Paradise Trinity Day

[Amigoo]

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 circle-squaring ratios:)

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

[Amigoo]

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 ... ...

[Amigoo]

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

[Amigoo]

Re: Caliperfection design

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

Rod

[Amigoo]

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 ... ...

[Amigoo]

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 ... ...

[Amigoo]

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)

[Amigoo]

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 ... ...

[Amigoo]

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

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.

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

Rod

[Amigoo]

Re: Cross of Gaia design

Long story short ...

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.

Rod

[Amigoo]

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 ... ...

[Amigoo]

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

[Amigoo]

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

Rod

[Amigoo]

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

Note: Dark blue coloring identifies gaps in circle-squaring 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)

[Amigoo]

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

[Amigoo]

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 ... ...

[Amigoo]

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 circle-squaring right triangle where ratio
of lengths of long side to short side equals the squared-circle
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

[Amigoo]

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?"

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)

[Amigoo]

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

[Amigoo]

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 ... ...

[Amigoo]

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..)

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

Rod

[Amigoo]

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 ... ...

[Amigoo]

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).

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 ! )

[Amigoo]

Re: Tri Phi Pi design

Who knew Twin portals!

Rod ... ...

[Amigoo]

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)