Paradise Trinity Day

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Re: Paradise Trinity Day
Re: E = mc^2 design
"Evidence = morbus cyclometricus ... squared"
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 circlesquaring ratios:)
1.1283791670955125738961589031215.. hypotenusetolongside
1.9130583802711007947403078280203.. longsidetoshortside
For D = 2.0, length of bottom line of trapezoid equals sqrt(Pi)
Rod
"Evidence = morbus cyclometricus ... squared"
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 circlesquaring ratios:)
1.1283791670955125738961589031215.. hypotenusetolongside
1.9130583802711007947403078280203.. longsidetoshortside
For D = 2.0, length of bottom line of trapezoid equals sqrt(Pi)
Rod

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Re: Paradise Trinity Day
Re: Symmetry of mc^2 design
(added to file: http://aitnaru.org/images/The_Right_Triangle.pdf )
Just another late night doodle ...
with focus on the isosceles trapezoid
(magenta lines have equal length).
Rod ... ...
(added to file: http://aitnaru.org/images/The_Right_Triangle.pdf )
Just another late night doodle ...
with focus on the isosceles trapezoid
(magenta lines have equal length).
Rod ... ...

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Re: Paradise Trinity Day
Re: Symmetry of mc^2 design
“Evidence of sanitas cyclometricus”
Impressive symmetry, this isosceles trapezoid
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
“Evidence of sanitas cyclometricus”
Impressive symmetry, this isosceles trapezoid
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

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Re: Paradise Trinity Day
Re: Caliperfection design
Exploration of the geometry of Symmetry of mc^2.
(aka, simple symmetrical salience)
Rod
Exploration of the geometry of Symmetry of mc^2.
(aka, simple symmetrical salience)
Rod

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Re: Paradise Trinity Day
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 ... ...
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 ... ...

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Re: Paradise Trinity Day
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!
When D = 2, what else is new
with "impossible" quadrature
o'er triangle rights, adjoined
once/twice, API Waterloo?
API = A Plane "Impossible"
Rod ... ...
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!
When D = 2, what else is new
with "impossible" quadrature
o'er triangle rights, adjoined
once/twice, API Waterloo?
API = A Plane "Impossible"
Rod ... ...

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Re: Paradise Trinity Day
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
"Replication Integration Perturbation" indeed!
Note: "sss" represents the three inner circles,
all squared, overlapping, and integrated.
Rod ... ... (off to buy a Locksss Pi Pan)
(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
"Replication Integration Perturbation" indeed!
Note: "sss" represents the three inner circles,
all squared, overlapping, and integrated.
Rod ... ... (off to buy a Locksss Pi Pan)

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Re: Paradise Trinity Day
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.
"Say what " Lotsa Locksss
Rod ... ...
(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.
"Say what " Lotsa Locksss
Rod ... ...

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Re: Paradise Trinity Day
Re: Locksss Pi design
(aka "Geometry Locksss Pi", "Pi Corral")
Long story short: This geometry proves ...
each of the sides of the two similar circlesquaring right triangles
(one set green, one magenta) has sqrt(2) line length relationship*
to the similar side of the adjacent** cs 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.
... Repeating mantra:
Either sqrt(2) is transcendental or Pi is not.
Rod
(aka "Geometry Locksss Pi", "Pi Corral")
Long story short: This geometry proves ...
each of the sides of the two similar circlesquaring right triangles
(one set green, one magenta) has sqrt(2) line length relationship*
to the similar side of the adjacent** cs 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.
... Repeating mantra:
Either sqrt(2) is transcendental or Pi is not.
Rod

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Re: Paradise Trinity Day
Re: Cross of Gaia design
Long story short ...
Exploration of the 2(sqrt(1/Pi)) geometry reveals a twobar cross with a shorter lower bar. Historically, twobar crosses have a shorter upper bar, so what would this new cross represent? A Matriarchal Cross (to complement the twobar 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.
Rod
Long story short ...
Exploration of the 2(sqrt(1/Pi)) geometry reveals a twobar cross with a shorter lower bar. Historically, twobar crosses have a shorter upper bar, so what would this new cross represent? A Matriarchal Cross (to complement the twobar 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.
Rod

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Re: Paradise Trinity Day
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;
the 14 right triangles contain this ratio!
Rod ... ...
"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;
the 14 right triangles contain this ratio!
Rod ... ...

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Re: Paradise Trinity Day
Re: Locksss Pi222 design
Who knew
Pi is evenly divisible by sqrt(2)
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
SoIS = Side of Inscribed Square,
SoCS = Side of Circle's Square
Rod
Who knew
Pi is evenly divisible by sqrt(2)
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
SoIS = Side of Inscribed Square,
SoCS = Side of Circle's Square
Rod

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Re: Paradise Trinity Day
Re: P.S. design
(conservative elaboration of Locksss Pi222)
This puissant symmetry is a good example of
"squared circle geometry that speaks for itself",
highlighting the two ratios* and sqrt(2).
* 1.1283791670955125738961589031215.. hypotenusetolongside
1.9130583802711007947403078280203.. longsidetoshortside
Rod
(conservative elaboration of Locksss Pi222)
This puissant symmetry is a good example of
"squared circle geometry that speaks for itself",
highlighting the two ratios* and sqrt(2).
* 1.1283791670955125738961589031215.. hypotenusetolongside
1.9130583802711007947403078280203.. longsidetoshortside
Rod

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Re: Paradise Trinity Day
Re: P.S. design
(conservative elaboration of Locksss Pi222)
Note: Dark blue coloring identifies gaps in circlesquaring concentricity
when sqrt(2) and 2(sqrt(1/Pi)) are increments/decrements of D, SoCS.
"Say what?" When going from here to there
in the local universe, expect to cross bridges.
Rod ... ... (using prepaid toll bridges)
(conservative elaboration of Locksss Pi222)
Note: Dark blue coloring identifies gaps in circlesquaring concentricity
when sqrt(2) and 2(sqrt(1/Pi)) are increments/decrements of D, SoCS.
"Say what?" When going from here to there
in the local universe, expect to cross bridges.
Rod ... ... (using prepaid toll bridges)

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Re: Paradise Trinity Day
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."
Remember the first color test patterns for television?
This might be a test pattern for a new universe circuit
"Can you hear me now?"
Rod
"When squared circle geometry speaks for itself
with puissant symmetry, pattern, and sqrt(2);
all finessed by 2.0, sqrt(Pi), and Pi/2."
Remember the first color test patterns for television?
This might be a test pattern for a new universe circuit
"Can you hear me now?"
Rod

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Re: Paradise Trinity Day
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).
* 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"
The dark blue infinity loop? Artistic embellishment.
Rod ... ...
"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).
* 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"
The dark blue infinity loop? Artistic embellishment.
Rod ... ...

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Re: Paradise Trinity Day
Re: Tri Phi of Pi design
"Geometric integration of iPhi in 3 circles squared."
D = 4(sqrt(1/Pi)), 2.0, sqrt(Pi)
"iPhi" alludes to the circlesquaring right triangle where ratio
of lengths of long side to short side equals the squaredcircle
defining ratio of 1.9130583802711007947403078280203..
'i' of iPhi is whimsical allusion to "impossible" squared circles.
The abstract Pi lines (magenta), portions of two squares,
highlight this convincing Cartesian presentation!
Rod
"Geometric integration of iPhi in 3 circles squared."
D = 4(sqrt(1/Pi)), 2.0, sqrt(Pi)
"iPhi" alludes to the circlesquaring right triangle where ratio
of lengths of long side to short side equals the squaredcircle
defining ratio of 1.9130583802711007947403078280203..
'i' of iPhi is whimsical allusion to "impossible" squared circles.
The abstract Pi lines (magenta), portions of two squares,
highlight this convincing Cartesian presentation!
Rod

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Re: Paradise Trinity Day
Re: Tri Phi of Pi design (shortened to "Tri Phi Pi")
"Geometric integration of iPhi in 3 circles squared."
Given: Diameter = 2.0 (side c; hypotenuse of right triangle),
SoCS = 1.7724538509055160272981674833411.. (side b; long side)
= Side of Circle's Square = sqrt(Pi)
Then, from the Pythagorean Theorem (a^2 + b^2 = c^2),
side a (short side) = 0.92650275035220848584275966758921..
and iPhi = b/a = 1.9130583802711007947403078280203..
Rod ... ... (calculating on the short side)
"Geometric integration of iPhi in 3 circles squared."
"Say what?! Whence this constant?"squaredcircle defining ratio of 1.9130583802711007947403078280203..
Given: Diameter = 2.0 (side c; hypotenuse of right triangle),
SoCS = 1.7724538509055160272981674833411.. (side b; long side)
= Side of Circle's Square = sqrt(Pi)
Then, from the Pythagorean Theorem (a^2 + b^2 = c^2),
side a (short side) = 0.92650275035220848584275966758921..
and iPhi = b/a = 1.9130583802711007947403078280203..
Rod ... ... (calculating on the short side)

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Re: Paradise Trinity Day
Re: Tri Phi Pi design
"Put a fork in it." (it's done!)
Lines of the red "fork" have unique relationship.
Savor the aroma of Baked Pi, ah la mode.
Rod
"Put a fork in it." (it's done!)
Lines of the red "fork" have unique relationship.
Savor the aroma of Baked Pi, ah la mode.
Rod

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Re: Paradise Trinity Day
Re: Tri Phi Pi design
"Put a fork in it." (it's done!)
Who knew
The portal is open! The circuit is live!
"Can you hear me now?"
Rod ... ...
"Put a fork in it." (it's done!)
Who knew
The portal is open! The circuit is live!
"Can you hear me now?"
Rod ... ...

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Re: Paradise Trinity Day
Re: Tri Phi Pi design
Who knew
When "squaredcircle 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..)
Get one today for a geometrically ethereal slice of Puissant Pi !
(sold only in Cartesian neighborhoods of iQuadrature status)
Rod
Who knew
When "squaredcircle 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..)
Get one today for a geometrically ethereal slice of Puissant Pi !
(sold only in Cartesian neighborhoods of iQuadrature status)
Rod

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Re: Paradise Trinity Day
Re: Tri Phi Pi design
Finally! (as occurs mysteriously ), 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..
'iPhi' is whimsical allusion to "impossible" squared circles
with 'Phi' allusion to the related (IMO) Golden Ratio.
Rod ... ...
Finally! (as occurs mysteriously ), 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..
'iPhi' is whimsical allusion to "impossible" squared circles
with 'Phi' allusion to the related (IMO) Golden Ratio.
Rod ... ...

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Re: Paradise Trinity Day
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 3pointed 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).
When drawing diagonal lines within the star,
note presence of sqrt(Pi) as line length ratios.
Rod ( reviewing Pythagorean myth
"My Pi! My Pi! My triangular Pi!", on page preceding
"Lines and triangles and squares! Oh, Pi ! )
Despite the iPhi geometric presence (and name),
2(sqrt(1/Pi)) is the real foundation of this composition
Given that the 3pointed 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).
When drawing diagonal lines within the star,
note presence of sqrt(Pi) as line length ratios.
Rod ( reviewing Pythagorean myth
"My Pi! My Pi! My triangular Pi!", on page preceding
"Lines and triangles and squares! Oh, Pi ! )

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Re: Paradise Trinity Day
Re: Tri Phi Pi design
Who knew Twin portals!
Rod ... ...
Who knew Twin portals!
Rod ... ...

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Re: Paradise Trinity Day
Re: Tri Phi Pi design
(updated in: http://aitnaru.org/images/The_Right_Triangle.pdf )
Behold the geometric lock
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,
implying that the two "portals" have complementary resonance.
Rod (done pickith the lock ... or not)
(updated in: http://aitnaru.org/images/The_Right_Triangle.pdf )
Behold the geometric lock
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,
implying that the two "portals" have complementary resonance.
Rod (done pickith the lock ... or not)