Paradise Trinity Day

[Amigoo]

Re: Thrice Phi of Pi design,
updated in: http://aitnaru.org/images/Yin_Yang_Pi.pdf

I.C. Not,

I agree that an isosceles right triangle is not easy to imagine!
... with all this "cacophony of Cartesian coloring" ...
especially since Quadrature is still "impossible".
(third line of the triangle was added)

Rod ... ...

[Amigoo]

Re: Thrice Phi of Pi design
and Phi Four of Pi design

Geometer's secret ...
In circle-squaring right triangle, ratio of short side to long side
is equal to the ratio of upper diagonal to the hypotenuse.

Rod ... ...

[Amigoo]

Re: One Phi 360 design

Geometer's secret ...
While this geometry speaks for itself, non-geometers may not know that
((Phi of Pi)^2)^2 defines relationship of largest and smallest rectangle.

1.7724538509055160272981674833411..
/ 0.92650275035220848584275966758914..
= 1.9130583802711007947403078280205.. Phi of Pi
^2^2 = 13.394080164614312715066168666409..

Rod

[Amigoo]

Re: Phi of Pi ^2 design

More Cartesian Neighborhood "chatter".
(apparent translation: "Quadrature 101")

Diameter of largest circle = 4
Diameter of next largest circle = 2(sqrt(Pi))
Length of diagonal dark blue line = 2(2/sqrt(Pi))

Rod

[Amigoo]

Re: Phi of Pi ^2 design

"Proof of Quadraturial balance!" they say.
"Seeing is believing!" I say.
"Believing is seeing!" they say.

Rod ... ...

[Amigoo]

Re: Phi of Pi ^2 design
"Believing is seeing!"

Geometer's secret ...
I long believed that juxtaposition of the two quadrilaterals (red lines, inscribed in light blue circles)
were necessary for proof of Quadrature, but only recently discovered that Phi of Pi revealed
the contrasting and important line length ratios.

Rod

[Amigoo]

Re: Phi of Pi ^2 design
"Believing is seeing!"

Geometer's secret ...

I still find no direct association between the golden ratio Phi = (1 + sqrt(5))/2 = 1.6180339887498948482045868343656..
and Phi of Pi = sqrt(Pi)/sqrt(4-Pi) = 1.9130583802711007947403078280205.., but note that Phi is based on straight line ratios
whereas Phi of Pi is based on the circle-squaring right triangle where even hypotenuse length is a significant quantity.

Rod

[Amigoo]

Re: Phi of Pi ^2 design
"Believing is seeing!"

Geometer's secret ...

Rectangle including Phi of Pi ratio (long side to short side of circle-squaring right triangle)
also includes iQ of Pi ratio* (hypotenuse to long side). These tandem ratios persist
for 360 degrees of that spiral ... and on into the next segment of the spiral.

* "iQ of Pi" was the beginning whimsical term for "impossible Quadrature"
when it was obvious that 2/sqrt(Pi) defined the circle-squaring right triangle,
where triangle's hypotenuse = circle's diameter.

Phi of Pi = sqrt(Pi)/sqrt(4-Pi) = 1.9130583802711007947403078280205..
iQ of Pi = 2/sqrt(Pi) = 1.1283791670955125738961589031215..

Rod (been up to my Phi in Pi for 3.14159.. years)

[Amigoo]

Re: iC Arc Arc design,
added to: http://aitnaru.org/images/Yin_Yang_Pi.pdf

Quadrature hosted by sqrt(2) ... obviously.

Rod ... ...

[Amigoo]

Satisfying plateau, this current portfolio.
New expression: "Go with the Phorce!"

Expression derived from Phi of Pi geometry that suggests ...
squared circles exist even if "the circle cannot be squared".

"Go with the Phorce!" and discover what's truly "impossible"
... or just more difficult.

Rod

[Amigoo]

Re: iC Arc Arc design

Go with the Phorce!

No matter how complex the sqrt(2) Neighborhood becomes,
the geometry of Quadrature always includes these two ratios
(and that Pythagorean, circle-squaring right triangle):

Phi of Pi = sqrt(Pi)/sqrt(4-Pi) = 1.9130583802711007947403078280205..
iQ of Pi = 2/sqrt(Pi) = 1.1283791670955125738961589031215..

Rod ... ... (cruisin' with the Phorce)

[Amigoo]

Re: iCircles of 'X' design

"Lines and triangles and squares, oh my!" (and circles!)
... and Quadraturially plane to see!

Rod ... ... (more force of Phorce, of course)

[Amigoo]

Re: Corresponding Quadrature design

Simplified to highlight geometric salience.

Rod

[Amigoo]

Re: Corresponding Quadrature design*
(simply simplified with salient salience)

In this new millennium, Quadrature ...
is as easy as a piece of cake ...
and intuitive as a box of Pi.

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

Rod ... ...

[Amigoo]

Re: CQ CQ design and "Go with the Phorce!"
(a few more lines)

Discover shared boundary of parallel universes on the ER bridge!
(of course, some thinkin' outside the box may be required)

Rod

[Amigoo]

Re: CQ CQ design and "Go with the Phorce!"
(s'more thinkin' outside the box)

Who can tell
This seems to reveal a new geometric shape!

Obviously, a polyhedron, perhaps ...
"Quadrahedron" (from "Quadrature")
"Cyclomehedron" (from "Cyclometricus")
"Impossihedron" (from "Impossible")

Rod

[Amigoo]

Re: CQ CQ design and "Go with the Phorce!"
(s'more thinkin' outside the box)

Geometer's secret ...
When the red cube is visualized as a three-dimensional object,
line connecting two points of the overlapping scalene triangles
has same length as lines in cube's two-dimensional perimeter.

"So ...what's the length?"
1.4142135623730950488016887242097..

Rod ... ...

[Amigoo]

Re: Arcs of Mark design
"Quadraturially different, but the same"

More exploration of CQ CQ geometry ...
highlighting obvious circle-squaring scalenity.

Rod

[Amigoo]

Re: Arcs of Mark design*
"Quadraturially different, but the same"

Geometer's secret ...
"Arcs of Mark" aka "The arcs that mark"

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

Rod ... ...

[Amigoo]

Re: Arcs of Mark design
"Quadraturially different, but the same
... with 'impossible' fractal potential"

"Go with the Phorce!" ...
Discover math that nurtures 'impossible' fractals.
(a certain scalene triangle will repeat forever ...
but not without its associated right triangles ...
and re-appearing similar objects)

Rod

[Amigoo]

Re: IAM Concentric design
(Impossible Arcs of Mark Concentric)

"Go with the Phorce!" ...
to discover sqrt(2)-hosted concentric Quadrature.

Rod ... ... (cruisin' with Gee! Phorce)

[Amigoo]

Re: IAM Concentric design
(Impossible Arcs of Mark Concentric)

"Go with the Phorce!"
(the shock & awe of the geometry of Quadrature ...
aka "Another new line in the Cartesian Neighborhood")

It's another new line in the Neighborhood ...
Would you be mine Could you be mine
Won't you be my Neighbor?

Rod

[Amigoo]

Re: IAM Concentric design
(Impossible Arcs of Mark Concentric)

"Impressive! ... but what's the point?!"

There are only 8 points on the circle
upon which rests the circle's area square.
Ask Pythagoras what happens next.

Rod ... ...

[Amigoo]

Re: IAM Concentric design*
(Impossible Arcs of Mark Concentric)

Who knew?!
Even today, it's possible to tap into your inner Neanderthal
and continually upscale your (man)cave painting.

Neanderthal memories inspired by the '8' ...
Those tough 8-day work weeks, hunting and foraging,
and dim-light painting would give you mitosis.

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

Rod

[Amigoo]

Re: Secret of 8 design

Displays two overlapping squares within the central circle,
squares that are inscribed in that circle squared.

Rod