Paradise Trinity Day

[Amigoo]

Re: Scalenitweet design
(aka "Sweet Scalenity")

No wonder squared circles are considered "impossible"
The 3 corner-to-corner magenta lines (connect these points
of the overlapping scalene triangles) all have equal length,
proving geometric balance in this simple composition.

Rod

[Amigoo]

Re: Scalenitweet design*
"Quietly messaging Sweet Scalenity"

Slightly recolored to highlight festive assembly of circle-squaring objects:
right triangle, scalene triangle, isosceles trapezoid, unique quadrilateral
... with honorable mention to the similar Point-to-Pointers triad.

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

Rod

[Amigoo]

Re: Scalenitweet design
"Quietly messaging Sweet Scalenity"

Chirp! Chirp!
Geometers' secret: Squared circles have building blocks (objects)
that, once discovered, permit unending Cartesian complexity.

And Pi/16? Who knew What's that

Rod

[Amigoo]

Re: Scalenitweet design
"Quietly messaging Sweet Scalenity"

Geometers' secret:

Similar objects that share one side and hypotenuse (respectively)
of an isosceles right triangle have line lengths with sqrt(2) relationship
regardless of the number of lines/angles in each object. In other words,
sqrt(2) defines all paired lines/angles from the two objects.

Conversely, the objects are similar if this sqrt2) relationship
exists for all paired lines/angles from the two objects.

Rod ... ... (prefer a slice of Pi cut on the hypotenuse)

[Amigoo]

Re: Scalenitweet design
"Quietly messaging Sweet Scalenity"

Geometers' dark blue secret:
Vesica Piscis is present at this Squared Circles Soirée.

Rod

[Amigoo]

Re: Scalenitweet design
"Quietly messaging Sweet Scalenity"

At this Squared Circles Soirée, Vesica reveals
that Pi is evenly divisible by sqrt(2)
... and therefore 2.0

Considering that (re: https://en.wikipedia.org/wiki/Transcendental_number )

"the square root of 2 is irrational but not a transcendental number,
since it is a solution of the polynomial equation x2 − 2 = 0" ...
and sqrt(2) is represented by an isosceles right triangle ...

... and that Pi can also be represented by a similar triangle,
then Pi's transcendental quality must devolve to sqrt(2)
at least, in these Cartesian Neighborhoods.

Rod ... ...

[Amigoo]

Re: Scalenitweet design
"Quietly messaging Sweet Scalenity"

Pi is evenly divisible by sqrt(2)

Several straight lines were added to support
when sqrt(2) teases squared circle geometers ...
"Can you hear me now "

Rod

[Amigoo]

For your next Squared Circles Soirée ...
decorate with Pi's distant relatives:

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

sqrt(Pi) x 1.1283791670955125738961589031215.. = 2.0
sqrt(Pi) / 1.1283791670955125738961589031215.. = Pi/2

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

Rod

[Amigoo]

Re: Scalenitwixt design

A new name was required but still related to "Scalenitweet"
... and considering that the geometric influence of sqrt(2)
seems betwixt and between the scalene triangles.

Besides, 'X' marks the spot(s).

Rod

[Amigoo]

Re: Scalenitwixt design

Betwixt and between juxtapositional scalene,
quadrature reigns, sans king and sans queen.
Princess and prince, oft yonder, draw nigh -
irrational square roots of 2 and of Pi.

Rod

[Amigoo]

Re: Scalenitwixt design

Ask not "What's inside the box?"
ask "What's not inside the box?"

Rod ... ...

[Amigoo]

Re: Scalenitwixt design

ask "What's not inside the box?"

Given: SoCS = Side of Circle's Square,
2/sqrt(Pi) = 1.1283791670955125738961589031215..

If Diameter = 2/sqrt(Pi), SoCS = 1 (aka "Pi = 1")

Calculation: Area = Pi x radius^2
D = 1.1283791670955125738961589031215..
/ 2 = 0.56418958354775628694807945156077.. radius
^2 = 0.31830988618379067153776752674503.. radius^2
x Pi = 1 (the Area), and sqrt(1) = 1 (the SoCS)

One Pi is enough for me when I'm really hungry

Rod

[Amigoo]

Re: Three Squares of Pi design
(SoCS = Side of Circle's Square)

Cartesian contrast of 3 squared circles
where SoCS = 2.0, sqrt(Pi), Pi/2
Diameters = 2(2/sqrt(Pi)), 2.0, sqrt(Pi)

Rod ... ... (shoppin' for 3 squares a la mode)

[Amigoo]

Re: Three Squares of Pi design
(SoIS = Side of Inscribed Square)

A smaller squared circle was added, using a SoIS of that golden circle
to illustrate how this geometry propagates with these quadrature
"building blocks" of objects & angles in CSC Neighborhoods.

Rod

[Amigoo]

Re: Perfect Slice design

This "Pi Fork" right triangle must be the definitive circle-squaring object
with it's two longer lines and two next shorter lines having 2/sqrt(Pi) ratio
... and the two shortest lines having sqrt(Pi) line length ratio.

Also, a longer line and it's complementary next shorter line
have sqrt(Pi) line length ratio. HCIT ... not to mention
that this object is composed of 3 similar right triangles.

Apparently, for the circle to be squared by the Greek rules,
first creating this right triangle may be Step One.

Tip: Hypotenuse is circle's diameter.

Rod ... ...

[Amigoo]

Re: Perfect Slice design

Geometers' secret:
The magenta lines in the Scalenitweet design have equal length
when the circle is squared. These 3 lines connect similar points
(angles) of two CSC-related, circle-squaring scalene triangles.

Rod ... ... (shoppin' for Tweet Tweet ring tone)

[Amigoo]

Re: PS Redux design

Propagation of Perfect Slice geometry ...
convincing that sqrt(2) rules the Neighborhood

The hypotenuse and side of an isosceles right triangle
easily propagate geometric objects by a factor of sqrt(2).

Rod

[Amigoo]

Re: PS Redux design*

Sqrt(2) rules the Neighborhood

Obviously, the 2/sqrt(Pi) right triangle ratio defines ...
a "detailed, recursive, and infinitely self-similar mathematical set",
a newly discovered Quadractal (quadrature + fractal)?

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

Rod

[Amigoo]

Re: PS Redux design*

Sqrt(2) rules the Neighborhood

Geometers' secret:

The magic of the Quadractal is that the hypotenuse of this right triangle defines a circle's diameter
and the long side the length of a side of that circle's area square. Since sqrt(2) is inherent in circles,
infinite propagation of geometric circle-squaring objects (this right triangle, scalene triangle,
isosceles trapezoid) becomes the "impossible" reality.

The boundaries of Heaven and Earth adjoin ... apparently.

Rod

[Amigoo]

Re: Scalenifiable design
"Sqrt(2) also rules this Neighborhood"

Geometers' secret:
Quadrature is confirmed by a squared circle's scalenifiability.

Rod (4-letter words are easier to pronounce)

[Amigoo]

Re: Scalenifiable design
"Sqrt(2) also rules this Neighborhood"

Adding more lines was easier than thinking of four 4-letter words
to substitute for the 16-letter tongue twister "scalenifiability".
Besides, these lines help confirm scalenifiabilty

The red 'H' is best associated with HCIT!

Rod

[Amigoo]

Re: Scalenifiable design
"Sqrt(2) also rules this Neighborhood"

Geometers' secret:
(sqrt(Pi) x sqrt(2)) / (sqrt(Pi) / sqrt(2)) = 2

(1.7724538509055160272981674833411..
x 1.4142135623730950488016887242097..)
/
(1.7724538509055160272981674833411..
/ 1.4142135623730950488016887242097..)
= 2

Rod

[Amigoo]

Re: Scalenifiable design
"Sqrt(2) also rules this Neighborhood"

Embellished to show that the hypotenuse of an isosceles right triangle
squares the largest circle and the side squares its sqrt(2) sibling.

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

Rod ... ...

[Amigoo]

Re: Trinitized Scalenity design
(exploration of Scalenifiable geometry)

Simply connecting available points revealed a higher-level,
circle-squaring scalene triangle and its squared circle.

Quadrature to infinity

Rod

[Amigoo]

Re: Trinitized Scalenity design
"Geometry that speaks for itself"

Simplified for better contrast of the circle-squaring scalene triangles.

The large red scalene is in the same Cartesian position as when
I first recognized that this triangle had quadraturial significance
... and it resolved my evolving pre-condition that quadrature
must display the "marriage" of sqrt(Pi) and sqrt(2).

Rod ... ...