Paradise Trinity Day

[Amigoo]

Re: Two Ps in a Pod design

Geometer's secret ...
Relationship of a radius to an arc (different circles);
of a straight line to half a circumference.

For D = 2 and r = 2/sqrt(Pi),
C/2 = (2/r)^2 = Pi, therefore ...
2(Pi)/2 = (2/(2/sqrt(Pi)))^2 = Pi

Ratio of radius to half circumference (different circles)
occurs when the ratio of the two radii = 2/sqrt(Pi).

Rod

[Amigoo]

Re: Two Ps in a Pod design*

The Neighborhood's unanimous vote today proclaims that "Two Ps in a Pod" refers to
the two overlapping, circle-squaring scalene triangles, triangles associated by sqrt(2).

* updated in: http://aitnaru.org/images/Yin_Yang_Pi.pdf

Rod ... ...

[Amigoo]

Re: Two Ps in a Pod design

The numbers for C1/2 = (2/r2)^2 = Pi
C1 = inner circle, C2 = outer circle, C = Pi(D)

C1 = 2(Pi) = 6.283185307179586476925286766559..
D1 = 2
r1 = 1

C2 = Pi(2(2/sqrt(Pi))) = 7.0898154036220641091926699333646..
D2 = 2(2/sqrt(Pi)) = 2.2567583341910251477923178062431..
r2 = 2/sqrt(Pi) = 1.1283791670955125738961589031215..

C1/2 = (2/r2)^2 = Pi
3.1415926535897932384626433832795..
= (2/1.1283791670955125738961589031215..)^2
= 3.1415926535897932384626433832795..
= Pi

Rod

[Amigoo]

Re: Encircled Squares of Pi (ESP) design

Geometers of Quadraturial persuasion often develop a certain ESP about 2/sqrt(Pi)
and then gain expertise for infinite expressions of similar Cartesian design.

Rod

[Amigoo]

Re: ABCyclometricus design
"A^2 + B^2 = C^2"

Who knew?! Pythagoras?

Rod ... ...

[Amigoo]

Re: ABCyclometricus design*
"A^2 + B^2 = C^2"

More detail (in the spirit of "Morbus Cyclometricus")

* updated in: http://aitnaru.org/images/Yin_Yang_Pi.pdf

Rod

[Amigoo]

Re: Pythagorean Qube design
"A^2 + B^2 = C^2"

The Pythagorean Cube of Quadrature (aka "The Qube")

Rod

[Amigoo]

Re: Pythagorean Qube design
"A^2 + B^2 = C^2"

New era symbolism for a "School of Meganta"
with the Pythagorean Cube of Quadrature presenting
"the new box from which to think outside of".

Rod ...

[Amigoo]

Re: Pythagorean Qube design*
(aka "The Qube")

If you truly comprehend The Qube,
you're qualified to cross the E-R bridge ...
if you dare to go so far outside the box.

* updated in: http://aitnaru.org/images/Yin_Yang_Pi.pdf

Rod

[Amigoo]

Re: Pythagorean Qube design
(aka "The Qube")

That's sum cube! (a few lines were recolored)
Good for thinkin' outside the box (and inside the box)
... when in Cartesian lockdown.

Who knew You have to exercise your Qube
as well as your body when in lockdown mode.

"What's a Qube?" Ask Herman Munster.

Rod ... ...

[Amigoo]

Re: Pythagorean Qube design

Who knew You have to exercise your Qube
as well as your body when in lockdown mode.
"What's a Qube?" Ask Herman Munster.

Herman claims that the E-R bridge is a bit warped
... when viewed from outside the box.

Rod

[Amigoo]

Re: Pythagorean Qube design

"What's a Qube?" Ask Herman Munster.

Herman claims that the E-R bridge is a bit warped
... when viewed from outside the box.

Herman also cautioned that when thinkin' far outside the box,
your Qube gets more neutrinos (you see more of the E-R bridge
... or you head to the local E-R for neuron-calming meds).

Today's best T-Shirt caption:

"My left and right hemispheres are better balanced
now that I'm outside the stifling coronavirus box."

Rod ... ...

[Amigoo]

Re: Pythagorean Qube design

Geometer's secret (as certain as square Pi) ...

As you advance through the inhabited universes,
there will always be a box to think outside of ...
until you really comprehend "infinity".

Rod

[Amigoo]

Re: The Qube design*
(new design when you mentally climb into the next box)

"Final, Final ^3" ... with new T-Shirt caption:
"I survived the 2020 coronavirus but warped my Qube
... then cubed my warp."

* updated in: http://aitnaru.org/images/Yin_Yang_Pi.pdf

Rod ... ...

[Amigoo]

Re: Totally Trapezoidal design
(The Qube reveals its trapezoidal foundation)

There's even more geometry at the E-R bridge!
A can of Quadraturial wigglies at the wormhole.

Rod

[Amigoo]

Re: Totally Trapezoidal design
(The Qube reveals its trapezoidal foundation)

Geometer's secret about golden "bookmark" ...
In upper and lower circles' inscribed isosceles trapezoid,
the ratio of base2 to base1 = sqrt(Pi) / sqrt(4-Pi)
= 1.9130583802711007947403078280205..

Rod ... ...

[Amigoo]

Re: Totally Trapezoidal design

ratio of base2 to base1 = sqrt(Pi) / sqrt(4-Pi)
= 1.9130583802711007947403078280205..

New T-Shirt caption ...

"With lockdown ending,
there are lines all over the Neighborhood!
And geometers of Quadraturial persuasion
travel on the Morbus Cyclometricus line."

Rod ... ...

[Amigoo]

Re: Totally Trapezoidal design

And geometers of Quadraturial persuasion
travel on the Morbus Cyclometricus line.

Chatter today on the MC line ...

"A certain Pythagorean right triangle defines both the circle and its square,
with the base2:base1 trapezoid then appearing to confirm Quadrature ...
not to mention the inscribed circle-squaring scalene triangles!

"Lines and triangles and squares! Oh my!"

Rod

[Amigoo]

Re: ATTIC design
(A Totally Trapezoidal Integrated Composition)

Geometry teacher: "Johnny, please explain the Quadrature on page 133."
Johnny: "I dunno - It's so geometrically mature and suggestive."
Teacher: "Well, what do you see? What form? What shape?"
Johnny: "A T T I C"

Rod

[Amigoo]

Re: ATTIC design
(A Totally Trapezoidal Integrated Composition)

Geometer's secret about the golden '77" (as in "seventy times and seven"):
The enclosing circles help identify the three lines that have sqrt(2) relationship.

Local universe secret about "seventy times and seven":
The geometry of Quadrature goes on forever (so it seems),
creating an artist's defined super fractal.

"From whence the pattern?" In your attic.

Rod ... ...

[Amigoo]

Re: ATTIC design*
(A Totally Trapezoidal Integrated Composition)

Geometer's secret about this research (and '77') ...

Foundation of this composition, created with ViaCAD software,
then edited with FastStone Editor for artistic presentation,
is the Pythagorean circle-squaring right triangle (a '7')
where ratio of hypotenuse to long side = 2/sqrt(Pi).

Triangle defines both circle and its square:
hypotenuse = 2 = diameter
sides = sqrt(Pi), sqrt(4-Pi)

* updated in: http://aitnaru.org/images/Yin_Yang_Pi.pdf

Rod

[Amigoo]

Re: ATTIC design
(A Totally Trapezoidal Integrated Composition)

Geometer's secret about the golden '707' ...

Diameters of those two circles = 0.79788456080286535587989211986878..
therefore long sides of the resulting right triangles = D/(2/sqrt(Pi)).

Obviously, sqrt(2) is the true MC of Morbus Cyclometricus.

Rod ... ...

[Amigoo]

Re: MC Thrice Circular design
"Pi/4 ... precisely!"

Simple Quadrature that speaks for itself.

Rod

[Amigoo]

Re: MC Thrice Circular design

Geometer's secret about "Pi/4 precisely" ...

The area of the smaller circle-squaring scalene triangle
is 1/4 area of larger circle-squaring scalene triangle.

Geometer's secret about calculations ...
Both triangles were divided into their adjoined
isosceles right and right triangles:

0.78539816339744830961566084581988..
+ 0.41054584193408096912860688490008..
= 1.19594400533152927874426773072..
/ 4 = 0.29898600133288231968606693267999..

0.19634954084936207740391521145497..
+ 0.10263646048352024228215172122502..
= 0.29898600133288231968606693267999..

"Say what?!" Areas of the 4 scalenes.

Rod

[Amigoo]

Re: MC Thrice Circular design,
updated in: http://aitnaru.org/images/Yin_Yang_Pi.pdf

Simplified with great sqrt(2) ambience!

Rod