Paradise Trinity Day

[Amigoo]

Re: Trapezoidal Bifurcation design
Quadrature is all about pattern!

Quadraturial reality ...
The chord of any Cartesian object inscribed in a circle directs the sqrt(2) increase or decrease
of a similar object, including one incorporating transcendental Pi (as shown in composition).

Rod

[Amigoo]

Re: Trapezoidally Cubic design*
Quadrature is all about pattern

Who knew
Quadrature, forevermore "out of the box"

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

Rod ... ...

[Amigoo]

Re: Trapezoidally Cubic design
Quadrature is all about pattern

"Say what?! There's another Q in town?! What's their angle

"anonym - fictitious name used when the person performs a particular social role"

Popular name and fictitious group! ... but believed to be real
Who knew?! The internet can create reality ... or so it seems.

Rod

[Amigoo]

Re: What's Where? design

Geometry nerds of Quadraturial persuasion always ask "What's where?"
(as in "Where's the next sqrt(2) iteration of these similar trapezoids?")

Quadrature, forevermore "out of the box"
... trapezoidally speaking.

Rod ... ...

[Amigoo]

Re: What's Where? design*

Quadrature, forevermore "out of the box(es)"
... trapezoidally speaking.

Now highlights (red) three circle-squaring objects of Quadrature:
Pythagorean right triangle, unique scalene triangle, isosceles trapezoid.

"So, what's where now?" OMG! What's not where

* updated in: http://aitnaru.org/images/Yin_Yang_Pi.pdf

Rod

[Amigoo]

Re: What's Where? design
Quadrature, forevermore "out of the box(es)"
... trapezoidally speaking.

Highlights (green) adjoined right and isosceles right triangles,
creating quadrilateral showing the union of sqrt(Pi) and sqrt(2):

1.7724538509055160272981674833411.. sqrt(Pi)
/ 1.4142135623730950488016887242097.. sqrt(2)
= 1.2533141373155002512078826424055.. (sqrt(Pi)sqrt(2)) / 2

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

/ 1.2533141373155002512078826424055.. (sqrt(Pi)sqrt(2)) / 2
= 2.0 = circle's diameter and hypotenuses

Rod

[Amigoo]

Re: Rule of Root 2 design

"Ask not 'What's the point?' "

Rod ... ...

[Amigoo]

Re: Rule of Root 2 design
"Ask not 'What's the point?' "

1. Geometer's secret about "secret"
"A secret shared is an oxymoron"

(once information is shared, it's no longer a "secret",
but not to say that information should not be shared)

2. Geometer's secret about Rule of Root 2

In 2009, when revealing to a friend that a circle's square
rests upon 8 points of its circumference, my friend asked
"What the point?". Cartesian exploration to date reflects
now countless attempts to answer the question.

3. Geometer's secret about closure: "What the point?"

(Quadrature is Pythagorean geometry about those 8 points
reflected by an infinite number of sqrt(2)-nested circles)

4. Geometer's secret about "transcendental"
"sqrt(2) is transcendental or Pi is not ... apparently"

Rod

[Amigoo]

Re: Rule of Root 2 design*
"Ask not 'What's the point?' "

Despite the increasingly esoteric geometry,
this Cartesian composition simply proves that
Pi must be evenly divisible by sqrt(2).

* updated in: http://aitnaru.org/images/Yin_Yang_Pi.pdf

Rod

[Amigoo]

Re: Rule of Root 2 II design
(sometimes pronounced "Rule of Root Tutu")

Two overlapping, circle-squaring isosceles trapezoids
with all lines having sqrt(2) association

Geometer's secret about "overlapping trapezoids":

The "all lines" sqrt(2) association may not be obvious
since there are more than two isosceles trapezoids
... with varying similarity.

Rod ... ...

[Amigoo]

Re: Rule of Root 2 II design*
(sometimes pronounced "Rule of Root Tutu" and suggesting
a Cartesian Tutu is required to comprehend Quadrature).

Three overlapping, circle-squaring isosceles trapezoids
with all lines having sqrt(2) association

Better visual identification of the 3 overlapping trapezoids,
showing the sqrt(2) transition within these nested circles.

Who knew
2/sqrt(Pi) = 1.1283791670955125738961589031215..
is still the challenging constant since this ratio**
defines both the circle and its square!

** circle-squaring right triangle with length of hypotenuse
(circle's diameter) divided by length of triangle's long side,
having length equal to a side of circle's area square.

* updated in: http://aitnaru.org/images/Yin_Yang_Pi.pdf


Who knew
This isosceles trapezoid seems to associate with ancient Egyptian fashion
and suggests a Cartesian kilt for Mighty Mathematical Men (MMM)
(MMW probably wear a kiltette).

Entrepreneur Alert
Design one for the inaugural Squared Circles Soirée

Rod

[Amigoo]

Re: Rule of Root 2 II design
(sometimes pronounced "Rule of Root Tutu")

Geometer's secret about the 2 green trapezoids,
one four times the size of the other. Go figure ...
4 similar trapezoids fit precisely in the larger.
("some assembly required")

And voted mo betta' design to date!
"Aw, honey, you lookin' mo betta' today!"
https://www.urbandictionary.com/define. ... mo%20betta

Rod

[Amigoo]

Re: Rule of Root 2 II design
(sometimes pronounced "Rule of Root Tutu")

More "What's where?" exploration, adding another trapezoid
and making it more challenging to find 3 overlapping trapezoids
with all lines having sqrt(2) association. Bon Appétit

Rod ... ...

[Amigoo]

Re: Rule of Root 2 II design
(sometimes pronounced "Rule of Root Tutu")

Even more "What's where?" exploration!
Only a geometry nerd of Quadraturial persuasion
will know how this latest design has changed ...
and what new object is highlighted.

Rod

[Amigoo]

Re: Scalenifiable design*

Until advanced math usurped the Cartesian domain,
morbus cyclometricus was considered checkmated whenever
a Cartesian composition presented convincing evidence of
"sqrt(2) scalenifiability".

* updated in: http://aitnaru.org/images/Yin_Yang_Pi.pdf

Rod

[Amigoo]

Re: Scalenifiable design
"The Three Secrets of Scalenifiability"

Geometer's secret about "scalenifiable" ...
Reference to a circle-squaring scalene triangle
where two sides have these unique lengths:
sqrt(2) - one side of the circle's inscribed square.
sqrt(Pi) - one side of the circle's area square.

Geometer's secret about Scalenifiable design ...
In this Cartesian composition, 2 of these triangles overlap
(one green, one yellow) where all 3 sides of one triangle
have sqrt(2) length relationship to the sides of the other.

Geometer's secret about the overlapping triangles ...
Each scalene triangle defines a squared circle. Go figure!
And these nested circles also have sqrt(2) relationship.

Thus, a revelation of "morbus cyclometricus"
that sqrt(2) and sqrt(Pi) indeed unite in Quadrature
and that Quadrature is "tri-scalenifiable". HCIT

"How can we explain Quadrature?" Tri-scalenifiable!
"Thanks! We'll try ... when we're 'out of the box' ".

Rod

[Amigoo]

Re: Scalenifiable 2020 design
"E FP TOZ LPED PECFD EDFCZP"

Different perspective with visual humor
( requires "out of the box" thinkin' )

Rod

[Amigoo]

Re: Scalenifiable 2020 design*
"E FP TOZ LPED PECFD EDFCZP"

More sqrt(2) correlation of circular objects, well-nested.

* updated in: http://aitnaru.org/images/Yin_Yang_Pi.pdf

Rod ... ...

[Amigoo]

Re: Scalenifiable 2020 design
"E FP TOZ LPED PECFD EDFCZP"

Geometer's secret ...

All three lines connecting similar-angle to similar-angle
of the overlapping scalene triangles have identical length!
(proves sqrt(2) association of these two scalene triangles
as enclosed by their sqrt(2)-associated circles).

"The inner triangle's circle is not shown!"
It's super simple geometry! Go figure!

Rod

[Amigoo]

Re: Scalenifiable 2020 design
"E FP TOZ LPED PECFD EDFCZP"

Geometer's secret of the little hat
on top of the circle-squaring right triangle:
a² + b² = c² (aka "Pi are square")

Rod ... ...

[Amigoo]

Re: Scalenifiable 2020 design
"E FP TOZ LPED PECFD EDFCZP"

Geometer's secret of the little hats*
on top of the circle-squaring right triangles:
a² + b² = c² (aka "Pi are square")

* Obvious presence of Pythagorean Theorem ...
twice and having sqrt(2) object relationship!

Rod

[Amigoo]

Re: Scalenifiable 2020 design*
"E FP TOZ LPED PECFD EDFCZP"

A Cartesian composition tweaked up, down, left and right, for days
... and sqrt(2) still refuses to take second place to transcendental π

So be it! ... in 2020 (but vaccines are on the way!)

* updated in: http://aitnaru.org/images/Yin_Yang_Pi.pdf

Rod ... ...

[Amigoo]

Color changed for one line ("So be it! ... in 2020")
for reasons known only to geometers of Quadraturial persuasion.

Rod

[Amigoo]

Re: Scalenifiable 2020 design
"E FP TOZ LPED PECFD EDFCZP"

Geometer's secret ...

This Cartesian composition visually links triangles that define Quadrature:
Pythagorean Right and a unique Scalene (which is composed of a similar
Pythagorean and the sqrt(2)-iconic Isosceles Right triangle).

Good evidence of persistent "morbus cyclometricus"
... but vaccines will be available in 2021.

Rod ... ...

[Amigoo]

Re: Scalenifiable 2020 design
"E FP TOZ LPED PECFD EDFCZP"

Despite the fast food of advanced math,
many geometers of Quadraturial persuasion
still prefer numeric fiber in their diet:

1.7724538509055160272981674833411.. sqrt(Pi)
/ 1.4142135623730950488016887242097.. sqrt(2)
= 1.2533141373155002512078826424055..

2 / 1.7724538509055160272981674833411..
= 1.1283791670955125738961589031215..

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

Rod