Paradise Trinity Day

[Amigoo]

Re: Bubble eQ16 design (in development)

Mysterious pattern of "Sweet 16" in circle-squaring right triangle
that can be divided into 16 similar triangles, where 16 = 1 + 3 + 5 + 7

That a triangle can be divided into 16 similar triangles is not so mysterious (to geometers),
but how does 16 (instead of other increments) relate to this squared circle geometry
Bubble eQ16 shows a meaningful relationship with further emphasis on sqrt(2).

An intriguing geometric clue (re: 16 / 2 = 8 ) ...
A circle and its square connect at 8 symmetric points ... only

Rod

[Amigoo]

Re: Bubble eQ16 design (in development)

Mysterious pattern of "Sweet 16" in circle-squaring right triangle
that can be divided into 16 similar triangles, where 16 = 1 + 3 + 5 + 7

Interesting quotient: 2.0 / 1.5 = 4/3
= 1.3333333333333333333333333333333..
therefore (4/3) x 1.5 = 2.0

"Say what?!"
Late night snackin' on "Sweet 16" patterns.

Rod

[Amigoo]

Re: Bubble eQ16 design

Mysterious pattern of "Sweet 16" in circle-squaring right triangle
that can be divided into 16 similar triangles, where 16 = 1 + 3 + 5 + 7

What fun! for geometers of Quadraturial persuasion:
This geometry wants to comment on the Pythagorean Theorem,
with focus on the sides of the circle-squaring right triangle.

"Say what?!" This triangle's long side, representing Pi,
is mathematically "constrained" by its hypotenuse-diameter.

"Say what?!" In these Cartesian Neighborhoods,
"transcendental" may be more ethereal than "real".

"Say what?!" What

Rod ... ...

[Amigoo]

Re: Bubble eQ16 design*
"Quadraturial triangulation, simplified"

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

Rod

[Amigoo]

Re: Bubble eQ16 design*
"Quadraturial triangulation, simplified"

Geometers' secret ...
Length of straight lines in red "caliper" = (Pi/2)-1,
confirming esoteric essence of Pythagorean Pi
... give or take some decimal digits ...
until proven precise.

Rod ... ...

[Amigoo]

Re: Bubble eQ16 design*
"Quadraturial triangulation, simplified"

Length of straight lines in red "caliper" = (Pi/2)-1,

Not close enough to justify keeping the "caliper"
(after recalculation of line lengths in right triangle,
then extrapolation to other lines).

3 .14159.. steps forward, 3 steps backward

Rod

[Amigoo]

Re: Bubble eQ16 design*
"Quadraturial triangulation, simplified"

3 .14159.. steps forward, 3 steps backward

No "caliper" but real Quadrature ...
showing the sqrt(2) right triangle sibling
of the large circle-squaring right triangle.

Rod

[Amigoo]

Re: Bubble eQ16 design
"Quadraturial triangulation, simplified"

Geometers' secret ...
The significance of "16" is that the circle-squaring right triangle beginning the srqt(2) spiral
has dimensions 16 times larger (or smaller) when the spiral completes a 360 degree turn,
with triangle ending size (begins next spiral) dependent on spiral rotation.

Rod ... ...

[Amigoo]

Re: Bubble eQ16 design
"Quadraturial triangulation, simplified"

Who knew ...
This geometric composition wants to be "impossibly" esoteric
... and now hosts the integrated right and scalene triangles
as well as the isosceles trapezoid, all circle-squaring!
(or geometric evidence of a squared circle)

"What happened to 'simplified'?"
Apparently, a Quadraturial oxymoron;
why Quadrature is believed "impossible".

Rod

[Amigoo]

Re: Bubble eQ16 design
"Quadraturial triangulation, once simplified"

Geometers' secret ...

The 3 integrated, circle-squaring geometric objects*
all have one side with the same length (SoCS).
And each of the 3 objects have a side with length
equal to the circle's diameter or its SoIS.

* right and scalene triangles, isosceles trapezoid

SoCS = Side of Circle's Square
SoIS = Side of Inscribed Square
(= Diameter / sqrt(2))

... making this secret "Triangles, An Aside"

"But a trapezoid is not a 'triangle' "
Au contraire This trapezoid is composed of 2 overlapping,
circle-squaring scalene triangles and a small right triangle.

Rod

[Amigoo]

Re: Primary Scalenity of Quadrature design

Two overlapping, circle-squaring trapezoids,
with 3 overlapping, circle-squaring scalene triangles.
It's a beautiful day in the Neighborhood!

P.S. Quadrature

Rod ... ...

[Amigoo]

Re: Primary Scalenity of Quadrature design

Geometers' secret ...
"P.S. Quadrature" refers to the smaller, dark blue, scalene triangle
that is similar to the three larger scalenes, but squares a circle
of smaller diameter ... by sqrt(2).

Rod

[Amigoo]

Re: Primary Scalenity of Quadrature design

P.S. Quadrature (“That's the truth”)

Geometer's whimsical secret about the blue face
with “That's the truth” righteousness: Edith Ann

Rod ... ...

[Amigoo]

Re: Primary Scalenity of Quadrature design*

P.S. Quadrature (“That's the truth”)

Geometers' secret - not whimsical ...

The red pentagon (five-sided polygon), composed of the circle-squaring
right and scalene triangles with overlapping trapezoids, is the sqrt(2)^2
sibling of the larger pentagon (golden yellow). This sqrt(2) association
of similar lines is confirmed with simple geometry!

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

Somehow, Quadrature's 2/sqrt(Pi) constant comes to mind
(= 1.1283791670955125738961589031215..)

Rod

[Amigoo]

Re: Primary Scalenity of Quadrature design*

Quadrature's 2/sqrt(Pi) constant comes to mind

Who knew
Pi itself defines Quadrature (defines circle-squaring right triangle):
2(sqrt(Pi))/Pi = 2/sqrt(Pi) = sqrt(Pi)/(Pi/2) = 2(1/sqrt(Pi))
= 1.1283791670955125738961589031215..

Rod ... ...

[Amigoo]

Re: Primary Scalenity of Quadrature design

Who knew
Quadraturial scalenity faciltates geometric division of a circle-squaring
isosceles trapezoid into four similar trapezoids, one divided.

"Say what?!" That's the truth.

Rod

[Amigoo]

Re: Exhibit Q design

Quadrature, simplified.

Rod

[Amigoo]

Re: Exhibit Q design

Geometers' secret of Exhibit Q ...
All respective lines of two similar geometric objects have sqrt(2) relationship
if the objects share either hypotenuse or one side of an isosceles right triangle.

"Say what?!" The line whose length represents Pi in a squared circle
has sqrt(2) association with the respective line In another squared circle
where both circles associate via the isosceles right triangle.

"Say what?!" Pi is divisible by 2 and sqrt(2).

Rod

[Amigoo]

Re: Exhibit Q design

Such sqrt(2) integration of the circle-squaring objects
(quadrilateral, isosceles trapezoid, scalene and right triangles)
that geometers of Quadraturial persuasion now ask:
"Can squared circles exist without sqrt(2)?"

Rod ... ...

[Amigoo]

Re: Exhibit Q design
"Quadrature, simplified … and hosted by sqrt(2)"

Quadrature, by the numbers ...
where D = Diameter, r = radius,
SoCS = Side of Circle's Square

2/sqrt(Pi) = 1.1283791670955125738961589031215..
^2 = 1.2732395447351626861510701069801.. = D
/ 2 = 0.63661977236758134307553505349006.. = r
^2 = 0.40528473456935108577551785283891.. = r^2

Area of D = Pi(r^2)
= Pi x 0.40528473456935108577551785283891..
= 1.2732395447351626861510701069801.. Pi(r^2)
sqrt( = 1.1283791670955125738961589031215.. SoCS

Rod

[Amigoo]

Re: Exhibit Q design
"Quadrature, simplified … and hosted by sqrt(2)"

Not that squared circles rule the universe (until revealed),
the sqrt(2) integration of CSCSC geometry is still notable
Did you know that the diagonal of the smaller trapezoid
has same length as midpoint-to-midpoint of the larger?

Rod ... ...

[Amigoo]

Re: Exhibit Q design*
"Quadrature, simplified … and hosted by sqrt(2)"

Geometers' secret:

There are 2 overlapping, circle-squaring isosceles trapezoids in each circle,
and geometric integration of the many objects of Quadraturial persusasion
in this composition creates the "impossible" geometric Blue Wedgie,
correlation of the larger trapezoids' midpoint-to-midpoint lines
with the smaller trapezoid's diagonal and bottom lines ...
correlation facilitated by circle-squaring scalenes.

"Say what?!" OMG! Pi gets a Blue Wedgie!

Rod

[Amigoo]

Re: Exhibit Q design
"Quadrature, simplified … and hosted by sqrt(2)"

Quality Control prefers these better line colors,
especially to highlight the center of this Quadrature,
(identified by the geometric "butterfly").

Rod ... ...

[Amigoo]

Re: Scalenity If,Then design

The circle-squaring scalene triangle in one circle
defines the diameter (etc.) of the opposite circle ...
with Quadrature in a Cartesian Neighborhood.

"How does it know?"
The magic of simple geometry, opposite angles,
and all that math stuff, faciltated by 2/sqrt(Pi).

"Where's the Blue Wedgie?"
Dunno! In a tasty Blueberry Pi?

Rod

[Amigoo]

Re: Scalenity If,Then design*
"If Quadrature here, then Quadrature there"

... and If Quadrature in that inhabited universe,
then Quadrature in this inhabited universe!

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

Rod ... ... (here, there, everywhere Quadrature ... apparently)