Paradise Trinity Day

[Amigoo]

Re: Here! There! (and everywhere!) design

The Revealing Ratios of Pi and Sqrt(2)

2.0
/ 1.4142135623730950488016887242097.. sqrt(2)
= 1.4142135623730950488016887242097.. sqrt(2)

3.1415926535897932384626433832795.. Pi
/ 1.4142135623730950488016887242097.. sqrt(2)
= 2.2214414690791831235079404950303..

3.1415926535897932384626433832795.. Pi
/ 2.0
= 1.5707963267948966192313216916398.. Pi/2

2.2214414690791831235079404950303..
/ 1.4142135623730950488016887242097.. sqrt(2)
= 1.5707963267948966192313216916398.. Pi/2

Since Pi and sqrt(2) are geometrically related
in Cartesian Neighborhoods of circles all squared,
either sqrt(2) is transcendental or Pi is not.

Rod ... ...

[Amigoo]

Re: Periphery design
"Where Here and There share overlapping boundaries"

While it's not possible to "get there from here"
it is possible to glimpse There from Here ...
even from this periphery of O.

Rod

[Amigoo]

Re: Periphery design
"At the Periphery
knit one, purl two ..."

"Say what?" What

Rod ... ...

[Amigoo]

Re: Periphery design
"At the Periphery, knit one, purl two ..."

Re: https://en.wikipedia.org/wiki/Crochet

"The salient difference between crochet and knitting, beyond the implements used for their production, is that each stitch in crochet is completed before proceeding with the next one, while knitting keeps a large number of stitches open at a time."

... And such is life upon our planet in the local universe
Our unfinished projects (stitches) may at times be many,
depending on the current capacity of our mortal vessel.

Salient lesson on How to Achieve the "Impossible":
One stitch at a time ... with many in mind.

Rod

[Amigoo]

Re: Periphery design
"At the periphery of O., knit one, purl two ..."

Revised to highlight sqrt(2) relationship of two diagonal green lines
and accompanied by a purl of wisdom from the Old Why?'s:

"A stitch in time saves thyme."
(thyme stays in the cupboard when yer busy stitchin')

Rod

[Amigoo]

Re: Periphery design
"At the periphery of O., knit one, purl two ..."

Notice that the re-coloring completes the "impossible" geometry:

As color-coded, each side of the two similar circle-squaring scalene triangles
has a geometrically-provable sqrt(2) relationship with the respective side
of the related scalene triangle.

And one of those 3 sides is a side of that circle's area square!
"Either sqrt(2) is transcendental or Pi is not."

Rod ... ...

[Amigoo]

Re: iQ design ("impossible" Quadrature)
with embellishment of an abstract eye (part of 'i') and the letter 'Q'

and highlighting constant 2(sqrt(1/Pi)) in a geometric progression.

Rod

[Amigoo]

Re: iQ Centromere design

Geometric juxtapostion of two sets of Pi/2, sqrt(Pi), and 2.0;
the "centromere" portion of the replicating iQ geometry,
confirming these similar circle-squaring ratios.

"Say what?" Symbolic early phase of iQ mitosis
(where geometers multiply ... or divide)

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

Rod ... ...

[Amigoo]

Re: iQ design ("impossible" Quadrature)
with an abstract eye (part of 'i') and the letter 'Q'

It's simply iQ in plane view, featuring defining angle:
acos(sqrt(Pi)/2) = 27.59711263569.. degrees

... for “impossible” circle-squaring right triangle
where 2/sqrt(Pi) = sqrt(Pi)/(Pi/2) = 2(sqrt(1/Pi))
= 1.1283791670955125738961589031215..

Re: iQ Centromere design

Juxtapostion of two line length sets: Pi/2, sqrt(Pi), 2.0;
"centromere" portion of the replicating iQ geometry,
confirming the similar circle-squaring ratios.

"Say what?" The symbolic early phase of iQ mitosis
where expert geometers divide ... or multiply.

Rod ... ... in Texas ...
where "The stars at night are big and bright ..."

[Amigoo]

Re: iQ design ("impossible" Quadrature)
where 2(sqrt(1/Pi)) = 2/sqrt(Pi) = sqrt(Pi)/(Pi/2)

Who knew
Pi itself defines the new circle-squaring constant:
2(sqrt(Pi)) / Pi = 1.1283791670955125738961589031215..

"When is the circle squared?" When Pi is well-baked.

Rod

[Amigoo]

Re: iQ design ("impossible" Quadrature)
where 2(sqrt(1/Pi)) = 2/sqrt(Pi) = sqrt(Pi)/(Pi/2)

Another spin of the record (Round Vinyl, Squared*) ...

2(sqrt(1/Pi)) is the ratio of hypotenuse to long side of a right triangle
where the hypotenuse has length equal to the circle's diameter
and long side has length equal to a side of the circle's square.

That Pi itself defines this ratio suggests "Pi is ready to eat!
... well-baked and cooled (for millennia)"

* next design, a golden record

Rod ... ... (still spinnin' squared circles)

[Amigoo]

Re: Round Vinyl, Squared design

"The circle in the center?" A Pi hole
(large circle's diameter = 20, hole = 3.14..)

Rod

[Amigoo]

Re: Round Vinyl, Squared design
(SoCS = Side of Circle's Square)

If SoCS = Pi/2, D = sqrt(Pi)
If SoCS = sqrt(Pi), D = 2.0
If SoCS = 2.0, D = 4(sqrt(1/Pi))

Analysis: If Pi is the ratio of circle's circumference to its diameter,
then 2(sqrt(1/Pi)) defines the right triangle that squares the circle.

Rod

[Amigoo]

Re: Round Vinyl, Squared design

Now, more representation for constant 1.9130583802711007947403078280203..
(ratio of long side to short side in the circle-squaring right triangles)

Calculated with Pythagorean Theorem:

hypotenuse = 2.0, long side = 1.7724538509055160272981674833411..
short side = 0.92650275035220848584275966758914..

1.7724538509055160272981674833411.. long side
/ 0.92650275035220848584275966758914.. short side
= 1.9130583802711007947403078280203.. constant

Rod ... ...

[Amigoo]

Re: Round Vinyl, Squared design

Changing the color of two "shorts" added dual, complementary, and meaningful symbolism:
"having an open mind", "thinking outside the box"; necessary to comprehend circles squared

Re: https://www.macys.com/shop/mens-clothin ... nk?id=3310
"classic, performance, raw edge, stretch" are enticing masculine themes ... for pink shorts,
all just above the knee (too much above the knee may change the theme).

Rod

[Amigoo]

Re: RVS 2T design
Seven years later, the Texas 'T' geometry is revealed:

The 'T' comprises circle-squaring right triangles that
have the long-side-to-short-side ratio as now calculated:

For D = 2.0 (the hypotenuse) ...
1.7724538509055160272981674833411.. long side
/ 0.92650275035220848584275966758914.. short side
= 1.9130583802711007947403078280203.. constant

Re: viewtopic.php?f=15&t=17885&start=400#p177232
"I sensed it as a prompt when I noticed it while driving on this north Texas tollway in late 2010"

Rod ... ... (been cruisin' for years)

[Amigoo]

Re: RVS 2T design
"Texas 'T' time redux"

Geometers' secret:
The dark blue 'X' shape has connecting lines at the top and bottom
where sqrt(Pi) is the ratio of the top length to the bottom length
and 2(sqrt(1/Pi)) is the ratio of lengths of the 'X' lines. HCIT

Rod

[Amigoo]

Re: RVS 2T design
"Texas 'T' time redux"

For the record (RVS = "Round Vinyl, Squared") ...
the NTTA logo ("Texas 'T'") was a clue that a unique 'T' shape
defines a squared circle. The NTTA 'T' is not the precise shape
displayed in the PTD geometry designs.

Rod

[Amigoo]

Re: RVS 2T design, "Texas 'T' time redux"
Updated in: http://aitnaru.org/images/The_Right_Triangle.pdf

Curious, why the RVS geometry seemed geometrically balanced,
I studied the two integrated inner circles (one green, one golden) ...
and finally noticed that a 'T' triangle squares both circles:
~ hypotenuse is side of golden circle's square.
~ long side is side of green circle's square.

Rod

[Amigoo]

Re: RVS 2T Lite design (simplification of RVS 2T)

Geometric focus on sqrt(Pi) and 2(sqrt(1/Pi))
aka “Pi Fork and 'T'” time. Bon appétit!

This geometry highlights the circle-squaring ratio,
2(sqrt(1/Pi)), and sets of sqrt(Pi) line length ratios.

Rod ... ...

[Amigoo]

Re: RVS 2T Lite design
w/ focus on sqrt(Pi) and 2(sqrt(1/Pi))

About the two-tine "Pi Fork" ...

- formed by the 'V' of that circle-squaring right triangle.
- dark blue 'X' shape creates two symbolic "fork tines".
- top line of 'T' and bottom of 'X' have sqrt(Pi) relationship.
- each side of 'V' and a blue line has sqrt(Pi) relationship.
- two lines of 'X' have 2(sqrt(1/Pi)) length relationship.
- side length of that circle's square (SoCS) = Pi/2.
- diameter length of circle squared = sqrt(Pi).

Bon appétit!

Rod

[Amigoo]

Re: RVS 2T Lite design
w/ focus on sqrt(Pi) and 2(sqrt(1/Pi))
aka “Pi Fork and 'T'”, a new T-Square

Who knew

Using Pi itself to define a squared circle
presents this new millennium's challenge:
Prove that the circle is not squared.

About "2T" ...
One 'T' shape but 2 right Triangles.

Rod ... ...

[Amigoo]

Re: Phi of Pi design (a "golden ratio" of squared circles)

For a circle-squaring right triangle,
ratio of long side to short side = 1.9130583802711007947403078280203..

For each 90-degree turn, hypotenuse increases/decreases by Phi of Pi,
creating an inspirational Golden Cross in a geometric spiral.

Rod

[Amigoo]

Re: Phi of Pi design (summary)

There’s a circle-squaring “golden ratio” in a unique right triangle
where the hypotenuse has length equal to the circle’s diameter:

For D = 2.0 (also unique triangle’s hypotenuse) …
1.7724538509055160272981674833411.. long side, sqrt(Pi)
/ 0.92650275035220848584275966758914.. short side
= 1.9130583802711007947403078280203.. “Phi of Pi”

About “0.9265027503522..” …
Calculated with the Pythagorean Theorem
since 2(sqrt(1/Pi)) = sqrt(Pi)/(Pi/2) = 2.0/sqrt(Pi)
= 1.1283791670955125738961589031215.., constant ratio
of this unique triangle’s hypotenuse to its long side.

… perhaps, the new millennium’s tasty Two-Phi Pi

Rod ... ...

[Amigoo]

Re: Two-Phi Pi design
(renamed from "Phi of Pi")

Who knew
A new millennium ichthys swims in this Cartesian pond.

Rod