Paradise Trinity Day

[Amigoo]

Re: eQuilibrium design
"The Seventh Heaven of Quadrature"

For this new millennium ...

qPi is "new & improved" Pi since qPi defines both circle and its square
and qPi incorporates both sqrt(Pi) (thus Pi) and sqrt(2) (thus 2.0),
where sqrt(2) is the host of Cartesian circles and sqrt(2) nesting.

"Say what?!"
If Q = Quadrature?, then A = qPi ...
for all serious Q&A about Morbus Cyclometricus.

"Say what?!"
Since 2/sqrt(Pi) = 1.1283791670955125738961589031215.. (qPi)
Circumference = Diameter((2/qPi)^2), etc., etc.
If D = 2, C = 6.283185307179586476925286766559..

Rod

[Amigoo]

Re: eQuilibrium design
"The Seventh Heaven of Quadrature"

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

Who knew?! Pi itself defines qPi

Rod

[Amigoo]

Re: eQuilibrium design*
"The Seventh Heaven of Quadrature"
* updated in: http://aitnaru.org/images/Quinale!.pdf

Given: Diameter = 2.0
qPi = 1.1283791670955125738961589031215..
= 2(sqrt(1/Pi)) = 2/sqrt(Pi) = sqrt(Pi)/(Pi/2)

Circumference = Diameter((2/qPi)^2)
= 6.283185307179586476925286766559..

Area = ((2/qPi) x Radius)^2
= 3.1415926535897932384626433832795..

Is it possible? qPi = a quantum of Pi.

Rod ... ...

[Amigoo]

Re: eQuilibrium design
"The Seventh Heaven of Quadrature"

Morning Numba Rumba ...

sqrt(2) / qPi
= 1.4142135623730950488016887242097..
/ 1.1283791670955125738961589031215..
= 1.2533141373155002512078826424055..
= (1.7724538509055160272981674833411..
x 1.4142135623730950488016887242097..)/2
= (sqrt(Pi)sqrt(2)) / 2

Rod

[Amigoo]

Re: eQuilibrium design
"The Seventh Heaven of Quadrature"

Evening Numba Rumba ...

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

(1.1283791670955125738961589031215..)^2
= 1.27323954473516268615107010698..

If Diameter = (qPi)^2, circle's Area = (qPi)^2
How does this happen

Rod

[Amigoo]

Re: eQuilibrium design
"The Seventh Heaven of Quadrature"

The internet never forgets (a previous year's version,
aka "Boxa Pi") - http://aitnaru.org/threepoints.html

Geometer's secret ...
the 2 "boxes" inscribe within 3 overlapping circles.

Rod

[Amigoo]

Re: eQuilibrium design
"The Seventh Heaven of Quadrature"

Textbook ready! ... at least, for discussion.

This version highlights one more geometric object that replicates in a sqrt(2) continuum:
a unique quadrilateral with adjoined isosceles right and circle-squaring right triangles.

"Lines, and triangles, and squares! Oh my!"

Rod ... ...

[Amigoo]

Golden Ratio of Quadrature (GRoQ)
(via angles of the flutterby)

B = acos(sqrt(Pi)/2) = 27.597112635690604451732204752339..
A = 45 - acos(sqrt(Pi)/2) = 17.402887364309395548267795247661..

45 / A = 2.5857778113468672482962141856361..
45 / B = 1.6306053678167298125377722958187..

2.5857778113468672482962141856361..
/ 1.6306053678167298125377722958187..
= 1.5857778113468672482962141856361..

(45 / A) = 1 + ((45 / A) / (45 / B))
2.5857778113468672482962141856361..
= 1 + (2.5857778113468672482962141856361..
/ 1.6306053678167298125377722958187..)

Rod

[Amigoo]

Re: Golden Ratio of Quadrature design (GRoQ)
(via angles of the flutterby)

"Been there! Done that!"

Rod

[Amigoo]

Re: Golden Ratio of Quadrature design (GRoQ)
(via angles of the flutterby)

the real Golden Ratio: (a+b)/a = a/b

Re: https://www.fastcompany.com/3044877/the ... ggest-myth

"We’re creatures who are genetically programmed to see patterns and to seek meaning ..."
victims to their natural desire to find meaning in the pattern of the universe"

Rod ... ... (fond of patterns)

[Amigoo]

Re: Transcendental Correlation design

Juxtapositional geometry that contrasts "transcendental" as a Pi value
(re: circle's square) and "transcendental" as a circle's circumference value
... and vice versa via the integrated Quadrature.

"Say what?" Geometric correlation via 2/sqrt(Pi).

Rod

[Amigoo]

Re: Transcendental Correlation design*
"Six of one - half dozen of the other"

The Pythagorean Theorem's a^2 + b^2 = c^2
geometrically explains why the two sides of the right triangle
always balance the hypotenuse (the circle's circumference
or its square can both reflect the transcendental Pi value
... but only one reflection per squared circle).

"Say what?!" Squared circles exist ...
but you can't get there from here.

* added to: http://aitnaru.org/images/Quinale!.pdf

Rod ... ...

[Amigoo]

Re: Transcendental Correlation design
"Six of one - half dozen of the other
... where C = 2.0, sqrt(Pi), Pi/2"

2/sqrt(Pi) comes to mind.

Rod

[Amigoo]

Re: Transcendental Correlation design
"C = 2(2/sqrt(Pi)), 2.0, sqrt(Pi), Pi/2"

An inscribed octagon is a good geometric object to prove
that Pi, by design, is transcendental: short side of these wedges
(side is a chord of the circle) reduces to nano length and less
as octagon's isosceles triangles multiply by 1,2,3, or more.

"Say what?!" Any triangle with a vertice at circle's center
and a side as circle's chord promotes this transcendentalism.

Rod ... ...

[Amigoo]

Re: Transcendental Correlation design
"C = 2(2/sqrt(Pi)), 2.0, sqrt(Pi), Pi/2"

Speaking of "nano"...
Pi is relative to the circle's diameter, a value easily reduced by sqrt(2),
reducing the related Pi value in the triangle's hypotenuse or two sides.

Which mathematically terminates first: Pi or sqrt(2)?

Rod

[Amigoo]

Re: Transcendental Correlation design
"C = 2(2/sqrt(Pi)), 2.0, sqrt(Pi), Pi/2"

An interesting pattern where D = 2
and half of circumference equal to Pi,
then values decremented by sqrt(2):

2.0000000000000000000000000000000.. 3.1415926535897932384626433832795..
1.4142135623730950488016887242097.. 2.2214414690791831235079404950303..
1.0000000000000000000000000000000.. 1.5707963267948966192313216916398..
0.7071067811865475244008443621048.. 1.1107207345395915617539702475152..
0.5000000000000000000000000000000.. 0.7853981633974483096156608458198..
0.3535533905932737622004221810524.. 0.5553603672697957808769851237575..
0.2500000000000000000000000000000.. 0.3926990816987241548078304229099..
0.1767766952966368811002110905262.. 0.2776801836348978904384925618787..

Note: Relationships remain equal to Pi/2

If C = 4, D = ((2/sqrt(Pi))^2
= 1.2732395447351626861510701069801..
= circle's Area with SoCS = 2/sqrt(Pi)
= 1.1283791670955125738961589031215..

"Say what?! Diameter = Area?!"

Rod

[Amigoo]

Re: Pi Fork 'n Shadow design

Circumference of outer circle and perimeter of inner square both equal 2.0.
... and the two circles are squared (with Pi Fork displayed in inner circle).

Rod

[Amigoo]

Re: Pi Fork 'n Shadow design
Final esotericism for 2019.

Who knew
Pi is defined as ratio of circumference to diameter ...
and diameter is a diagonal of an inscribed square ...
identifying Pi's direct relationship to sqrt(2) ...
proving Pi is divisible by sqrt(2) ... and 2.

Who knew?! '3' is a nice round number ...
when representing a triangle, inscribed in a circle.

Q: Which came first: the circle or the square
A: "What's the point?" Indeed!

Rod ... ...

[Amigoo]

Re: Pi Fork 'n Shadow design

Who knew
Geometric equilibrium appeared in the final minutes of 2019!

Long story short (where Diameter = 2.0) ...
With half the circumference equal to Pi, the three green lines (left to right)
associate with Pi/2, sqrt(Pi), and Pi where sqrt(Pi)/(Pi/2) = 2/sqrt(Pi)
= 1.1283791670955125738961589031215..

Rod

[Amigoo]

Re: A Pi Baked design*

Stick a fork in it. It's done!

* added to: http://aitnaru.org/images/Quinale!.pdf

Rod ... ...

[Amigoo]

Re: Yin Yang Pi design

Obviously, Quadrature is all about Cartesian balance.

YYPi hints of a sqrt(2) correlation between circle's diameter and circumference,
correlation that might be more transcendental that the current Pi model(s).

In 2020, will 'YYPi' become "Why? Why? Pi" ... or "Yippee!"

Rod

[Amigoo]

Re: Yin Yang Pi design

Quadrature is all about Cartesian balance.

Apparently, sqrt(2) has more to say about Yin Yang and Quadrature,
but the bedraggled mouse needs quality time away from the pad.
There's plenty of 2020 time for this curious Cartesian chatter.

Rod ... ...

[Amigoo]

Re: Yin Yang Pi design

Quadrature is all about Cartesian balance.

... but what's the geometric secret of Yin Yang Pi ?

Every point on the circle exists between the midpoint of a side of the circle's inscribed square
and a corner of the circle's enclosing square, as these points are rotated around the circle.

Rod

[Amigoo]

Re: Yin Yang Pi design

Quadrature is all about Cartesian balance.

In these Cartesian Neighborhoods ...
it matters not that you go East first, then West
as long as you also go North, then South ...
or vices versas versas vices.

"Say what?!" That's life on Route 606.

Rod ... ... (why is every exit an entrance?)

[Amigoo]

Re: Yin Yang Pi design

Quadrature is all about Cartesian balance.

Há Pi New year !!
May you continue to go North, South, East, and West all year
... but not at the same time.

2019 recap (and worthy design finale):
http://aitnaru.org/images/Yin_Yang_Pi.pdf

Rod ... ...