11:11 Angels Message Board

Re: Diagonals, Juxtapositional design*

circle-squaring scalene triangle, uniting sqrt(Pi) and sqrt(2)

Scalene triangle inscribed in circle where D = 2

1.7724538509055160272981674833411.. sqrt(Pi), side a
1.4142135623730950488016887242097.. sqrt(2), side b

1.2533141373155002512078826424055.. sqrt(Pi) / sqrt(2)
+ 0.65513637756203355309393588562466.. sqrt(4-Pi) / sqrt(2)
= 1.9084505148775338043018185280301.. side c

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

Rod

Re: Diagonals, Juxtapositional design

Today's popular subtitle: "Quadrature as perceived from Superuniverse Seven"

Rod ... ...

Re: Diagonals, Juxtapositional design

1.2533141373155002512078826424055.. sqrt(Pi) / sqrt(2)
+ 0.65513637756203355309393588562466.. sqrt(4-Pi) / sqrt(2)
= 1.9084505148775338043018185280301.. side c

"So, what's this sqrt(4-Pi)?"

Length of short side of the circle-squaring right triangle
where long side = sqrt(Pi), hypotenuse = circle's diameter = 2.

"Who says?!"

Pythagoras: a^2 + b^2 = c^2 [Pi + (4-Pi) = 4]

Rod

Re: Diagonals, Juxtapositional design

Geometer's secret ...

Light blue arc added to identify sqrt(2) transition
of two circle-squaring right triangles.

Rod ... ...

Re: Diagonals, Juxtapositional design

Geometer's secret ...

Another line added to better identify a sqrt(2)-hosted Center of Quadrature (CQ),
dividng a plane into four sectors, two 90 degrees and two 135 degrees.

CQ CQ ?

Rod

Re: Centre of Quadrature design
"Infinity, a finity aflight"

"Say what?!"
Ask a geometer of Quadraturial persuasion.

Rod ... ...

Re: CS + SC design

"Say what?!" Go figure!

Rod

Re: CS + SC design*

When Quadrature waxes poetic with “je ne sais quoi”

"Say what?!" “je ne sais quoi”
"Say what?!" I don't know what
"Say what?!" What

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

Rod ... ...

Re: CS + SC design

Now ... Quadrature waxes Picasso
(to be displayed at the House of Pi).

Rod

Re: CS + SC design

Now ... Quadrature waxes Picasso

First, Marketing insisted on a better “je ne sais quoi”,
then Design upped the ante with their uncountable Omega
(hidden in this Cartesian composition).

Rod ... ...

Re: CS + SC design (+ “je ne sais quoi”)

"Final, final" ... even more esoteric

A good MD compromise ...
when Cartesian objects of Quadraturial persuasion are properly entangled.

Look! Up in the sky! It's a Byrd! It's a plane! ...

Or for a mundane challenge, try this: https://www.cnn.com/2020/07/25/us/krypt ... index.html

Rod

Re: CS + SC design (+ “je ne sais quoi”),
updated in: http://aitnaru.org/images/Yin_Yang_Pi.pdf

Geometer's secret regarding green '777' (circle-squaring right triangles):
Length of largest hypotenuse divided by lengths of smaller hypotenuses
= 1.3333333333333333333333333333333333333333333333333333333..

0.63661977236758134307553505349003
/ (0.31830988618379067153776752674501..
+ 0.15915494309189533576888376337251..)
= 1.33333333333333333333333333333333..

Same ratio exists for long sides (SoCS):
0.56418958354775628694807945156077..
/ (0.28209479177387814347403972578039..
+ 0.14104739588693907173701986289019..)
= 1.33333333333333333333333333333333..

Note: 2/sqrt(Pi) = 1.1283791670955125738961589031215..

Quick analysis: 1/3 of a Pi is a tasty serving,
albeit an irrational rational oft rationalized ...
especially when "á la mode".

Rod ... ... (off to the store since ...
a Sunday is oft "á la mode" day)

Re: CS + SC design (+ “je ne sais quoi”)

Geometer's secret ...
Area of circle surrounding '777' = 1 where D = 2/sqrt(Pi)
Area of smallest golden circle = 1/64 = 0.015625

Rod

Re: Ratio of Quadrature design

The Golden Ratio of Quadrature (geometrically speaking).

"Come and you will see!"

Rod

Re: Ratio of Quadrature design*

Even geometers of Quadraturial persuasion
sometimes ask "WTF?" ("Whence The Finesse?")
to identify the Golden Ratio of Quadrature.

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

Rod ... ...

Re: Golden Ratio of Quadrature design

"Whence The Finesse?"

A Cartesian display of the Golden Ratio of Quadrature
would be easier without all those squared circles!

Rod

Re: Golden Ratio of Qfin design ("fin" from "finesse")

Thence the finesse! (more sqrt(Pi)/sqrt(4-Pi) ratios)

Geometer's secrets ...
Dark blue line in lower left quadrant is a clue
on how to calculate the length of the red diagonal.

Length ratio of two adjoined red lines =
length ratio of two adjoined lines of golden rectangle
= sqrt(Pi)/sqrt(4-Pi) = 1.9130583802711007947403078280205..

Rod ... ...

Re: Golden Ratio of Qfin design

Thence the finesse!

While a geometric spiral of Quadrature is necessarily esoteric,
mystery begins fading when overlapping patterns are identified.

Who knew?! Sqrt(2) is host of this "Pi Corral" (green line)

Rod

Re: Golden Ratio of Qfin design
(lines show next step of spiral expansion)

Thence the finesse!

Geometers of Quadraturial persuasion always hafta know ...
"What's the next step to geometric infinity?"

Rod ... ...

Re: Golden Ratio of Qfin design*

Sqrt(2) wanted the last word ("the circuits are open")

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

Rod

Re: Golden Ratio of Qfin design

the circuits are open

Geometer's secret (about spiral of circles squared) ...

Inscribe any right triangle with 90-degree angle at center and lengths of sides
(not the hypotenuse) will maintain a sqrt(Pi)/sqrt(4-Pi) relationship.
( = 1.9130583802711007947403078280205.. = "Phi of Pi" )

Rod ... ...

Re: Golden Ratio of Qfin design

the circuits are open

Geometer's "Mystery of Qfin Zero" (relative to Transcendental Pi) ...

At various rotations of the inscribed right triangle, each side (independently)
will have whole digit length (no decimal points). That this right triangle forever
maintains its line length relationships and constant angles begs ...
"Whence the transcendence of Pi?"

Rod

Re: Golden Ratio of Qfin design

Mystery of Qfin Zero

EVERY point on the spiral is represented by a circle-squaring right triangle
(two points per triangle) with every hypotenuse having length of circle's diameter
and every long side having the length of a side of the circle's area square.

Rod

Re: Golden Ratio of Qfin design
(Q = Quadrature, "fin" from "finesse")

Given: Side of Circle's Square = SoCS = long side,
circle-squaring right triangle where hypotenuse = circle's diameter
and short side = sqrt(4-Pi), long side = sqrt(Pi), hypotenuse = 2.0

Relationship of short side to hypotenuse:
sqrt(4-Pi) x sqrt(Pi)/sqrt(4-Pi) x 2/sqrt(Pi) = 2.0

0.92650275035220848584275966758914.. sqrt(4-Pi)
x 1.9130583802711007947403078280205.. sqrt(Pi)/sqrt(4-Pi)
x 1.1283791670955125738961589031215.. 2/sqrt(Pi)
= 2.0

Rod ... ...

Re: Golden Ratio of Qfin design,
updated in: http://aitnaru.org/images/Yin_Yang_Pi.pdf

"Textbook ready" version.

"So, the circle can be squared!"
No, but squared circles exist! Go figure!

Rod