Paradise Trinity Day

[Amigoo]

Re: Three Pi Vise design
"The crown awaits the king (of a geometric proof?)"

2(sqrt(1/Pi)) can hardly be called "revelation"
when 2.0 / sqrt(Pi) = sqrt(Pi) / Pi/2 ...

But it was long unknown (apparently) that this "revealed" constant
defined hypotenuse and long side of a circle-squaring right triangle.

Rod ... ...

[Amigoo]

Re: Three Pi Vise design
"The crown awaits the king (of a geometric proof?)"

Such geometric integration in the Three Pi "crown".
Between here and there (the two sides, one a diagonal), the length calculates
as (2(sqrt(1/Pi))^3 ... making the crown a very unique geometric object.

Rod (experiential crown)

[Amigoo]

Re: Three Pi Vise design
"The crown awaits the king (of a geometric proof?)"

Before roundin' the next corner of the largest squared circle,
I thought it best to include geometry that would help prove
that all four circles associate in quadraturial essence.

Geometers' riddle: A short line from the diameter would
create a right angle with the once-"fuchsine" line that, in music,
would be called a "harmonic set of pitches". The resulting
right triangle notes such "quadraturial essence".

Rod

[Amigoo]

Re: Three Pi Vise design

With geometers clamorin' to know the Pi value of the line
representing the left side of the golden crown, Pi(sqrt(Pi)/4),
the side-to-side length ratio became an easy calculation:

2.0 / Pi(sqrt(Pi)/4)
= 1.1283791670955125738961589031215..^3
= 1.4366969769991930158060821722332..

Note: R/L diameters = 2.0, sqrt(Pi), Pi/2, Pi(sqrt(Pi)/4)

Rod ... ... (off to get hot Pi cubes à la mode)

[Amigoo]

Re: Three Pi Vise design (concise summary)

A geometric composition of three adjoined, circle-squaring right triangles where hypotenuse is that circle's diameter. Long side of each right triangle has length equal to a side of that circle's square. "Three Pi" of "Three Pi Vise" refers to the three adjoined right triangles, defined by increments of Pi; "Vise" refers to the symbolic Pi Corral (re: "transcendental" Pi) of this composition.

Length of adjoined diameters (hypotenuse): 2.0, sqrt(Pi), Pi/2, Pi(sqrt(Pi)/4)

Integrated by constant: 1.1283791670955125738961589031215..
= 2(sqrt(1/Pi)) = 2.0/sqrt(Pi) = sqrt(Pi)/(Pi/2) = (Pi/2)/(Pi(sqrt(Pi)/4))

Second range: 2.0/(Pi/2) = sqrt(Pi)/(Pi(sqrt(Pi)/4))
= 1.1283791670955125738961589031215..^2

Third range: 2.0/(Pi(sqrt(Pi)/4))
= 1.1283791670955125738961589031215..^3

Rod : :

[Amigoo]

Re: Three Pi Vise design (more stuff)
(update: http://aitnaru.org/images/Golden_rPi.pdf )

Since this geometry permits the creation of a large circle-squaring right triangle,
it was expanded to include those lines (but image was cropped to fit page).

Now, composition well-highlights interaction of the two constants:
1.1283791670955125738961589031215.. 2(sqrt(1/Pi))
1.9130583802711007947403078280203.. "iPhi"

About Pi(sqrt(Pi)/4) ... How it was derived is still a mystery
I knew that it had to be an increment of Pi and probably involved
a square root; a 30-minute "shuffling" of numbers revealed it
(decrements of 2(sqrt(1/Pi)) gave the numeric value).

Rod ... ...

[Amigoo]

Re: Three Pi Vise design
The two Pi-derived constants:

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

sqrt(Pi)/sqrt(4-Pi) = 1.9130583802711007947403078280203.. "iPhi"

Rod

[Amigoo]

Re: Three Pi Vise design

The two Pi-derived constants
(define the circle-squaring right triangle):

2/sqrt(Pi) = 1.1283791670955125738961589031215.. = sqrt(Pi)/(Pi/2)
[ratio of hypotenuse (circle's diameter) to long side of triangle]

sqrt(Pi)/sqrt(4-Pi) = 1.9130583802711007947403078280203.. "iPhi"
[ratio of long side of triangle to short side; a^2 + b^2 = c^2]

Rod

[Amigoo]

Re: TPV Indeed design

Geometers' Bonus:
How to contrast the two circle-squaring right triangles
for circles having a diameter of 2.0 and sqrt(Pi)

... effectively confirming the constants,
2(sqrt(1/Pi)) and sqrt(Pi)/sqrt(4-Pi)

Rod ... ...

[Amigoo]

Re: The Right Triangle design
"Like father, like son"

Embellished and integrated sqrt(2) sibling to show
similarity of geometric circle-squaring objects
(especially scalene and right triangles).

Rod

[Amigoo]

Re: The Right Triangle(s) design
"Like father, like son"

Although intuitive to experienced geometers, this 2(sqrt(1/Pi)) equation
is at least numerically reassuring when written: (c/aa)/sqrt(2) = c/a.

Rod

[Amigoo]

Re: T-Squares design
"Like fathers, like sons"

Highlights the geometric 'T' of the "iPhi" ratio
(= 1.9130583802711007947403078280203..)
... and notes this squared circles spiral.

Lines t,b,a define this replicating 'T'.
So, what's "to be announced"

Rod ... ...

[Amigoo]

Re: T-Squares design
"Like fathers, like sons"

About the circle-squaring scalene triangle with sides h,t+b,c ...
angle c' = 45 degrees and c = side of circle's inscribed square.

If side h (hypotenuse of an isosceles right triangle) is a side
of the circle's area square, h/(sqrt(Pi)/sqrt(2)) = c.

Say what The simple abc's of quadrature.

Rod

[Amigoo]

Re: T-Squares design
"Like fathers, like sons"

Proof that sqrt(2) is Owner/Manager (OM) of the Pi Corral
as well as overcontrol of this Cartesian neighborhood.

sqrt(Pi/2) = 1.2533141373155002512078826424055..
sqrt(Pi)/(Pi/2) = 1.1283791670955125738961589031215..

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

Rod ... ... (cruisin' with OM on)

[Amigoo]

Re: T-Squares abc design

Re: http://www.telegraph.co.uk/science/2017 ... ths-could/
(T-Squares abc design* is simplification of T-Squares for reference to Plimpton 322's right triangles)
* http://aitnaru.org/images/The_Right_Triangle.pdf

My posted comment:
"Is it possible ... in the beginning was the right triangle and that prehistoric geometric object
begat both circle and square (circle's area square)? Consider a right triangle with sides a,b,c,
where a is the long side, b is the short side, and c is the hypotenuse (also circle's diameter).
This right triangle squares the circle when these line length ratios exist:

a/b = 1.9130583802711007947403078280203..
c/a = 1.1283791670955125738961589031215..
= 2/sqrt(Pi) = sqrt(Pi)/(Pi/2) = 2(sqrt(1/Pi))

See also:
https://en.wikipedia.org/wiki/Plimpton_322
https://en.wikipedia.org/wiki/Edgar_James_Banks

Analysis: This circle-squaring right triangle may not be one of the P322,
but this Babylonian focus on right triangles is very interesting!
And note how the sqrt(2) spiral encloses the "Pi Corral".

Rod

[Amigoo]

Re: "ace ... in the hole" design
( http://aitnaru.org/images/The_Right_Triangle.pdf )

A study of the adjoined right triangles defining a squared circle,
with particular focus on the lines a,c,e: a = e(sqrt(Pi)/sqrt(2))

Who knew . Pi is all about the circle
with diameter and radius all about right triangles.

Rod

[Amigoo]

Re: The Right Ratios design

Apparently, a reminder lesson that squared circle geometry is complex! "Period. End of story."
Lesson learned (again): Search for simplicity and you'll be challenged with complexity.

Regarding this new geometry (further study of T-Squares abc) ...
The largest circle now contains two smaller circles with the largest
(and objects) having dimensions 1.913.. greater than the smallest.

The 1.913..? 1.9130583802711007947403078280203..,
defining ratio of one circle-squaring right triangle.
Of course, 2/sqrt(Pi) is a Cartesian co-star.

Rod

[Amigoo]

Re: The Right Ratios design

squared circle geometry is complex! Period. End of story.

This Right Ratios geometry hints of the years-elusive "geometric lock",
that Cartesian Neighborhood with such balanced geometric objects that ...
The circle must be squared! (despite the transcendental digits of Pi)

Rod ... ...

[Amigoo]

Re: The Right Ratios design
More evidence of the "geometric lock" ...

The defining circle-squaring ratio*, 1.9130583802711007947403078280203..,
also identifies the geometric relationship of the two blue diagonal lines
that each form the circle-squaring scalene triangle in their circle.

The combined length of these lines equals the length of the longer,
parallel diagonal (red and green lines) of the largest circle.

* one of the pair which includes 2(sqrt(1/Pi))

Rod

[Amigoo]

Re: The Right Ratios design
More evidence of the "geometric lock" ...

Extended the arc of the lower circle to emphasize that four similar quadrilaterals,
with their circle-squaring right and scalene triangles, exist in this Neighborhood.

Another circle and its quadrilateral suddenly appeared at the bottom,
just like rabbits . Now 5 similar quadrilaterals in the Neighborthood!
"Right Rabbitoid Replication" comes to mind.

Rod ... ...

[Amigoo]

Re: The Right Ratios design

"The portal is open."
(squared-circle geometry that speaks for itself)

Rod (dusting off the steno pads)

[Amigoo]

Re: The Right Ratios design

"The portal is open."
(squared-circle geometry that speaks for itself)

Curious about the alignment of the portal openings,
I dared to peak inside and discovered more complexity:

Geometers' secret: The large arc to the left defines a circle
whose diameter is an increment of 2(sqrt(1/Pi)) and encloses
íts own circle-squaring scalene triangle (red lines).

Rod

[Amigoo]

Re: The Right Ratios design

"The portal is open."
(squared-circle geometry that speaks for itself)

Geometric magic of The Right Ratios:
(comparison of largest and next largest squared circles;
sqrt(1/Pi) = 0.56418958354775628694807945156077..;
D = Diameter, SoCS = Side of Circle's Square)

D = 4.5135166683820502955846356124862.. 8(sqrt(1/Pi)), SoCS = 4.0
D = 2.0, SoCS = 1.7724538509055160272981674833411.. sqrt(Pi)

Independent geometry confirms the ratios:
1.1283791670955125738961589031215..
1.9130583802711007947403078280203..

Rod

[Amigoo]

Re: The Right Ratios design

"The portal is open."
(squared-circle geometry that speaks for itself)

Geometers' secret: Small right triangle in lower right of design
identifies center of line that represents SoCS* of the largest circle.
Right triangle (hypotenuse not displayed; sides part of longer lines)
contains the two circle-squaring ratios.

* SoCS = Side of Circle's Square

Rod ... ... (cruisin' the portal)

[Amigoo]

Re: The Right Ratios design

"The portal is open."
(squared-circle geometry that speaks for itself)

Such mysterious geometry, this contrast of largest and next largest circles,
both squared! Apparently, the inner complexity is the required starting point
because the outer simplicity, as starting point, would not lead a geometer
to the inner complexity (geometric concentricity would be assumed
but would not be found ... if goal is circle (diameter) and square (SoCS)
having the same dimension (or relative increment).

"Say what?"
You can't get here from there, but you can get there from here!
... suggesting that the long journey is one-way only.

Rod