Paradise Trinity Day

[Amigoo]

Re: "Ro of Fourths" design
"Squared-circle patterns extraordinaire"

And the winner is ... sqrt(2)
(ratios of both the hypotenuses and long sides
of the two overlapping, golden right triangles)

Rod

[Amigoo]

Re: RoFlutterby design

Further exploration of the Ro of Fourths geometry (and because squared circles
are enticingly complex and seemingly from a bottomless geometry toybox).

Rod ... ... (off chasin' a RoFlutterby)

[Amigoo]

Re: ... ... (off chasin' a RoFlutterby)

Found two more! Now five circles, all squared (some displayed as arcs)

This tightly integrated geometry does not claim that "the circle is squared",
but the patterns present convincing evidence that we're in the ballpark.

Rod

[Amigoo]

Re: RoFlutterby design

Would you believe, six circles squared?!
... and earning the whimsical sub-title "three strikes of Pi at the old ball game".

Rod ... ... (boot scootin' the bases)

[Amigoo]

Re: Trapezoidal Triumph design
(further exploration of the RoFlutterby
with circle-inscribed trapezoids)

More "geometry that speaks for itself"
... and may chatter for decades ...
(if not centuries and millennia)

Geometers' tip:
Note how the diagonal of a smaller trapezoid
has same length as side midpoint-to-midpoint
of the next larger trapezoid.

Rod

[Amigoo]

Re: Trapezoidal Triumph design
(further evolution of RoFlutterby geometry)

"Squared-circle geometry that speaks for itself
… with three tightly integrated trapezoids."

Such chatter! Sounds like Greek to me!
(but trapezoidal integration is obvious: two sides of trapezoid
have length equal to sides of inscribed squares; longest line
of trapezoid has length equal to side of circle's square).

See online: http://aitnaru.org/lighttruth.html
(click on image to view attached PDF)

The quote? UB (110:6.4)

Rod

[Amigoo]

Re: Rule of Trapezoids design

Proffering ...
Rule of Trapezoids - mathematical "noun of assembly"
Re: seven major trapezoids in a squared circle.
(technically, 7 trapezoids in 5 circles squared,
hosted by 5 similar scalene triangles)

Rod

[Amigoo]

Re: Rule of Trapezoids design
(aka "Trapezoids Rule!")

"In squared circles, trapezoids rule ...
if a unique scalene triangle is present."

Geometers' tip: The inner four light blue circles enclose
the four similar scalene triangles of larger scalene (red).

Rod ... ... (still bikin' by the Rules
in Cartesian squared-circle neighborhoods)

[Amigoo]

Re: Rule of Trapezoids design
(aka "Trapezoids Rule!")

Now with more complexity and increasingly ESOTERIC,
but "long story short", especially for our inner YoLF.

Rod

[Amigoo]

Re: Dissimilar Paired design

Highlights the two dissimilar (and tightly integrated)
trapezoids that define a squared circle ...
with both objects inscribed in the circle.

Rod

[Amigoo]

Re: Dissimilar Paired design
“Trapezoids Rule!”

Simplified version (so I've been informed),
but this geometry jungle may simply inspire
“Lines and triangles and squares, oh Pi!”

Rod

[Amigoo]

Re: Dissimilar Paired design
“Trapezoids Rule!”

Geometry eye candy for squared circle geometers:
http://aitnaru.org/images/Dissimilar_Paired.pdf

iCSCi = impossible CSC integration

Rod

[Amigoo]

Re: Le Pi Doptera design

Irrational emergence of Pi from its long-enchanting transcendental cocoon?

Rod ... ... (off to find a long-enchanting Le Pi Net)

[Amigoo]

Re: Le Pi Doptera design
“Emanation from a transcendental cocoon."
(aka “Counsel of Elders”)

If geometry would represent the voice of the elders.
“Emanation” (not plural) refers to the ONE golden voice.

Rod

[Amigoo]

Re: Le Pi Doptera design
“Emanation from a transcendental cocoon.”
(aka the “Counsel of Elders”)

How interesting that the geometry hints that e II
might be a future math constant as is Euler's Number.
(Euler's number e = 2.71828183.. )

Maybe 2(sqrt(1/Pi)) has such potential!
e II = 1.12837916..

Rod ... ... (still cruisin', looking for the e II exit)

[Amigoo]

Re: Le Pi Doptera design (final,final)^2

aka "Conservative, long-term navigation for universe departure."
aka "Greetings from that periphery: Wish you were here!"
aka "New perspective for these final days of 2016."

Enchanting and mystifying integration of squared-circle objects.
Impossible, imagination, or illumination? Or "all of the above"?

Rod

[Amigoo]

Re: Le Pi Doptera design

Makes sense to me ...
If the circle is squared, the sphere is cubed.

Rod ... ... (off to find the tourist map
for exploration of the 8 corners of the world)

[Amigoo]

Re: Le Pi Doptera design (final,final)^3

If a box of freshly baked Pi suddenly opens in the forest
where no one is around to smell it, does it have aroma?


When the circle is squared, a related sphere is cubed.
For cube displayed, diameter of its cubed circle = D2:

D1 = 2(sqrt(1/Pi))
D1 = 1.1283791670955125738961589031215..

D2 = D1/sqrt(2)
D2 = 0.79788456080286535587989211986876..
D2 area = 0.5
D2 side = 0.70710678118654752440084436210485..
= sqrt(2)/2

Rod

[Amigoo]

Re: Le Pi Doptera design (final,final)^3

This morning's word~twist, begging assumptions and
making this a good discussion question for many venues:

If a box of freshly baked plum Pi suddenly opens in the forest
where no one is around to smell it, does it have a roma?

What's intriguing about this question, "anatomically" related to a popular muse*,
is that one question seems more intuitive than the other. How is this possible?
And which question? which assumptions?

* "If a tree falls in a forest and no one is around to hear it, does it make a sound?"


Rod ... ... (off to buy small, red plums for my salad)

[Amigoo]

Re: Le Pi Doptera design (final,final)^3
(see online: http://aitnaru.org/lighttruth.html
click on image to view design in PDF)

"(a question more intuitive than its Chautauquan muse?)
If a box of freshly baked plum Pi suddenly opens in the forest
where no one is around to smell it, does it have a roma?"

Note: "Chautauquan" is mentioned in this article:
https://en.wikipedia.org/wiki/If_a_tree ... n_a_forest

Rod

[Amigoo]

Re: Le Pi Doptera design (re: in the forest)

I read on the internet that generation of a sound or smell is only part of the event;
sound waves or aroma molecules must be sensed by a person (or animal, insect)
to fully qualify the sound or smell. However, placing the event in a forest is good
riddle strategy since animals and insects are always present!

Rod

[Amigoo]

Re: Sqrt(Pi) Entwined design "WYSIWYG"

Do you see two veggie leaves or two alien eyes at the top
... or just arc, arc, arc ... ?

Rod ... ...

[Amigoo]

Re: Sqrt(Pi) Entwined design

Hmmm ... the Texas 'T' appeared in this update of the design!
Now, the geometry is starting to suggest "alien" influence.

Rod

[Amigoo]

Re: Sqrt(Pi) EnTwined design

Finally! Math that explains the significance of the Texas 'T'.
(see in PDF: http://aitnaru.org/images/Concentrizity_Squares.pdf )

Integration of two right triangles within the light blue circle (an arc)
is highlighted, with length of 'T's vertical line equal to Pi/4 and
2(sqrt(1/Pi)) the shared ratio of hypotenuse to long side.

Rod ... ...

[Amigoo]

Re: Geometry of the Cross design (D = 1, sqrt(Pi)/2)

Discovery of the Pi/4 line segment inspired more exploration of Sqrt(Pi) EnTwined
and suggests that certain "geometry of the cross" is inherent in circles squared.

Rod