Paradise Trinity Day

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Re: Paradise Trinity Day

Post by Amigoo » Mon Sep 04, 2017 2:22 am

:flower: Re: E = mc^2 design
"Evidence = morbus cyclometricus ... squared" :roll:

:geek: Constructed upon the Portal geometry, the center's isosceles trapezoid
includes the two ratios with the ratio of the lengths of bottom and top lines
equal to 1.9130583802711007947403078280203.., a trapezoid composed of
two isosceles right triangles and 4 similar right triangles (each containing
both circle-squaring ratios:) 8)

1.1283791670955125738961589031215.. hypotenuse-to-long-side
1.9130583802711007947403078280203.. long-side-to-short-side
For D = 2.0, length of bottom line of trapezoid equals sqrt(Pi)

Rod :D

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Re: Paradise Trinity Day

Post by Amigoo » Mon Sep 04, 2017 8:11 am

:flower: Re: Symmetry of mc^2 design
(added to file: http://aitnaru.org/images/The_Right_Triangle.pdf )

:roll: Just another late night doodle ...
with focus on the isosceles trapezoid :roll
(magenta lines have equal length).

Rod ... :bike: ...

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Re: Paradise Trinity Day

Post by Amigoo » Tue Sep 05, 2017 2:37 pm

:flower: Re: Symmetry of mc^2 design
“Evidence of sanitas cyclometricus”

Impressive symmetry, this isosceles trapezoid :!:

:geek: Diameter of largest circle = 2.0;
length of sides of trapezoid = sqrt(2);
perpendicular magenta lines have equal length:
sqrt(Pi) + sqrt(Pi)/1.9130583802711007947403078280203..

Rod :D

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Re: Paradise Trinity Day

Post by Amigoo » Wed Sep 06, 2017 6:46 am

:flower: Re: Caliperfection design

Exploration of the geometry of Symmetry of mc^2. 8)
(aka, simple symmetrical salience) :roll

Rod :D

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Re: Paradise Trinity Day

Post by Amigoo » Wed Sep 06, 2017 3:14 pm

:flower: Re: Caliperfection design
Who knew :?: :!: Trapezoids Rule :!:

Caliperfection Quietus

When D = 2, what else is new
with "impossible" quadrature
and trapezoids askew, part
square of 1.0 amidst D 2?


Rod ... :bike: ...

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Re: Paradise Trinity Day

Post by Amigoo » Wed Sep 06, 2017 5:22 pm

:flower: Re: Caliperfection design
Who knew :?: :!: Trapezoids Rule :!:

Caliperfection Quietus (first draft)

:?: When D = 2, what else is new
with "impossible" quadrature
and trapezoids askew, part
square of 1.0 amidst D 2?

:!: When D = 2, what else is new
with "impossible" quadrature
+ points of 8 - no more nor less -
upon the plate, circular too!

:shock: When D = 2, what else is new
with "impossible" quadrature
o'er triangle rights, adjoined
once/twice, API Waterloo?

API = A Plane "Impossible" :roll:

Rod ... :bike: ...

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Re: Paradise Trinity Day

Post by Amigoo » Fri Sep 08, 2017 11:23 am

:flower: Re: Locksss Pi design
(aka "Geometric Locksss Pi")

Conjecturing that a "geometric lock" of squared circles
would be constructed upon two overlapping circles
whose inscribed squares share one side, Locksss Pi
quickly evolved ... with challenging complexity :!:

:roll: "Replication Integration Perturbation" indeed!

Note: "sss" represents the three inner circles,
all squared, overlapping, and integrated.

Rod ... :bike: ... (off to buy a Locksss Pi Pan) :roll

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Re: Paradise Trinity Day

Post by Amigoo » Sat Sep 09, 2017 12:22 pm

:flower: Re: Locksss Pi design
(aka "Geometric Locksss Pi")

Recolored to highlight isosceles right triangle that controls
locking mechanism of the circle squaring right triangles
in two circles, one having a diameter larger by sqrt(2);
similar mechanism exists for adjacent set of circles. 8)

"Say what :?: :!: " Lotsa Locksss :lol:

Rod ... :bike: ...

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Re: Paradise Trinity Day

Post by Amigoo » Sat Sep 09, 2017 9:11 pm

:flower: Re: Locksss Pi design
(aka "Geometry Locksss Pi", "Pi Corral") ;)

:geek: Long story short: This geometry proves ...

each of the sides of the two similar circle-squaring right triangles
(one set green, one magenta) has sqrt(2) line length relationship*
to the similar side of the adjacent** c-s right triangle :!:

* the two diameters also have sqrt(2) line length relationship.
** adjacent via the small adjoining isosceles right triangle,
consistent indicator of sqrt(2) geometric influence.

:duh :duh ... Repeating mantra:
Either sqrt(2) is transcendental or Pi is not. :shock:

Rod :rambo:

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Re: Paradise Trinity Day

Post by Amigoo » Sun Sep 10, 2017 2:57 pm

:flower: Re: Cross of Gaia design

Long story short ...

:farao: Exploration of the 2(sqrt(1/Pi)) geometry reveals a two-bar cross with a shorter lower bar. Historically, two-bar crosses have a shorter upper bar, so what would this new cross represent? A Matriarchal Cross (to complement the two-bar Patriarchal Cross)? ... Or even better, a cross representing the "mother" of squared circle geometry! Since Gaia has no cross (apparently), this geometry proffers a first offering. 8)

Rod :D

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Re: Paradise Trinity Day

Post by Amigoo » Sun Sep 10, 2017 9:07 pm

:flower: Re: Cross of Gaia design (a new "World of the Cross")
"2(sqrt(1/Pi)) = sqrt(Pi)/(Pi/2) = 1.12837916709551257.."

2(sqrt(1/Pi)) is easier to see in this current version ...
From top to bottom of the two sets of right triangles,
line lengths* decrease by 2(sqrt(1/Pi)).

* hypotenuse to long side, then repeating; 8)
the 14 right triangles contain this ratio!

Rod ... :bike: ...

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Re: Paradise Trinity Day

Post by Amigoo » Mon Sep 11, 2017 5:33 am

:flower: Re: Locksss Pi222 design

Who knew :?: :!:
Pi is evenly divisible by sqrt(2) :roll:

3.1415926535897932384626433832795.. Pi
/ 1.4142135623730950488016887242097.. sqrt(2)
= 2.2214414690791831235079404950303.. a diameter
/ 1.4142135623730950488016887242097.. sqrt(2)
= 1.5707963267948966192313216916398.. Pi/2, SoIS

If D = 1.7724538509055160272981674833411.. sqrt(Pi)
SoCS = 1.5707963267948966192313216916398.. Pi/2

:geek: SoIS = Side of Inscribed Square,
SoCS = Side of Circle's Square

Rod :stars:

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Re: Paradise Trinity Day

Post by Amigoo » Tue Sep 12, 2017 6:17 am

:flower: Re: P.S. design
(conservative elaboration of Locksss Pi222) :roll:

This puissant symmetry is a good example of
"squared circle geometry that speaks for itself",
highlighting the two ratios* and sqrt(2). 8)

* 1.1283791670955125738961589031215.. hypotenuse-to-long-side
1.9130583802711007947403078280203.. long-side-to-short-side

Rod :D

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Re: Paradise Trinity Day

Post by Amigoo » Tue Sep 12, 2017 6:33 pm

:flower: Re: P.S. design
(conservative elaboration of Locksss Pi222) :roll:

Note: Dark blue coloring identifies gaps in circle-squaring concentricity
when sqrt(2) and 2(sqrt(1/Pi)) are increments/decrements of D, SoCS.

:scratch: "Say what?" When going from here to there
in the local universe, expect to cross bridges. ;)

Rod ... :bike: ... (using prepaid toll bridges)

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Re: Paradise Trinity Day

Post by Amigoo » Fri Sep 15, 2017 8:57 am

:flower: Re: P.S. design (more artistic)

"When squared circle geometry speaks for itself
with puissant symmetry, pattern, and sqrt(2);
all finessed by 2.0, sqrt(Pi), and Pi/2." 8)

Remember the first color test patterns for television?
This might be a test pattern for a new universe circuit :!:
"Can you hear me now?" :roll:

Rod :D

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Re: Paradise Trinity Day

Post by Amigoo » Fri Sep 15, 2017 4:47 pm

:flower: Re: P.S. design ("Final, final")
"When squared circle geometry speaks for itself"

To get your mathematical mind around this natural* geometric magnificence,
note that the diameters of the dark blue circles = 2.0 and sqrt(2). 8)

* :geek: the geometry has always existed, being recently discovered
after many thousands of hours of CAD exploration (literally!),
aka "my insatiable busyness for the first 8 years of retirement" :roll:

The dark blue infinity loop? Artistic embellishment. :roll

Rod ... :bike: ...

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Re: Paradise Trinity Day

Post by Amigoo » Sat Sep 16, 2017 2:22 pm

:flower: Re: Tri Phi of Pi design
"Geometric integration of iPhi in 3 circles squared."

:geek: D = 4(sqrt(1/Pi)), 2.0, sqrt(Pi)

"iPhi" alludes to the circle-squaring right triangle where ratio
of lengths of long side to short side equals the squared-circle
defining ratio of 1.9130583802711007947403078280203.. 8)

:roll: 'i' of iPhi is whimsical allusion to "impossible" squared circles.

The abstract Pi lines (magenta), portions of two squares,
highlight this convincing Cartesian presentation! :roll

Rod :D

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Re: Paradise Trinity Day

Post by Amigoo » Sun Sep 17, 2017 1:57 am

:flower: Re: Tri Phi of Pi design (shortened to "Tri Phi Pi")
"Geometric integration of iPhi in 3 circles squared."
squared-circle defining ratio of 1.9130583802711007947403078280203..
"Say what?! Whence this constant?" :scratch:

:geek: Given: Diameter = 2.0 (side c; hypotenuse of right triangle),
SoCS = 1.7724538509055160272981674833411.. (side b; long side)
= Side of Circle's Square = sqrt(Pi) ;)

:idea: Then, from the Pythagorean Theorem (a^2 + b^2 = c^2),
side a (short side) = 0.92650275035220848584275966758921..
and iPhi = b/a = 1.9130583802711007947403078280203..

Rod ... :bike: ... (calculating on the short side) :roll:

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Re: Paradise Trinity Day

Post by Amigoo » Sun Sep 17, 2017 9:22 am

:flower: Re: Tri Phi Pi design
"Put a fork in it." (it's done!)

:geek: Lines of the red "fork" have unique relationship.
Savor the aroma of Baked Pi, ah la mode. :roll:

Rod :D

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Re: Paradise Trinity Day

Post by Amigoo » Sun Sep 17, 2017 4:11 pm

:flower: Re: Tri Phi Pi design
"Put a fork in it." (it's done!)

:scratch: Who knew :?: :!:
The portal is open! The circuit is live! :cheers:
"Can you hear me now?"

Rod ... :bike: ...

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Re: Paradise Trinity Day

Post by Amigoo » Mon Sep 18, 2017 12:34 pm

:flower: Re: Tri Phi Pi design

Who knew :?: :!:
When "squared-circle geometry that speaks for itself" waxes loquacious.

;) Geometers' secret: Diameters of largest and smallest circles (largest displayed as an arc)
that form a symbolic hand scythe have an "impossible" squared circle ratio: Pi (3.14159..) 8)

Get one today for a geometrically ethereal slice of Puissant Pi !
(sold only in Cartesian neighborhoods of iQuadrature status) :roll

Rod :D

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Re: Paradise Trinity Day

Post by Amigoo » Mon Sep 18, 2017 1:47 pm

:flower: Re: Tri Phi Pi design

Finally! (as occurs mysteriously :o ), the completed geometry
reflects the design name chosen weeks earlier. "Tri Phi" refers to
the iPhi ratios of the three integrated right triangles (red lines).

iPhi = ratio of lengths of long line to short line
= 1.9130583802711007947403078280203..

:roll: 'iPhi' is whimsical allusion to "impossible" squared circles
with 'Phi' allusion to the related (IMO) Golden Ratio. 8)

Rod ... :bike: ...

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Re: Paradise Trinity Day

Post by Amigoo » Tue Sep 19, 2017 4:44 pm

:flower: Re: Tri Phi Pi design

Despite the iPhi geometric presence (and name),
2(sqrt(1/Pi)) is the real foundation of this composition :!:

Given that the 3-pointed star (within large right triangle;
red lines) contains identical angles (at the points) and that
the hypotenuse and long side (both diameters of circles)
have 2(sqrt(1/Pi)) relationship, the entire composition
can be constructed (try long side with D = 2.0). 8)

:geek: When drawing diagonal lines within the star,
note presence of sqrt(Pi) as line length ratios.

Rod :study: ( reviewing Pythagorean myth :roll:
"My Pi! My Pi! My triangular Pi!", on page preceding
"Lines and triangles and squares! Oh, Pi ! ) :D

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Re: Paradise Trinity Day

Post by Amigoo » Tue Sep 19, 2017 10:56 pm

:flower: Re: Tri Phi Pi design

Who knew :?: :!: Twin portals! 8)

Rod ... :bike: ...

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Re: Paradise Trinity Day

Post by Amigoo » Wed Sep 20, 2017 1:07 pm

:flower: Re: Tri Phi Pi design
(updated in: http://aitnaru.org/images/The_Right_Triangle.pdf )

Behold the geometric lock :?: :scratch:

The two lines appearing as the top and bottom of the large red 'X'
have sqrt(Pi) length relationship only when the circle is squared, :roll
implying that the two "portals" have complementary resonance.

Rod (done pickith the lock ... or not) :roll:

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