Paradise Trinity Day

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Re: Paradise Trinity Day
Re: Golden rPi design
Re: The Golden Ratio by Mario Livio, 2002, p. 85
"The Golden Rectangle is the only rectangle with the property
that cutting a square from it produces a similar rectangle."
Impressive, however ...
The rPi Golden Rectangle is the only rectangle with the property
that its diagonal has length equal to the diameter of a circle and
sides a+b have length equal to a side of that circle's square.
Given that the diagonal is side d, d / (a + b) = 2(sqrt(1/Pi)),
consistent ratio of these sides in all related right triangles.
Rod
Re: The Golden Ratio by Mario Livio, 2002, p. 85
"The Golden Rectangle is the only rectangle with the property
that cutting a square from it produces a similar rectangle."
Impressive, however ...
The rPi Golden Rectangle is the only rectangle with the property
that its diagonal has length equal to the diameter of a circle and
sides a+b have length equal to a side of that circle's square.
Given that the diagonal is side d, d / (a + b) = 2(sqrt(1/Pi)),
consistent ratio of these sides in all related right triangles.
Rod

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Re: Paradise Trinity Day
Re: Golden rPi design
I finally comprehend that this geometry is revealing a new "golden ratio(s)"
with another Cartesian dimension beyond the linesegmented Golden Ratio:
relationships of the lines* in a circlesquaring right triangle ... lines that
define a new spiral for consumption in believing math circles.
* one line is segmented (to allude to the old Golden Ratio?)
Rod ... ...
I finally comprehend that this geometry is revealing a new "golden ratio(s)"
with another Cartesian dimension beyond the linesegmented Golden Ratio:
relationships of the lines* in a circlesquaring right triangle ... lines that
define a new spiral for consumption in believing math circles.
* one line is segmented (to allude to the old Golden Ratio?)
Rod ... ...

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Re: Paradise Trinity Day
Re: Golden rPi design
Who knew That a new Phi ratio would exist for the rPi rectangle!
(repeating ratio observable in similar objects/lines of Golden rPi)
Current Phi = 1.618033988749894848204586834 ..
New iPhi = 1.913067540823067208163119533.. (approx)
'i' alludes to "impossible" squaring of the circle.
Rod
Who knew That a new Phi ratio would exist for the rPi rectangle!
(repeating ratio observable in similar objects/lines of Golden rPi)
Current Phi = 1.618033988749894848204586834 ..
New iPhi = 1.913067540823067208163119533.. (approx)
'i' alludes to "impossible" squaring of the circle.
Rod

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Re: Paradise Trinity Day
Re: Golden rPi design
New iPhi = 1.913058380271100794740307828.. (approx)
Revised calculation starting with hypotenuse of right triangle
= 2,000,000,000 units and downsizing via 2(sqrt(1/Pi))
and the Pythagorean Theorem to obtain line lengths
of the golden cross in the largest rectangle.
Say what
Just ask a squared circles geometer.
Rod
New iPhi = 1.913058380271100794740307828.. (approx)
Revised calculation starting with hypotenuse of right triangle
= 2,000,000,000 units and downsizing via 2(sqrt(1/Pi))
and the Pythagorean Theorem to obtain line lengths
of the golden cross in the largest rectangle.
Say what
Just ask a squared circles geometer.
Rod

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Re: Paradise Trinity Day
Re: Golden rPi design (the wiggly numbers)
New iPhi = 1.913058380271100794740307828.. (approx)
downsize right triangles via 2(sqrt(1/Pi)):
Find length of shortest side of largest right triangle
2,000,000,000.0 = largest hypotenuse (length of longer golden line)
/ 1.1283791670955125738961589031215.. 2(sqrt(1/Pi))
= 1772453850.9055160272981674833411.. sqrt(Pi) x 1,000,000,000
Apply Pythagorean Theorem
2,000,000,000.0^2  1772453850.9055160272981674833411..^2
= 858407346410206761.53735661672066..
sqrt( ) = 926502750.35220848584275966758923.. = shortest side of large right triangle
Find length of shorter golden line (another hypotenuse)
926502750.35220848584275966758923.. = shortest side of large right triangle
x 2(sqrt(1/Pi)) = 1045446401.7541266302735942239055.. length of smaller golden line
Calculate new iPhi ratio (approx)
2,000,000,000.0 / 1045446401.7541266302735942239055..
= 1.9130583802711007947403078280203.. iPhi (approx)
Rod
New iPhi = 1.913058380271100794740307828.. (approx)
downsize right triangles via 2(sqrt(1/Pi)):
Find length of shortest side of largest right triangle
2,000,000,000.0 = largest hypotenuse (length of longer golden line)
/ 1.1283791670955125738961589031215.. 2(sqrt(1/Pi))
= 1772453850.9055160272981674833411.. sqrt(Pi) x 1,000,000,000
Apply Pythagorean Theorem
2,000,000,000.0^2  1772453850.9055160272981674833411..^2
= 858407346410206761.53735661672066..
sqrt( ) = 926502750.35220848584275966758923.. = shortest side of large right triangle
Find length of shorter golden line (another hypotenuse)
926502750.35220848584275966758923.. = shortest side of large right triangle
x 2(sqrt(1/Pi)) = 1045446401.7541266302735942239055.. length of smaller golden line
Calculate new iPhi ratio (approx)
2,000,000,000.0 / 1045446401.7541266302735942239055..
= 1.9130583802711007947403078280203.. iPhi (approx)
Rod

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Re: Paradise Trinity Day
Re: Golden rPi design
HCIT! After calculating the iPhi value, use it to confirm
length of shortest side from the Pythagorean Theorem:
Apply Pythagorean Theorem
2,000,000,000.0^2  1772453850.9055160272981674833411..^2
= 858407346410206761.53735661672066..
sqrt( ) = 926502750.35220848584275966758923.. = shortest side of large right triangle
Confirm PT calculation using iPhi
1772453850.9055160272981674833411.. length of long side
/ 1.9130583802711007947403078280203.. iPhi constant
= 926502750.35220848584275966758919.. = shortest side of large right triangle
The difference between ..8923.. and ..8919.. ?
4 (early rounding of squared circle corners)
Rod ... ... (bumping curb on quick right turns)
HCIT! After calculating the iPhi value, use it to confirm
length of shortest side from the Pythagorean Theorem:
Apply Pythagorean Theorem
2,000,000,000.0^2  1772453850.9055160272981674833411..^2
= 858407346410206761.53735661672066..
sqrt( ) = 926502750.35220848584275966758923.. = shortest side of large right triangle
Confirm PT calculation using iPhi
1772453850.9055160272981674833411.. length of long side
/ 1.9130583802711007947403078280203.. iPhi constant
= 926502750.35220848584275966758919.. = shortest side of large right triangle
The difference between ..8923.. and ..8919.. ?
4 (early rounding of squared circle corners)
Rod ... ... (bumping curb on quick right turns)

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Re: Paradise Trinity Day
Re: Golden rPi design
Signature iPhi added to design (lower left, green lines) showing
two circlesquaring right triangles adjoined at their 90 degree angle
with all line pair lengths having iPhi ratio, including hypotenuse
(blue lines) of the two triangles.
Rod
Signature iPhi added to design (lower left, green lines) showing
two circlesquaring right triangles adjoined at their 90 degree angle
with all line pair lengths having iPhi ratio, including hypotenuse
(blue lines) of the two triangles.
Rod

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Re: Paradise Trinity Day
Re: Golden rPi design ("final, final")
Golden rPi “Alpha to Omega” (Aom)
"Get a clue and devour the concept"
New millennium “golden rectangle” featuring
the circlesquaring ratio 2(sqrt(1/Pi)) = d/(a+b)
and supporting ratio iPhi = c/a = (a+b)/c
= 1.91305838027110079474030782802..
About the "critter" in lower left of design ...
"The bulldog who ate Pi" (almost daily for years)
Current collection:
http://aitnaru.org/images/Squarely_Entwined.pdf
Rod ... ... (off to get a looking glass)
Golden rPi “Alpha to Omega” (Aom)
"Get a clue and devour the concept"
New millennium “golden rectangle” featuring
the circlesquaring ratio 2(sqrt(1/Pi)) = d/(a+b)
and supporting ratio iPhi = c/a = (a+b)/c
= 1.91305838027110079474030782802..
About the "critter" in lower left of design ...
"The bulldog who ate Pi" (almost daily for years)
Current collection:
http://aitnaru.org/images/Squarely_Entwined.pdf
Rod ... ... (off to get a looking glass)

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Re: Paradise Trinity Day
Re: Golden rPi design ("final, final")
About the blue trapezoid in the lower right of design ...
Probably, a doghouse from which to escape occasional
brimstone attacks on the concept of squared circles.
Rod
Trimmed its tail a bit to not look like a cat's."The bulldog who ate Pi"
About the blue trapezoid in the lower right of design ...
Probably, a doghouse from which to escape occasional
brimstone attacks on the concept of squared circles.
Rod

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Re: Paradise Trinity Day
Re: Golden rPi design
A quick study of the iPhi concept:
http://aitnaru.org/images/Golden_rPi.pdf
Rod
A quick study of the iPhi concept:
http://aitnaru.org/images/Golden_rPi.pdf
Rod

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Re: Paradise Trinity Day
Re: Golden rPi design
(quick study of the iPhi concept)
Another page added to the PDF showing the mathematical correlation
of the circlesquaring triangle's line lengths (hypotenuselong_side vs
hypotenuseshort_side). Who knew ... that 2(sqrt(1/Pi))
would give the most convincing evidence of this correlation!
Rod
(quick study of the iPhi concept)
Another page added to the PDF showing the mathematical correlation
of the circlesquaring triangle's line lengths (hypotenuselong_side vs
hypotenuseshort_side). Who knew ... that 2(sqrt(1/Pi))
would give the most convincing evidence of this correlation!
Rod

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Re: Paradise Trinity Day
Re: Golden rPi design
(quick study of the iPhi concept)
Golden line 'e' is now identified. Despite all this Cartesian busyness,
the circle is not squared until a geometric proof is created.
The chess pieces are positioned  let the game begin!
Rod
(quick study of the iPhi concept)
Golden line 'e' is now identified. Despite all this Cartesian busyness,
the circle is not squared until a geometric proof is created.
The chess pieces are positioned  let the game begin!
Rod

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Re: Paradise Trinity Day
Re: Golden rPi design
(quick study of the iPhi concept)
Line 'f' is now identified in Golden rPi Simplified design
(to illuminate the golden diagonals object d/e):
a = f = (a+b)/(iPhi^2). Say what iPhi^2
iPhi = c/a = (a+b)/c = d/e
= 1.9130583802711007947403078280203..
^2 = 3.6597923663254876944787072692565..
Rod
(quick study of the iPhi concept)
Line 'f' is now identified in Golden rPi Simplified design
(to illuminate the golden diagonals object d/e):
a = f = (a+b)/(iPhi^2). Say what iPhi^2
iPhi = c/a = (a+b)/c = d/e
= 1.9130583802711007947403078280203..
^2 = 3.6597923663254876944787072692565..
Rod

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Re: Paradise Trinity Day
Re: GPSR Calculator design
(Golden Pi Square Root Calculator from Golden Pi rectangle geometry)
"Get a clue and devour the concept"
The new clue? The d/e object giving (a+b)/(iPhi^2)
iPhi supports a geometric square root calculator!
(that "calculates" both square and square root)
While GPSR geometry shows that such a calculator could exist,
its operation is a mystery (easier than creating a geometric proof
of a squared circle?) Let this next game begin!
Rod ... ...
(Golden Pi Square Root Calculator from Golden Pi rectangle geometry)
"Get a clue and devour the concept"
The new clue? The d/e object giving (a+b)/(iPhi^2)
iPhi supports a geometric square root calculator!
(that "calculates" both square and square root)
While GPSR geometry shows that such a calculator could exist,
its operation is a mystery (easier than creating a geometric proof
of a squared circle?) Let this next game begin!
Rod ... ...

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Re: Paradise Trinity Day
Re: GPSR Calculator design
Research indicates that its existence is also questionable;
the pattern locks on iPhi^2 and not just any square/root.
"Say what " A flash in the pan.
Rod
its operation is a mystery
Research indicates that its existence is also questionable;
the pattern locks on iPhi^2 and not just any square/root.
"Say what " A flash in the pan.
Rod

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Re: Paradise Trinity Day
Re: GPSR Calculator design
"Purpose and operation unknown!
"Begs the question: 'What's the point?'"
Indeed
Rod ... ...
"Purpose and operation unknown!
"Begs the question: 'What's the point?'"
Indeed
Rod ... ...

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Re: Paradise Trinity Day
Re: GPSR Calculator design II
Purpose and operation unknown!
Begs the question: "What's the point?"
... but answers: "What's the point?"
Rod OIC
Purpose and operation unknown!
Begs the question: "What's the point?"
... but answers: "What's the point?"
Rod OIC

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Re: Paradise Trinity Day
Re: Golden rPi design
1. Prove that squared circles exist (geometrically)*
2. Square the circle by the Greek rules.
Methinks that Part 1 is easier than Part 2 ...
and should have the greater priority.
* Not to worry that, mathematically, this is "impossible"
(making Part 2 a futile journey ... also).
Rod
Actually, this is a twopart challenge:the circle is not squared until a geometric proof is created
1. Prove that squared circles exist (geometrically)*
2. Square the circle by the Greek rules.
Methinks that Part 1 is easier than Part 2 ...
and should have the greater priority.
* Not to worry that, mathematically, this is "impossible"
(making Part 2 a futile journey ... also).
Rod

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Re: Paradise Trinity Day
Re: Golden rPi design
Actually, this is a twopart challenge:
1. Prove that squared circles exist (geometrically)
2. Square the circle by the Greek rules.
Constant 2(sqrt(1/Pi))* suggests that sqrt(Pi) can be
represented as the long side of a circlesquaring
right triangle whose hypotenuse = 2.0 (diameter).
* 2(sqrt(1/Pi)) = 2(sqrt(Pi))/Pi = sqrt(Pi)/(Pi/2)
= 1.1283791670955125738961589031215..
Rod ... ...
Actually, this is a twopart challenge:
1. Prove that squared circles exist (geometrically)
2. Square the circle by the Greek rules.
Constant 2(sqrt(1/Pi))* suggests that sqrt(Pi) can be
represented as the long side of a circlesquaring
right triangle whose hypotenuse = 2.0 (diameter).
* 2(sqrt(1/Pi)) = 2(sqrt(Pi))/Pi = sqrt(Pi)/(Pi/2)
= 1.1283791670955125738961589031215..
Rod ... ...

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Re: Paradise Trinity Day
Re: The Right Triangle design
Long Story Short (aka, "Period. End of Story"):
If this right triangle does not exist in a squared circle,
the circle is not squared according to this geometry.
Rod
Long Story Short (aka, "Period. End of Story"):
If this right triangle does not exist in a squared circle,
the circle is not squared according to this geometry.
Rod

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Re: Paradise Trinity Day
Re: The Right Triangle design
Embellished design with geometric LOIN (Lines Of Interest Noteworthy)
since the integrated scalene triangle also squares the circle.
Rod
Embellished design with geometric LOIN (Lines Of Interest Noteworthy)
since the integrated scalene triangle also squares the circle.
Rod

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Re: Paradise Trinity Day
Re: Three Pi Vise design
"See how they run! ... in the 'impossible' House of Pi.
Diameters = Pi/2, sqrt(Pi), 2.0; Paramount ratio: 2(sqrt(1/Pi))"
Also online: http://aitnaru.org/threepoints.html
"Answers 'What’s the point?' (about squaring circles). Right triangle (3 points)
contains defining hypotenusetolongside ratio 1.128379167095512573896..
= 2(sqrt(1/Pi)) = sqrt(Pi)/(Pi/2)"
Golden diagonals of flutterby also have 2(sqrt(1/Pi)) relationship.
Rod ... ...
"See how they run! ... in the 'impossible' House of Pi.
Diameters = Pi/2, sqrt(Pi), 2.0; Paramount ratio: 2(sqrt(1/Pi))"
Also online: http://aitnaru.org/threepoints.html
"Answers 'What’s the point?' (about squaring circles). Right triangle (3 points)
contains defining hypotenusetolongside ratio 1.128379167095512573896..
= 2(sqrt(1/Pi)) = sqrt(Pi)/(Pi/2)"
Golden diagonals of flutterby also have 2(sqrt(1/Pi)) relationship.
Rod ... ...

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Re: Paradise Trinity Day
Re: Three Pi Vise design
"The crown awaits the king (of a geometric proof?)"
Geometers' tip:
These line length ratios exist in the "crown":
1.1283791670955125738961589031215.. 2(sqrt(1/Pi))
^2 = 1.2732395447351626861510701069801..
^3 = 1.4366969769991930158060821722332..
Rod
"The crown awaits the king (of a geometric proof?)"
Geometers' tip:
These line length ratios exist in the "crown":
1.1283791670955125738961589031215.. 2(sqrt(1/Pi))
^2 = 1.2732395447351626861510701069801..
^3 = 1.4366969769991930158060821722332..
Rod

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Re: Paradise Trinity Day
Re: Three Pi Vise design
"The crown awaits the king (of a geometric proof?)"
Geometers' tip #2:
With Diameters = Pi/2, sqrt(Pi), 2.0
and the recently revealed constants:
1.1283791670955125738961589031215.. 2(sqrt(1/Pi))
1.9130583802711007947403078280203.. "iPhi"
and the Pythagorean Theorem: a^2 + b^2 = c^2
Length of flutterby diagonals can be calculated
with their ratio then proven mathematically
= 1.1283791670955125738961589031215..
Easy enough once the right triangles are selected
for this short mathematical journey!
Rod
"The crown awaits the king (of a geometric proof?)"
Geometers' tip #2:
With Diameters = Pi/2, sqrt(Pi), 2.0
and the recently revealed constants:
1.1283791670955125738961589031215.. 2(sqrt(1/Pi))
1.9130583802711007947403078280203.. "iPhi"
and the Pythagorean Theorem: a^2 + b^2 = c^2
Length of flutterby diagonals can be calculated
with their ratio then proven mathematically
= 1.1283791670955125738961589031215..
Easy enough once the right triangles are selected
for this short mathematical journey!
Rod

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Re: Paradise Trinity Day
Re: Three Pi Vise design
"The crown awaits the king (of a geometric proof?)"
Who knew
The longer diagonal is the diameter of a circle
and the shorter diagonal has length equal
to a side of that circle's square!
That's why their length ratio is 2(sqrt(1/Pi))
"Three Pi Vise ... See how they run!"
Did you ever see such a sight in your life!
Rod
"The crown awaits the king (of a geometric proof?)"
Who knew
The longer diagonal is the diameter of a circle
and the shorter diagonal has length equal
to a side of that circle's square!
That's why their length ratio is 2(sqrt(1/Pi))
"Three Pi Vise ... See how they run!"
Did you ever see such a sight in your life!
Rod