Paradise Trinity Day

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Re: Paradise Trinity Day

Post by Amigoo » Tue Jul 04, 2017 7:17 pm

:flower: 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 ... :bike: ...

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Re: Paradise Trinity Day

Post by Amigoo » Thu Jul 06, 2017 2:22 pm

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

Squared circle geometry 101
(objects of galactic force) ;)

Rod :D

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Re: Paradise Trinity Day

Post by Amigoo » Fri Jul 07, 2017 2:22 pm

:flower: 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 8)
:hithere (Cartesian landscape familiar to advanced geometers)

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

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Re: Paradise Trinity Day

Post by Amigoo » Sat Jul 08, 2017 12:34 am

:flower: Re: Counterpoint-3 Post Op design (C-3PO)
alludes to the familar a^2 + b^2 = c^2
:geek: 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 :D

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Re: Paradise Trinity Day

Post by Amigoo » Mon Jul 10, 2017 2:48 pm

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

:geek: 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))

:scratch: Portal to where? "Galactic" comes to mind
... and "Reel 2, Dialog 2". ;)

Rod :stars:

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Re: Paradise Trinity Day

Post by Amigoo » Tue Jul 11, 2017 7:27 am

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

Completed at warp speed. :roll:

Rod :stars:

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Re: Paradise Trinity Day

Post by Amigoo » Fri Jul 14, 2017 5:55 pm

:flower: 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.

:geek: 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 ... :bike: ...

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Re: Paradise Trinity Day

Post by Amigoo » Sat Jul 15, 2017 1:57 pm

:flower: Re: Squared Circles 101 design

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

Rod :D

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Re: Paradise Trinity Day

Post by Amigoo » Sat Jul 15, 2017 11:44 pm

:flower: Re: Squared Circles (pinpoint) Precision design

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

(the three "pins" in the two geometrically-concentric
circle-squaring scalene triangles have equal length) 8)

Rod :D

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Re: Paradise Trinity Day

Post by Amigoo » Sun Jul 16, 2017 2:57 pm

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

L a T a S, o, P ! ;) ;)

Rod ... :bike: ...

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Re: Paradise Trinity Day

Post by Amigoo » Mon Jul 17, 2017 3:11 am

:flower: 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." [ :idea: sqrt(2), sqrt(Pi), 2(sqrt(1/Pi)) ]

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

Rod :D

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Re: Paradise Trinity Day

Post by Amigoo » Tue Jul 18, 2017 9:33 pm

:flower: 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"). :roll:

:geek: In this layered geometry, 2(sqrt(1/Pi)), is the defining constant
that deternines the dimensions of the next right triangle. 8)

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

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Re: Paradise Trinity Day

Post by Amigoo » Wed Jul 19, 2017 3:29 am

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

:geek: 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. :shock:

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

Rod :rambo:

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Re: Paradise Trinity Day

Post by Amigoo » Thu Jul 20, 2017 12:22 am

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

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

Rod :D

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Re: Paradise Trinity Day

Post by Amigoo » Sat Jul 22, 2017 7:07 am

:flower: Re: Syzygial Scalenity design

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

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

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Re: Paradise Trinity Day

Post by Amigoo » Sat Jul 22, 2017 10:44 pm

:flower: Re: Syzygial Scalenity design

:idea: 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". :roll

Rod :D

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Re: Paradise Trinity Day

Post by Amigoo » Sun Jul 23, 2017 11:44 pm

:flower: Re: Syzygial Scalenity design
Cartesian action of squared circles is "outside the box"
:geek: About the golden '8' (and the reference to 'box') ...
The circle's square rests upon 8 points of the circle ... only. 8)

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

Rod ... :bike: ...

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Re: Paradise Trinity Day

Post by Amigoo » Tue Jul 25, 2017 12:34 pm

:flower: 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; :roll:
the overlapping scalenes can speak for themselves.

Rod :D

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Re: Paradise Trinity Day

Post by Amigoo » Tue Jul 25, 2017 11:22 pm

:flower: Re: Syzygial Scalenity design'

:roll 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 ... :bike: ...

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Re: Paradise Trinity Day

Post by Amigoo » Thu Jul 27, 2017 2:22 pm

:flower: 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." :roll:

Note the 45 degree angles of lines representing the short side
of the four integrated, circle-squaring right triangles. 8)

Rod :D

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Re: Paradise Trinity Day

Post by Amigoo » Wed Aug 02, 2017 3:47 am

:flower: Re: “WTP?” RT design

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

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

"Mark of the beast" slayer? :roll:

Rod ... :bike: ...

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Re: Paradise Trinity Day

Post by Amigoo » Wed Aug 02, 2017 3:33 pm

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."

:shock: 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). 8)

This Geometry 101 design is now in development. :finger:

Rod :D

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Re: Paradise Trinity Day

Post by Amigoo » Wed Aug 02, 2017 4:44 pm

:flower: Re: Golden rPi design
Move over Phi, there's another Golden Rectangle
A quick sketch of this new Cartesian neighborhood. :roll:
Updated: http://aitnaru.org/images/Squarely_Entwined.pdf

Rod :stars:

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Re: Paradise Trinity Day

Post by Amigoo » Wed Aug 02, 2017 6:57 pm

:flower: 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. :roll:

Rod :?

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Re: Paradise Trinity Day

Post by Amigoo » Wed Aug 02, 2017 10:10 pm

:flower: Re: Golden rPi design (updated)

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

Rod :D

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