Paradise Trinity Day

[Amigoo]

Re: Point of Quadrature design*
"What's the real point?!" Indeed

You would think that Quadrature would eventually reveal itself ...
instead of these replicating circles all squared ... with concentricity.

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

Rod

[Amigoo]

Re: Point of Quadrature design
"What's the real point?!" Indeed

You would think that Quadrature would eventually reveal itself ...
instead of these replicating circles all squared ... with concentricity
and obvious sqrt(2) integration Where's the '4' after 1,2,3,
... or maybe "Where's the '0' ?" is the real point.

After all, infinity has to start somewhere ...
and it already started ... apparently.

Rod ... ...

[Amigoo]

Re: Point of Quadrature design
"What's the real point?!" Indeed

Geometers' secret ...

For each of the three two-circle sets, all concentric,
the ratio of the outer diameter to the inner diameter
= 2/sqrt(Pi) = 1.1283791670955125738961589031215..
= sqrt(Pi)/(Pi/2) = 2(sqrt(1/Pi))

And this is the ratio of the lengths of each respective side
of those circles' area squares; as well as the ratio of the lengths
of the respective sides of those circles' inscribed squares.

Of course, sqrt(2) is the MC of this Three-Rings Concentricity.

Rod

[Amigoo]

Re: Point of Quadrature design

sqrt(2) is the MC of this Three-Rings Concentricity

And this reveals that the Pi values of the outer set of rings
are precisely twice the dimensions of the values of the inner set
(geometrically confirmed by the dark blue area squares).

The transcendental mystery of "impossible" Quadrature.

Rod ... ...

[Amigoo]

Re: Point of Quadrature design

Who knew that a lop-sided quadrilateral* (dark blue)
would add geometric balance to this colorful composition?

* circle-squaring right and isosceles right triangles.

Rod

[Amigoo]

Re: Point of Quadrature design

Who knew
Quadrature is as simple as 3,4,5 (triangle, quadrilateral, pentagon).

During this morning's review of points available for a circle-squaring scalene triangle,
the inner pentagon revealed itself to claim inclusion of the other two objects.

Who can tell what other objects square the circle. Trapezoid?
(also included in that pentagon)

Rod

[Amigoo]

Re: Right on Q design (in progress)

New arrangement of the circle-squaring objects (right and scalene triangles,
trapezoid, pentagon) constructed upon remaining portion of Point of Quadrature.
Not at all complicated since these objects "mix 'n match" in various overlays.

"What's Right on Q?" Geometrically obvious, but 'Q' is really a circle.

Rod

[Amigoo]

Re: Right on Q design (where 'Q' is a circle)

"What's with the '7' on the Q ?"

Who can tell Concentric circle-squaring right triangles
... or a symbolic "Hello!" from HQ of your local universe
(where "7 is fundamental to universe organization").

Rod ... ...

[Amigoo]

Re: Right on Q design

Then sqrt(2) insisted on obvious representation, thus one more circle (arc).
Now, 'C70' seems to display. Who knew Quadrature speaks chemistry.
(or it's a topical "Hello!" from Buck)

See also: https://en.wikipedia.org/wiki/C70_fullerene

Rod

[Amigoo]

Re: Right on Q design*

It always happens
Every circle-squaring object wants to attend the Squared Circles Soiré.
(if one's not displayed, you can connect the points)

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

Rod ... ...

[Amigoo]

Re: Right on Q design

Geometers' secret ...
Sqrt(2) and its isosceles right triangle propagates the circle-squaring quadrilateral
(adjoined right and isosceles right triangles) from circle to circle to circle,
geometric magic visible at the top of the '7' ... and confirmed because
the diameters of these circles associate by sqrt(2). Who knew?!

Rod

[Amigoo]

Re: Right on Q design

Geometers' secret ...
The integrated scalene triangles, when adjoined, create parallelograms
that further integrate and beg: "How can the circles not be squared?"

Rod ... ...

[Amigoo]

Re: Right on Q+P design

scalene triangles, when adjoined, create parallelograms

"Been there! Done That!"

Rod

[Amigoo]

Re: Right on Q II
"Quadrature as revealed by sqrt(2)"

Defines the mathematical limit of Pi ... apparently.

Rod ... ...

[Amigoo]

Re: Right on Q II
"Quadrature as revealed by sqrt(2)"

The only thing "impossible" about Quadrature ...
is that Quadrature cannot be proven without sqrt(2),
obviously hosting in a Cartesian Neighborhood.

Rod

[Amigoo]

Re: Right on Q II
"Quadrature as revealed by sqrt(2)"

Simplified slightly to highlight the parallelogram (green)
that unites the many right triangles in this composition;
a parallelogram that's also "circle-squaring" vis-à-vis
its two unique (and similar) scalene triangles.

Rod ... ...

[Amigoo]

Re: Right on Q II
"Quadrature as revealed by sqrt(2)"

Better highlighting of the coordinating parallelogram* (green)
and the two primary isosceles right triangles (top and bottom)
that facilitate this sqrt(2) hosting of circles all squared.

* composed of two similar circle-squaring scalene triangles.

Rod

[Amigoo]

Re: Right on Q II
"Quadrature as revealed by sqrt(2)"

Given circle-squaring scalene triangle with
Side of Circle's Square (SoCS) = sqrt(Pi),
Side of Inscribed Square (SoIS) = sqrt(2)
opposite a 45 degree angle, Diameter = 2.0

1.7724538509055160272981674833411.. SoCS, sqrt(Pi)
/ 1.4142135623730950488016887242097.. SoIS, sqrt(2)
= 1.2533141373155002512078826424055..

(1.7724538509055160272981674833411.. SoCS, sqrt(Pi)
x 1.4142135623730950488016887242097..) SoIS, sqrt(2)
/ 2 = 1.2533141373155002512078826424055..

Note: 2/ sqrt(Pi) = 1.1283791670955125738961589031215..
(Diameter to Long Side of circle-squaring right triangle)

Rod ... ...

[Amigoo]

Re: iScalenity design (infinite Scalenity)
(propagation of "Right on Q" scalene triangles;
a few steps in the direction of infinity)

Rod

[Amigoo]

Re: iConcentric design*
"One in ten million"

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

Rod ... ...

[Amigoo]

Re: iConcentric design
"One in ten million"

Geometers' secret:
Concentricity of circles squared = 2/sqrt(Pi)
= 1.1283791670955125738961589031215..
diameters increase/decrease by 2/sqrt(Pi)

Rod

[Amigoo]

Re: iConcentric design

Today's most popular sub-title ...
"Keep your eye on the ball ... or it's Pi in the sky"

"What's Peye in the sky?"

Rod ... ...

[Amigoo]

Re: iConcentric design

Geometry humor ...

The Finnish men of decades past had a similar mantra (and sauna ceremony) as this:
"Keep your eye on the ball ... or it's Pi in the sky"

In the winter, after thrashing themselves with their handmade vihta* in the hot sauna,
they would jump into the cold lake, then shout "I See Ball" (sounds like ; - )

* https://www.youtube.com/watch?v=k_6oqICxUlMSAftern

Rod

[Sandy]

Hi Rod,
That was fascinating. Why do Finnish men(women too?) thrash themselves in the Sauna? Does it help blood flow or something?
xxSandy

[Amigoo]

Sandy,

The thrashing is generally light and improves blood flow all over the body.
However, the aroma of the young birch leaves in the hot sauna is "to die for".

Interestingly, Americans (and others) not familiar with this Finnish custom
often use the vihta as a fig leaf when disrobing and entering a co-ed sauna.
The "fig leaf" doesn't need to be large since a dip in the icy water is next.

Rod