Paradise Trinity Day

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Re: Paradise Trinity Day
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 ... ...
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 ... ...

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Re: Paradise Trinity Day
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
"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

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Re: Paradise Trinity Day
Re: Periphery design
"At the Periphery
knit one, purl two ..."
"Say what?" What
Rod ... ...
"At the Periphery
knit one, purl two ..."
"Say what?" What
Rod ... ...

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Re: Paradise Trinity Day
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
"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

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Re: Paradise Trinity Day
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
"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

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Re: Paradise Trinity Day
Re: Periphery design
"At the periphery of O., knit one, purl two ..."
Notice that the recoloring completes the "impossible" geometry:
As colorcoded, each side of the two similar circlesquaring scalene triangles
has a geometricallyprovable 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 ... ...
"At the periphery of O., knit one, purl two ..."
Notice that the recoloring completes the "impossible" geometry:
As colorcoded, each side of the two similar circlesquaring scalene triangles
has a geometricallyprovable 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 ... ...

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Re: Paradise Trinity Day
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
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

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Re: Paradise Trinity Day
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 circlesquaring 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 ... ...
Geometric juxtapostion of two sets of Pi/2, sqrt(Pi), and 2.0;
the "centromere" portion of the replicating iQ geometry,
confirming these similar circlesquaring 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 ... ...

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Re: Paradise Trinity Day
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” circlesquaring 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 circlesquaring 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 ..."
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” circlesquaring 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 circlesquaring 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 ..."

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Re: Paradise Trinity Day
Re: iQ design ("impossible" Quadrature)
where 2(sqrt(1/Pi)) = 2/sqrt(Pi) = sqrt(Pi)/(Pi/2)
Who knew
Pi itself defines the new circlesquaring constant:
2(sqrt(Pi)) / Pi = 1.1283791670955125738961589031215..
"When is the circle squared?" When Pi is wellbaked.
Rod
where 2(sqrt(1/Pi)) = 2/sqrt(Pi) = sqrt(Pi)/(Pi/2)
Who knew
Pi itself defines the new circlesquaring constant:
2(sqrt(Pi)) / Pi = 1.1283791670955125738961589031215..
"When is the circle squared?" When Pi is wellbaked.
Rod

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Re: Paradise Trinity Day
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!
... wellbaked and cooled (for millennia)"
* next design, a golden record
Rod ... ... (still spinnin' squared circles)
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!
... wellbaked and cooled (for millennia)"
* next design, a golden record
Rod ... ... (still spinnin' squared circles)

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Re: Paradise Trinity Day
Re: Round Vinyl, Squared design
"The circle in the center?" A Pi hole
(large circle's diameter = 20, hole = 3.14..)
Rod
"The circle in the center?" A Pi hole
(large circle's diameter = 20, hole = 3.14..)
Rod

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Re: Paradise Trinity Day
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
(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

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Re: Paradise Trinity Day
Re: Round Vinyl, Squared design
Now, more representation for constant 1.9130583802711007947403078280203..
(ratio of long side to short side in the circlesquaring 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 ... ...
Now, more representation for constant 1.9130583802711007947403078280203..
(ratio of long side to short side in the circlesquaring 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 ... ...

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Re: Paradise Trinity Day
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/mensclothin ... 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
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/mensclothin ... 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

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Re: Paradise Trinity Day
Re: RVS 2T design
Seven years later, the Texas 'T' geometry is revealed:
The 'T' comprises circlesquaring right triangles that
have the longsidetoshortside 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)
Seven years later, the Texas 'T' geometry is revealed:
The 'T' comprises circlesquaring right triangles that
have the longsidetoshortside 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)

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Re: Paradise Trinity Day
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
"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

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Re: Paradise Trinity Day
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
"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

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Re: Paradise Trinity Day
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
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

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Re: Paradise Trinity Day
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 circlesquaring ratio,
2(sqrt(1/Pi)), and sets of sqrt(Pi) line length ratios.
Rod ... ...
Geometric focus on sqrt(Pi) and 2(sqrt(1/Pi))
aka “Pi Fork and 'T'” time. Bon appétit!
This geometry highlights the circlesquaring ratio,
2(sqrt(1/Pi)), and sets of sqrt(Pi) line length ratios.
Rod ... ...

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Re: Paradise Trinity Day
Re: RVS 2T Lite design
w/ focus on sqrt(Pi) and 2(sqrt(1/Pi))
About the twotine "Pi Fork" ...
 formed by the 'V' of that circlesquaring 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
w/ focus on sqrt(Pi) and 2(sqrt(1/Pi))
About the twotine "Pi Fork" ...
 formed by the 'V' of that circlesquaring 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

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Re: Paradise Trinity Day
Re: RVS 2T Lite design
w/ focus on sqrt(Pi) and 2(sqrt(1/Pi))
aka “Pi Fork and 'T'”, a new TSquare
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 ... ...
w/ focus on sqrt(Pi) and 2(sqrt(1/Pi))
aka “Pi Fork and 'T'”, a new TSquare
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 ... ...

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Re: Paradise Trinity Day
Re: Phi of Pi design (a "golden ratio" of squared circles)
For a circlesquaring right triangle,
ratio of long side to short side = 1.9130583802711007947403078280203..
For each 90degree turn, hypotenuse increases/decreases by Phi of Pi,
creating an inspirational Golden Cross in a geometric spiral.
Rod
For a circlesquaring right triangle,
ratio of long side to short side = 1.9130583802711007947403078280203..
For each 90degree turn, hypotenuse increases/decreases by Phi of Pi,
creating an inspirational Golden Cross in a geometric spiral.
Rod

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Re: Paradise Trinity Day
Re: Phi of Pi design (summary)
There’s a circlesquaring “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 TwoPhi Pi
Rod ... ...
There’s a circlesquaring “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 TwoPhi Pi
Rod ... ...

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Re: Paradise Trinity Day
Re: TwoPhi Pi design
(renamed from "Phi of Pi")
Who knew
A new millennium ichthys swims in this Cartesian pond.
Rod
(renamed from "Phi of Pi")
Who knew
A new millennium ichthys swims in this Cartesian pond.
Rod