres Pi Nice,
added to: http://aitnaru.org/images/Tasty_Pi.pdf

Voted best sub-title for a math protest T-shirt:
"Either sqrt(2) is transcendental or Pi is not!"

Tres Pi Nice, Tres Pi Nice!
See how they sum, see how they sum!
They all ran twixt the "impossible" fife
that 'brevied their tails with a rational right!
Ever see such a sight in your life as Tres Pi Nice?

Diagonals of smallest inscribed square (dark blue)
and the largest enclosing square (dark blue)
differ by Pi/sqrt(2). The two related circles
have finite position in this Cartesian space
via Diameter = 2 and Circumference = 4.
Thus, the enclosing and inscribed squares
also have finite position! Go figure


Proof that sqrt(2) is directly related to Pi ...
Pi / (2(sqrt(2)) = 1.1107207345395915617539702475152..

For every Circumference, Pi/(2(sqrt(2)) is the ratio
of 1/4 of that circle's Circumference to a side
of that circle's inscribed square (SoIS).

Relationship of 5 concentric circles ...

Code: Select all

 D         C         A           Arc/SoIS
1/2       π/2       π/16        π/8 / √2/4
 1        π         π/4         π/4 / √2/2
 2        2π        π           π/2 / √2
 4        4π        4π          π / 2(√2)
 8        8π        16π         2π / 4(√2)

The Numbas of Tres Pi Nice
(sqrt(Pi)sqrt(2))/2 = sqrt(Pi/2)
= 1.2533141373155002512078826424055..

Outer Squared Circle
D = 2.0
C = 6.283185307179586476925286766559..
A = 3.1415926535897932384626433832795..
IA = 2.0

Center Squared Circle
D = 1.5957691216057307117597842397375..
C = 5.013256549262001004831530569622..
A = 2.0
IA = 1.2732395447351626861510701069801..

Inner Squared Circle
D = 1.2732395447351626861510701069801..
C = 4.0
A = 1.2732395447351626861510701069801..
IA = 0.8105694691387021715510357056778..

Diameters [via (sqrt(Pi)sqrt(2))/2 ]
D = 2.0
D = 1.5957691216057307117597842397375..
D = 1.2732395447351626861510701069801..

Circumferences [via (sqrt(Pi)sqrt(2))/2 ]
C = 6.283185307179586476925286766559..
C = 5.013256549262001004831530569622..
C = 4.0

Area Squares [via Pi/2 ]
A = 3.1415926535897932384626433832795..
A = 2.0
A = 1.2732395447351626861510701069801..

Inscribed Area Squares [via Pi/2 ]
IA = 2.0
IA = 1.2732395447351626861510701069801..
IA = 0.8105694691387021715510357056778..

Ro ... ...

[Amigoo]

Pi Nice 2-2-2
(another version of Tres Pi Nice)
added to: http://aitnaru.org/images/Tasty_Pi.pdf

Pythagorean Quadrature in 5 circles squared

Code: Select all

   D           C          A           Arc/SoIS
  4/π          4         4/π        1 / (4/π)/√2
4/√π(√2)    4(√π/√2)      2      (√π(√2))/2 / 2/√π         
   2           2π         π          π/2 / √2
 √π(√2)     π(√π(√2))    π2/2    (π√π)/(2√2) / √π
   π           π2        π3/4       π2/4 / π/√2

Diameter
1.2732395447351626861510701069801.. 4/π
1.5957691216057307117597842397375.. 4/√π(√2)
2.0
2.506628274631000502415765284811... √π( √2)
3.1415926535897932384626433832795.. π

Circumference
4.0
5.0132565492620010048315305696221.. 4(√π/√2)
6.283185307179586476925286766559... 2π
7.8748049728612098721453229972335.. π(√π(√2))
9.8696044010893586188344909998762.. π2

Area
1.2732395447351626861510701069801.. 4/π
2.0
3.1415926535897932384626433832795.. π
4.9348022005446793094172454999379.. π2/2
7.7515691700749550438690787667753.. π3/4

Arc/SoIS (proof sqrt(2) directly related to Pi)
1.1107207345395915617539702475152.. 1 / (4/π)/√2
1.1107207345395915617539702475152.. (√π(√2))/2 / 2/√π
1.1107207345395915617539702475152.. π/2 / √2
1.1107207345395915617539702475152.. π(√π)/(2√2) /√π
1.1107207345395915617539702475152.. π2/4 / π/√2

Long story short (sqrt(2) directly related to π):
π / (2√2) = 1.1107207345395915617539702475152..
= 1/4 Circumference / Side of Circle's Square (SoIS)

Obviously, Pythagorean Quadrature is all about perfect balance
between sqrt(Pi) and sqrt(2) in the Cartesian Neighborhood.
"Are We There Yet?"

Ro ... ...

[Amigoo]

Tres Pi 4Pi2 (D = 4, Pi, 2)
added to: http://aitnaru.org/images/Tasty_Pi.pdf

Tasty Pi Sandwich

D = 4.0
A = 12.566370614359172953850573533118..
SoCS = 3.5449077018110320545963349666823..
<> 1.2732395447351626861510701069801.. 2/(Pi/2)
D = 3.1415926535897932384626433832795.. Pi
A = 7.7515691700749550438690787667753..
SoCS = 2.7841639984158539226424089910594..
<> 1.5707963267948966192313216916397.. Pi/2
D = 2.0
A = 3.1415926535897932384626433832795..
SoCS = 1.7724538509055160272981674833411..

1.2732395447351626861510701069801... 2/(Pi/2)
x 1.5707963267948966192313216916397.. Pi/2
= 2.0 Who knew

More about the Pi/2(sqrt2) constant ...

3.1415926535897932384626433832795..... Diameter
^2 = 9.8696044010893586188344909998762.. Circumference
/ 4 = 2.467401100272339654708622749969... 1/4 of Circumference

3.1415926535897932384626433832795.... Diameter
/ 1.4142135623730950488016887242097... sqrt(2)
= 2.2214414690791831235079404950303... Side of Inscribed Square
/ 2 = 1.1107207345395915617539702475152.. 1/2 of SoIS

2.467401100272339654708622749969.... 1/4 of Circumference
/ 2.2214414690791831235079404950303... Side of Inscribed Square
= 1.1107207345395915617539702475152... Pi/2(sqrt2)

Ro ... ...

[Amigoo]

Triangular Fourths,
added to: http://aitnaru.org/images/Tasty_Pi.pdf

At the Cartesian Beach of Quadraturial Persuasion,
mental clothing is optional (you are what you are)
and sunset fifths oft devolve to sunrise fourths.
"May the fourths be with you!"


Sqrt(2) maps a Cartesian sandbox with Pi/(2(sqrt2))
where Diameters = 4, 2(sqrt(2)), 2, sqrt(2).

If Quadrature is the marriage of sqrt(Pi) and sqrt(2),
sqrt(Pi) is the patriarch and sqrt(2) the matriarch. And since
sqrt(2) is obvious host of this Cartesian home, mama rules!

Apparently, Pythagorean Quadrature is real ... and Pi is very good estimate
of relationship between Circumference and Diameter. However, only Matrix knows
(especially since sqrt(2) is biased in this Cartesian Neighborhood).

Given: Diameter = 2, Circumference = 2Pi
2 / sqrt(2) = 1.4142135623730950488016887242097.. sqrt(2)
2Pi / 4 = 1.5707963267948966192313216916398.. Pi/2

(Pi/2) / sqrt(2) = Pi / (2(sqrt(2))
= 1.1107207345395915617539702475152..
= (Circumference/4) / (Diameter/sqrt(2))


Ro ... ...

[Amigoo]

Re: Triangular Fourths,
nested in: http://aitnaru.org/images/Tasty_Pi.pdf

Theoretically(?), in Pythagorean Quadrature,
Pi/sqrt(2) x sqrt(2) would make New Pi just irrational.


Common denominator for sqrt(Pi) and sqrt(2)?

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

1.7724538509055160272981674833411.. sqrt(Pi)
/ 1.5707963267948966192313216916398.. Pi/2
= 1.1283791670955125738961589031215.. 2/sqrt(Pi)
= Diameter/Side of Circle's Square (SoCS)


Olé! Sunrise numbas at the Beach

2.8284271247461900976033774484194.. Diameter
/ 2.506628274631000502415765284811.. Side of Circle's Square
= 1.1283791670955125738961589031215.. 2/sqrt(Pi)

2.506628274631000502415765284811.. Side of Circle's Square
/ 2.2214414690791831235079404950303.. 1/4 Circumference
= 1.1283791670955125738961589031215.. 2/sqrt(Pi)

Yum! Tasty Pi Sandwich at the Beach

2.8284271247461900976033774484194.. Diameter
2.506628274631000502415765284811.. Side of Circle's Square
2.2214414690791831235079404950303.. 1/4 Circumference

2.8284271247461900976033774484194.. Diameter
/ 2.2214414690791831235079404950303.. 1/4 Circumference
= 1.2732395447351626861510701069801.. (2/sqrt(Pi))^2


Summary: This geometry proves ...

- Pi rules! (relationship of circumference/diameter accurately calculated by Pi)
... if Quadrature is defined as only one circle and its square

- Otherwise, Sqrt(2) rules! when Quadrature defined as sqrt(2)-nested circles
... since dimensions of similar objects increase/decrease by sqrt(2).

- And since every circle is inherently part of sqrt(2) nesting,
Pi rules! only in a mathematical vacuum. Long live solo Pi !
(but everybody consumes more than one ... eventually)


Obviously, mathematical life's a Beach! (with lotsa Pi)

At the Cartesian Beach of Quadraturial Persuasion,
mental clothing is optional (you are what you are)
and sunset fifths oft devolve to sunrise fourths!
"May the fourths be with you!"

Yum!! Perfect Pi Sandwich at the Beach (diameter = area)
Good sunrise breakfast when the fourths be with you.

1.2732395447351626861510701069801.. Diameter
1.1283791670955125738961589031215.. Side of Circle's Square
1.0 = 1/4 Circumference

The "impossible" geometric balance
of circle and its square ... at the Beach


Ro ... ...

[Amigoo]

Re: Wheel Pi 4:4:4*, where D = 4, A = 4, C = 4.
Highlights scalene triangle that squares every circle.
* in: http://aitnaru.org/images/Tasty_Pi.pdf


They say, the new Wheel Pi is just irrational!

Code: Select all

4.0                                    Diameter
2.8284271247461900976033774484194..    Side of Inscribed Square (SoIS) 
12.566370614359172953850573533118..    Circumference = Area = 4(Pi)
3.1415926535897932384626433832795..    1/4 Circumference = Pi
/ SoIS =  1.1107207345395915617539702475152.. 
3.5449077018110320545963349666823..    Side of Circle's Square = 2(sqrt(Pi))
12.566370614359172953850573533118..    Circle's Area = 4(Pi)

2.2567583341910251477923178062431..    Diameter = 2(2/sqrt(Pi))
1.5957691216057307117597842397375..    Side of Inscribed Square (SoIS)
7.0898154036220641091926699333648..    Circumference = 4(sqrt(Pi)) 
1.7724538509055160272981674833411..    1/4 Circumference = sqrt(Pi)
/ SoIS =  1.1107207345395915617539702475152.. 
2.0                                    Side of Circle's Square
4.0                                    Circle's Area

1.2732395447351626861510701069801..    Diameter = Area = 4/Pi
0.90031631615710606955519919100673..   Side of Inscribed Square (SoIS)
4.0                                    Circumference
1.0                                    1/4 Circumference
/ SoIS =  1.1107207345395915617539702475152.. 
1.1283791670955125738961589031215..    Side of Circle's Square = 2/sqrt(Pi)
1.2732395447351626861510701069801..    Circle's Area = 4/Pi

Wheel-to-Wheel Pi, just like an old movie
Distance between Wheels? Pi (really!)

And like an old movie, supporting objects want credit!
Not to worry! Pi is infinite ... as infinite as sqrt(2).

Later update ...
This new millennium's "transcendental",
revealed by Pythagorean Quadrature!

Another Beach doodle, showing square root of Pi,
circular "halfway" point between the enclosing Wheels.
Pi-nested circles (not sqrt(2)-nested) where Pi creates
3 different objects with same whole-digit dimension

Geometry humor: Protractor -
Big Wheel tractor used by geometers of Quadraturial persuasion
when plowing under the insignificant digits of infinite Pi.


Ro ... ...

[Amigoo]

Re: Square One, as in "Back to square one!"
added to: http://aitnaru.org/images/Tasty_Pi.pdf

Highlights scalene triangle that squares every circle
... and sqrt(2), host of Pythagorean Quadrature

Given: Diameter = 2 Go figure the rest.
Tip: Two sides of the scalene = sqrt(2), sqrt(Pi)
(Pythagorean Quadrature is their geometric marriage)

A tasty Pi divided by sqrt(2) is so irrational,
but a Pi infinitely divisible by 2 is transcendental
May the fourths be with you!

Ro ... ...

[Amigoo]

Re: X of Why,
added to: http://aitnaru.org/images/Tasty_Pi.pdf

Geometric evidence of Quadrature.
Ratio of Diameter to Side of Inscribed Square: sqrt(2)
Precise angles of X: 90, 135, 90, 135


Transition to EPOS (Equidistant Points Of Scalenity)

Pythagorean Quadrature is like language modeling of Artificial Intelligence:
No one knows how it works, but resulting geometry proves circles squared
(via equidistant points of sqrt(2)-nested, circle-squaring, scalene triangles).

(they say the equidistant points are the diagonal of a trapezoid ...
and the equidistant points are also the diagonal of a parallelogram)
Sqrt(2) host of so many geometric objects with Quadraturial attitude!
... attitude about this "impossible" geometry! What fun!

a = 0.70710678118654752440084436210485.. sqrt(2)/2
b = 1.2533141373155002512078826424055.. (sqrt(Pi)sqrt(2))/2
h = 0.62665706865775012560394132120275.. (sqrt(Pi)sqrt(2))/4

q = 1.7005552065908.., K = 0.78539816339745.. Area = Pi/4

Calculated with https://www.calculatorsoup.com/calculat ... logram.php

A = C = 62.402887364309.. °
B = D = 117.59711263569.. °
a = 0.70710678118655..
b = 1.2533141373155..
p = 1.1179018932477..
q = 1.7005552065908..
h = 0.62665706865775..
P = 3.9208418370041..
K = 0.78539816339745..


Ro ... ...

[Amigoo]

Re: EPOS (Equidistant Points Of Scalenity),
added to: http://aitnaru.org/images/Tasty_Pi.pdf

In Pythagorean Quadrature,
sqrt(2) controls relationships throughout the Neighborhood
... and probably into other universes!

Clue: Long red line has sqrt(2) length relationship
to the two identical red lines (equidistant points).

Today's choice of T-shirt sub-title:
"Geometry is like a box of chocolates -
one point always leads to another!"

Ro ... ...