Paradise Trinity Day

[Amigoo]

Re: Los Dos design

Geometry so esoteric that I can't decide which way is up!
So, I compromised and selected the obvious (but abstract)
"Smile of Pythagoras" for Cartesian positioning.

Rod ... ...

[Amigoo]

Re: Counterpoint-3 Post Op design (C-3PO)

Squared circle geometry 101
(objects of galactic force)

Rod

[Amigoo]

Re: Counterpoint-3 Post Op design (C-3PO)

Who knew (more post op contemplation)
"Counterpoint-3" alludes to the familar a^2 + b^2 = c^2
(Cartesian landscape familiar to advanced geometers)

Rod ... ...
Cruisin' in squared circles is challenging enough ...
but the Einstein-Rosen Bridge is "out of this world".

[Amigoo]

Re: Counterpoint-3 Post Op design (C-3PO)

alludes to the familar a^2 + b^2 = c^2

The subtlety of a^2 + b^2 = c^2 analysis of the three integrated squared circles
is that this applies to the sides of a right triangle as diameters of the three circles
as well as to sides of a similar right triangle as sides of the squares of the circles.

A Cartesian neighborhood extraordinaire - even "impossible"!
... and quite worthy of a "Smile of Pythagoras".

Rod

[Amigoo]

Re: Concentrizity Portal design (in development;
SoCS = Side of Circle's Square)

Two sets of 4 "concentric" circles all squared:

D = Pi/4, sqrt(Pi)/2, 1.0, 2(sqrt(1/Pi))
D = Pi/2, sqrt(Pi), 2.0, 4(sqrt(1/Pi))

... inscribed in a larger squared circle, where:
D = 2(sqrt(2)), SoCS = 2(sqrt(Pi)/sqrt(2))

Portal to where? "Galactic" comes to mind
... and "Reel 2, Dialog 2".

Rod

[Amigoo]

Re: Concentrizity Portal design
"The Force, long-awakening"

Completed at warp speed.

Rod

[Amigoo]

Re: Squared Circles 101 design

Several years ago I was convinced that a squared circle would need to display
the geometric integration of sqrt(2) and sqrt(Pi). Recently, I've learned that
2(sqrt(1/Pi)), nicknamed "rPi", is the transcendental matchmaker.

Line length ratios in Squared Circles 101:
(Diameter = 2(sqrt(1/Pi)), SoCS = 2.0)

red lines: 2(sqrt(1/Pi))
yellow lines: 2(sqrt(1/Pi))
green lines: sqrt(Pi)

Notes: SoCS = Side of Circle's Square,
Length of longer yellow line = 2(sqrt(1/Pi)) ^2

Rod ... ...

[Amigoo]

Re: Squared Circles 101 design

Added the complementary scalene triangle that is also circle-squaring.
These right and scalene triangles have intriguing similarity of angles.

Rod

[Amigoo]

Re: Squared Circles (pinpoint) Precision design

“w/ defining replication Pertubation integration (rPi)”
“Answers the question: 'What's the point?' ”

(the three "pins" in the two geometrically-concentric
circle-squaring scalene triangles have equal length)

Rod

[Amigoo]

Re: LaTaSoP design
(pronounced "lotta S O P",
a lot of Standard Operating Procedure)

L a T a S, o, P !

Rod ... ...

[Amigoo]

Re: LaTaSOP design

"That's the ticket!" (geometric proof of squared squared circles can be derived from pertubation)

Re: https://en.wikipedia.org/wiki/Perturbation_theory

"This general procedure is a widely used mathematical tool in advanced sciences and engineering:
start with a simplified problem and gradually add corrections that make the formula that the corrected problem
becomes a closer and closer match to the original formula. It is the natural extension to mathematical functions
of the "guess, check, and fix" method first used by older civilisations to compute certain numbers,
such as square roots." [ sqrt(2), sqrt(Pi), 2(sqrt(1/Pi)) ]

... assuming that a portion of the geometry of a squared circle can be the "simplified problem".

Rod

[Amigoo]

Re: Pythagorean Triples design
D = Pi/2, sqrt(Pi), 2.0, 4(sqrt(1/Pi))

Technically, Squared Circles Essence of Pythagorean Triples
(SCEPT, pronounced as second syllable of "except" ...
since squared circles are "impossible").

In this layered geometry, 2(sqrt(1/Pi)), is the defining constant
that deternines the dimensions of the next right triangle.

Rod ... ... (cruisin' in squared circles, scept ...)

[Amigoo]

Re: Pythagorean Triples design
D = Pi/2, sqrt(Pi), 2.0, 4(sqrt(1/Pi))

Given the Pythagorean Theorem: a^2 + b^2 = c^2

When Diameter (hypotenuse, c^2) = 2.0, the squares of the other two sides
must sum to the integer 4.0. Therefore, the increment of Pi (the long side)
is effectively "corraled" (is no longer transcendental) by the short side.

Or maybe the short side is also transcendental ...
and two transcendentals cancel by multiplication.

Rod

[Amigoo]

Re: Pythagorean Triples design
"w/ D = 2.0, verdant ratios: sqrt(Pi), 2(sqrt(1/Pi))"

All diameters: Pi/2, sqrt(Pi), 2.0, 4(sqrt(1/Pi))
Green 'X' reveals integration of the "verdant ratios".

Rod

[Amigoo]

Re: Syzygial Scalenity design

Cartesian illumination for the U.S. solar eclipse (August 21 ;-)

Rod ... ... "Been there! Done that!"
(been cruisin' on many paths, briefly darkened)

[Amigoo]

Re: Syzygial Scalenity design

Speaking of "syzygy" and other alignments (re: August 21 ;-) ...
design was enhanced to herald that the exciting Cartesian action
of squared circles is obviously "outside the box".

Rod

[Amigoo]

Re: Syzygial Scalenity design

Cartesian action of squared circles is "outside the box"

About the golden '8' (and the reference to 'box') ...
The circle's square rests upon 8 points of the circle ... only.

... or maybe the '8' alludes to the month of syzygy
and other alignments (re: August 21)

Rod ... ...

[Amigoo]

Re: Syzygial Scalenity design

"Circle-squaring scalene and right triangles
w/ supporting line length ratio 2(sqrt(1/Pi))
= sqrt(Pi)/(Pi/2) = 1.1283791670955125.."

The other ratio (same value) wanted equal exposure;
the overlapping scalenes can speak for themselves.

Rod

[Amigoo]

Re: Syzygial Scalenity design'

New geometers' clue:

The small golden circle identifies two similar, circle-squaring right triangles
having line length differences of sqrt(2), causing astute geometers to wonder
"Whence transcendental Pi in the geometric association of these triangles?"
(especially since the long side represents an increment of Pi)

Pi remains transcendental after multiplication (or division) by sqrt(2)

Rod ... ...

[Amigoo]

Re: Sqrt(2) OIC design (pronounced "Oh, I see!")

Sqrt(2) has such command over this pattern of squared circles
that Sqrt(2) is Obviously In Control. "Period. End of story."

Note the 45 degree angles of lines representing the short side
of the four integrated, circle-squaring right triangles.

Rod

[Amigoo]

Re: “WTP?” RT design

“ 'WTP?' sqrt(Pi)/(Pi/2) = 2(sqrt(1/Pi))
= 1.1283791670955125738961589..”

So esoteric! And what's with the 7s
(777 for the 3 circle-squaring right triangles)

"Mark of the beast" slayer?

Rod ... ...

[Amigoo]

Re: The Golden Ratio by Mario Livio, 2002, p. 85

"The Golden Rectangle is the only rectangle with the property
that cutting a square from it produces a similar rectangle."

Move over Phi, there's another Golden Rectangle with the same property!
The foundation of this rectangle is the ratio 2(sqrt(1/Pi)) = sqrt(Pi)/(Pi/2).

This Geometry 101 design is now in development.

Rod

[Amigoo]

Re: Golden rPi design

Move over Phi, there's another Golden Rectangle

A quick sketch of this new Cartesian neighborhood.
Updated: http://aitnaru.org/images/Squarely_Entwined.pdf

Rod

[Amigoo]

Re: Golden rPi design

Move over Phi, there's another Golden Rectangle

Not necessarily THE Golden Rectangle nor its related Golden Ratio.
Just lines of interest ... until proven uninteresting.

Rod

[Amigoo]

Re: Golden rPi design (updated)

Oops! Forgot the abc's: c / a = ( a + b ) / c

Rod