Paradise Trinity Day

[Amigoo]

Re: OO Flutterby! design

Two circles sharing a side of their inscribed squares (SoIS).
Another Pi "sandwich" enclosed by sqrt(2) and defining a "flutterby"
(includes SoIS, circle-squaring right triangle, and circle's diameter)

Given: Diameter = 2(sqrt(2)), SoIS = 2.0

2.0
x 1.2533141373155002512078826424055.. (sqrt(Pi)sqrt(2))/2
= 2.506628274631000502415765284811.. sqrt(Pi)sqrt(2)
x 1.1283791670955125738961589031215.. 2/sqrt(Pi)
= 2.8284271247461900976033774484193.. 2(sqrt(2))

2.8284271247461900976033774484193.. 2(sqrt(2))
/ 2.0
= 1.4142135623730950488016887242097.. sqrt(2)

Geometers' secret ...
The "flutterby" incorporates a 45-degree angle
divided into 27.597.. and 17.402.. wedgies.
(27.597.. is smallest angle of that right triangle
and is the Quadraturial cosine angle)

Rod ... ...

[Amigoo]

Re: OO Flutterby! design,
added to: http://aitnaru.org/images/Pi_Fork_n_Lute.pdf

Quadraturial simplicity

Geometers' secret ...
All three circles are effectively squared.

Rod

[Amigoo]

Re: OO! Flutterby design

Quadraturial simplicity

... relatively speaking (now there are two).

Two circles sharing a side of their inscribed squares (SoIS).
Another Pi "sandwich" enclosed by sqrt(2) and defining
two "flutterby" (includes circle-squaring right triangle,
SoIS, and circle's Diameter)

Given: Diameters = sqrt(2), 2(sqrt(2)), SoIS = 2.0
(SoIS = Side of Inscribed Square)

1.4142135623730950488016887242097.. sqrt(2)
x 1.2533141373155002512078826424055.. (sqrt(Pi)sqrt(2))/2
= 1.7724538509055160272981674833411.. sqrt(Pi)
x 1.1283791670955125738961589031215.. 2/sqrt(Pi)
= 2.0
x 1.2533141373155002512078826424055.. (sqrt(Pi)sqrt(2))/2
= 2.506628274631000502415765284811.. sqrt(Pi)sqrt(2)
x 1.1283791670955125738961589031215.. 2/sqrt(Pi)
= 2.8284271247461900976033774484193.. 2(sqrt(2))

2.8284271247461900976033774484193.. 2(sqrt(2))
/ 1.4142135623730950488016887242097.. sqrt(2)
= 2.0

Rod

[Amigoo]

Re: OO! Flutterby design
(now more "simplified")

What to expect when crop circles know something about Quadrature.
And that should be interesting since human circle makers will probably
not yet comprehend the "simple" secrets of Quadrature.

Rod ("That's sum Pi!")

[Amigoo]

Re: OO! Flutterby design
(now more "simplified")

Without a doubt, 2/sqrt(Pi) is the alpha of "simple secrets",
for it defines the right triangle that squares the circle (aka Quadrature)
... and right triangles can perform Cartesian magic.

2/sqrt(Pi) = 1.1283791670955125738961589031215..

Rod

[Amigoo]

Re: Phi of Pi design

sqrt(Pi)/sqrt(4-Pi) = Phi of Pi
= 1.9130583802711007947403078280203..

The "golden rectangle" associated with the 2/sqrt(Pi) constant
finally arrived ... and the geometry simply speaks for itself!
(let's first hear what it has to say in social circles squared)

Rod

[Amigoo]

Re: Phi of Pi design

sqrt(Pi)/sqrt(4-Pi) = Phi of Pi

Regarding circle-squaring right triangle where D = 2 ...
1.7724538509055160272981674833411.. sqrt(Pi), long side
/ 0.92650275035220848584275966758914.. short side
= 1.9130583802711007947403078280203.. Phi of Pi
= sqrt(Pi)/sqrt(4-Pi)

Geometers' secret ...
Ratio of rectangle's long side to short side is similar to ratio of perpendicular diagonals.

Re: https://www.independent.co.uk/news/scie ... 04354.html

"Theories that the Parthenon in Athens and Great Pyramid in Egypt were built according to the golden ratio
have also been disproved, he said. 'The golden ratio stuff is in the realm of religious belief. People will argue
it is true because they believe it, but it’s just not fact.'"

"Mathematicians have their heads up their asses about half the time."

Probably poor cart design, allowing it to suddenly advance toward the towing donkey.
Fortunately, some critics are familiar with asses and will report even de tail.

Rod ... ...

[Amigoo]

Re: Phi of Pi design

sqrt(Pi)/sqrt(4-Pi) = Phi of Pi

= 1.9130583802711007947403078280203..
^2 = 3.6597923663254876944787072692565..

While not qualified to own a cart (with donkey), but good observer of Cartesian patterns,
I'm happy to report more de tail: The difference of the largest and smallest circle is Phi of Pi ^2.
"Come, and you will see."

Rod

[Amigoo]

Re: Phi of Pi design*

sqrt(Pi)/sqrt(4-Pi) = Phi of Pi
= 1.9130583802711007947403078280203..

Phi is impressive but Phi of Pi follows Quadrature to infinity.

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

Rod ... ...

[Amigoo]

Re: Phi of Pi design

Phi of Pi = sqrt(Pi) / sqrt(4-Pi)
= 1.9130583802711007947403078280203..
Qcue = 2/sqrt(Pi) = sqrt(Pi)/(Pi/2)
= 1.1283791670955125738961589031215..

Recipe perfected! - now served with a Pi Fork.
Not yet on any menu - ask for "Pi Fork 'n Phi".

Rod

[Amigoo]

Re: Phi of Pi design

Phi of Pi = sqrt(Pi) / sqrt(4-Pi)
= 1.9130583802711007947403078280203..

"Phi exists when a line is divided into two parts and the longer part (a)
divided by the smaller part (b) is equal to the sum of (a) + (b) divided by (a)"
(from: https://www.canva.com/learn/what-is-the-golden-ratio/ )

Such evaluation of Phi of Pi requires consideration of rectangle/circle integration,
suggesting that the right triangle - not straight lines - is the necessary focus.

Rod

[Amigoo]

Re: Phi of Pi design

evaluation of Phi of Pi requires consideration of rectangle/circle integration

Since Phi of Pi is all about Quadrature, "Qphi" is more descriptive ...
and "Qcue" refers to the Quadrature-defining, circle-squaring right triangle:

Qphi = sqrt(Pi) / sqrt(4-Pi)
= 1.9130583802711007947403078280203..

Qcue = 2/sqrt(Pi) = sqrt(Pi)/(Pi/2)
= 1.1283791670955125738961589031215..

Who can tell Contemplation of Qphi
is tantamount to contemplation of infinity
... and he/she who created it.

Rod (still mindin' my Ps and Qs)

[Amigoo]

Re: Phi of Pi design

"Phi exists when a line is divided into two parts and the longer part (a)
divided by the smaller part (b) is equal to the sum of (a) + (b) divided by (a)"

First draft of this concept as applied to Phi of Pi:

"The circle-squaring right triangle's long side is to its short side
as its hypotenuse is to the diagonal of the upsized or downsized
(via Qphi) 'golden rectangle' of Quadrature."

Rod

[Amigoo]

Re: Phi of Pi design

More lines added ... in the spirit of Morbus Cyclometricus.

Rod ... ...

[Amigoo]

Re: Phi of Pi design

... in the spirit of Morbus Cyclometricus ...
"The circle cannot be squared, but squared circles exist ... apparently."

Re: https://kiwihellenist.blogspot.com/2019 ... 4345231487

"there are other ratios in the neighbourhood of 1.6.
Always be alert for cherry-picking."

Where's the supporting geometry for showComment=156..
... since you might discover a new recipe for Cherry Pi!

"My Pi! My Pi!" - anonymous (again)

Rod ... ...

[Amigoo]

Re: Phi of Pi design

... in the spirit of Morbus Cyclometricus ...
"The circle cannot be squared, but squared circles exist ... apparently."

There's an excellent discussion of Pi in Chapter Four,
"Life of Pi", in "Here's looking at Euclid" by Alex Bellos:
https://www.amazon.com/Heres-Looking-Eu ... 1416588280

The discussion of Pi's transcendental aspect contrasts with Phi:
unlike Pi, Phi is not constrained by the circle (the definition of Pi)
... yet Pi is transcendental and Phi is not.

Rod

[Amigoo]

Re: Phi of Pi design

... in the spirit of Morbus Cyclometricus ...
"The circle cannot be squared, but squared circles exist ... apparently."

Qphi = Phi of Pi = sqrt(Pi)/sqrt(4-Pi) = sqrt(Pi/4-Pi)
= 1.9130583802711007947403078280203..

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

The obvious when contemplating the correlation of Qphi and Qcue:
These two constants both define the same circle-squaring right triangle
... of course, with similar angles: 90, 27.597.., 62.402..

Rod

[Amigoo]

Re: Phi of Pi design

... in the spirit of Morbus Cyclometricus ...
"The circle cannot be squared, but squared circles exist ... apparently."

Qphi and Qcue both define same circle-squaring right triangle
... and trigonometry can prove this ("Go figure!"):

Given: Pythagorean right triangle inscribed in circle where D = 2

Quadrature Angles (degrees):
27.597112635690604451732204752339.. (ref. angle)
62.402887364309395548267795247661..
90.0

Quadrature Sides (hypotenuse, opposite, adjacent)
h = 2.0 (= diameter)
a = 0.92650275035220848584275966758914.. sqrt(4-Pi)
b = 1.7724538509055160272981674833411.. sqrt(Pi)

Qphi = sqrt(Pi) / sqrt(4-Pi)
= 1.9130583802711007947403078280203..

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

Rod ... ...

[Amigoo]

Re: Phi of Pi design

Quadrature Sides (hypotenuse, opposite, adjacent)
h = 2.0 (= diameter), a = sqrt(4-Pi), b = sqrt(Pi)

Given: a^2 + b^2 = c^2, therefore (4-Pi) + Pi = 4, then 2 = 2.
Who knew That a circle-squaring right triangle eats Pi.
Pythagoras?

Rod

[Amigoo]

Re: Pythagorean Quadrature design
"... but you can't get there from here!"

However ... notice how intuitive this geometry is
A circle whose Diameter equals 2, must have SoCS equal to sqrt(Pi).

Since sqrt(Pi)^2 = Pi, the remaining square (re: Pythagorean Theorem)
must equal 2^2 - Pi. Geometry 101!

Rod ... ... (headin' there but still here)

[Amigoo]

Re: Pythagorean Quadrature design
"You can't get there from here! (inside the box)"

Green diagonals added to show (a+b)/sqrt(2) of circle-squaring right triangle
defines the trapezoid that includes Quadrature's unique scalene triangle.

Rod

[Amigoo]

Re: Pythagorean Quadrature design*
"You can't get there from here! (inside the box)"

Green diagonals now show confirmation of trapezoids
(the right triangles actually define the trapezoids).

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

Rod ... ...

[Amigoo]

Re: Pythagorean Quadrature design
"You can't get there from here! (inside the box)"

Who knew
The three squares of the Pythagorean Theorem define a trapezoid!
... of course, the effective square of a larger circle.

Rod

[Amigoo]

Re: Pythagorean Quadrature design

You can't get there from here! (from inside the box)

2 and sqrt(2) is at least Ethereal
if Pi and sqrt(Pi) is Transcendental ...
suggesting a Quadraturial E.T.

Rod ... ... (off to get Pieces for the next visit)

[Amigoo]

Re: Pythagorean Quadrature design

You can't get there from here! (from inside the box)

Unexpected late night E.T. enlightenment ...
This Cartesian composition defines a trapezoid
having a bottom-to-top line length ratio of 1.6

Fascinating and ethereal - even transcendental

Rod