Paradise Trinity Day

[Amigoo]

Who knew In this local universe, Patterns Rule!

Fourteen decimal digits of Pi is a better symbolic plateau!
A different pattern alignment: https://ufile.io/s6a6x

Regardless of numeric pattern implications,
this .70.. divisor is both entertaining and Quadraturial.

Rod ("Been there! Done that!")

[Amigoo]

Re: Cartesian Squares Inscribed design
(updated in: http://aitnaru.org/images/Khristos_Voskrese.pdf )

That 1/7 is Quadraturial - even mystical and ethereal -
suggests potential of this pattern: https://ufile.io/s6a6x

That Orvonton is seventh and one of seven (re: UB 15:3)
might allude to first ethereal greeting from our distant HQ.

Rod

[Amigoo]

Re: Cartesian Squares Inscribed design
and Quadraturial, mystical, ethereal ratio 1/7

Why stop at 16 decimal digits of Pi

Re: https://www.jpl.nasa.gov/edu/news/2016/ ... ally-need/

"For JPL's highest accuracy calculations, which are for interplanetary navigation, we use
3.141592653589793."
3.141592653589793 2384626433832795.. (more Pi)

From decimal digits pattern at https://ufile.io/x1n6f
Pi = 3.1415926535897932 384626433832795..
991148575128553 / 7000000000000000 = 0.1415926535897932 8571428571428571..
355/113 = 3.1415929203539823 008849557522124..

Rod ... ... (Project odometer just rolled over!)

[Amigoo]

Re: Cartesian Squares Inscribed design
and Quadraturial, mystical, ethereal ratio 1/7

Geometers' late night secret (re: https://ufile.io/dwilt ) ...
The original pattern table is "mystical" only because the pattern is not obvious.

The precise table is simply the decimal digits of Pi multiplied by 70.. (one or more zeros)
... although it's still "Quadraturial" (because Pi is Quadraturial).

Rod

[Amigoo]

Regarding sqrt(Pi) and sqrt(2) ...
(calculate up)

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

= 1.4142135623730950488016887242097.. sqrt(2)
x 1.1283791670955125738961589031215..
1.2533141373155002512078826424055..

aka "Pi Sandwich" (between sqrt(2) and 2.0)

Rod

[Amigoo]

Regarding "Pi in the sky" ...

Pi = 3.1415926535897932384 626433832795..
= 3.0
+ 1 / 10
+ 4 / 100
+ 1 / 1000
+ 5 / 10000
+ 9 / 1000000
+ 2 / 10000000
+ 6 / 100000000
+ 5 / 1000000000
+ 3 / 10000000000
+ 5 / 100000000000
+ 8 / 1000000000000
+ 9 / 10000000000000
+ 7 / 100000000000000
+ 9 / 1000000000000000
+ 3 / 10000000000000000
+ 2 / 100000000000000000
+ 3 / 1000000000000000000
+ 8 / 10000000000000000000
+ 4 / 100000000000000000000

If a circle's diameter = 2 inches, final + 4 decimal digit
= 0.00000000000000000008 inch (which cannot be drawn with paper and pencil).
So, increase diameter to 200000000000000000000 inches and + 4 length = 16 inches,
and giving circle's diameter = 3156565656565656.56.. miles!
(distance from earth to sun = 92960000 miles)

Analysis: Ethereal Pi (aka "Pi in the sky") becomes "apples and oranges"
when represented by Cartesian geometry! Besides, more than a handful
of Pi's decimal digits alludes to atomic sizes (where measurement
of boundaries is theoretical ... as is Pi).

See also: https://www.rd.com/culture/why-5280-feet-in-mile/
(when "feet" were real feet)

Caution: If there's not a "typo" in these figures,
then I'll be an uncle's monkey.

Rod

[Amigoo]

The upwards math of squared circles
(calculate up in each group)

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

as supported by ...

= 2.0
x 1.4142135623730950488016887242097.. sqrt(2)**
1.4142135623730950488016887242097.. sqrt(2)

** because this sqrt(2) is the product of:

= 1.4142135623730950488016887242097.. sqrt(2)
x 1.1283791670955125738961589031215.. *
1.2533141373155002512078826424055.... *

* because multiplicand/multiplier are quotients of:

= 1.1283791670955125738961589031215..
/ 1.7724538509055160272981674833411.. sqrt(Pi)
2.0

= 1.2533141373155002512078826424055..
/ 1.4142135623730950488016887242097.. sqrt(2)
1.7724538509055160272981674833411.. sqrt(Pi)

Rod ... ... (headin' down the upwards)

[Amigoo]

Regarding sqrt(Pi) and sqrt(2) ...
"The Very Short Story of Two Pi"

As two right triangles, inscribed in the same circle of diameter = 2.0, the circle-squaring right triangle contains Pi (as represented by sqrt(Pi), long side with length equal to a side of the circle's area square) and a hypotenuse as the circle's diameter. The other triangle is an isosceles right, also with its hypotenuse as the circle's diameter.

Since the length of the Pi line must be greater than sqrt(2) and less than 2.0 (re: isosceles right triangle), any focus on the transcendental digits of Pi is akin to comparing two similar four-legged creatures, but contrasting their tails not their heads! Because the Pi line must have constrained length, Pi's long-touted transcendence, apparently, is all about splitting hairs on the tale of two Pi (IMO).

Rod

[Amigoo]

"The Very Short Story of Two Pi"

Re: Tale of Two Pi (from Cartesian Squares Inscribed)
illustrates these right triangles' Pi line length constraint,
proving that sqrt(2) is a resident of this Neighborhood.

Rod

[Amigoo]

"The Very Short Story of Two Pi"

Chapter 1 - Pi in the Sqrt(2) Continuum

Code: Select all

Sq    Sqrt
---   -----
4.0___2.0
3.61__1.9
3.24__1.8
3.14__1.7724538509055160272981674833411..  
3.0___1.7320508075688772935274463415059..
2.89__1.7
2.56__1.6
2.25__1.5
2.0___1.4142135623730950488016887242097..

Rod

[Amigoo]

"The Very Short Story of Two Pi"

Chapter 1 - Pi in the Sqrt(2) Continuum

Code: Select all

  Sq     Sqrt
------  ------
4.0_____2.0
3.61____1.9
3.24____1.8
3.2041__1.79  
3.1400__1.78  
3.1415__1.7724538509055160272981674833411..
3.1684__1.76  
3.0625__1.75  
3.0276__1.74  
3.0_____1.7320508075688772935274463415059..
2.9584__1.72
2.9241__1.71    
2.89____1.7
2.56____1.6
2.25____1.5
2.0_____1.4142135623730950488016887242097..

__7 / 22  = 0.31818181818181818181818181818182..  
__1 / Pi  = 0.31830988618379067153776752674503..
113 / 355 = 0.31830985915492957746478873239437..

Rod

[Amigoo]

Re: Three Squares, Pi II design
(updated: http://aitnaru.org/images/Khristos_Voskrese.pdf )

A salient year-end project summary
"Squared-circle geometry that speaks for itself" ...
about the unique relationship of 2.0, sqrt(Pi), Pi/2

Rod ... ...

[Amigoo]

Re: Three Squares, Pi II design

Now, with a Quadraturial "lock".

Rod

[Amigoo]

Re: Three Squares, Pi II design

Now, with a Quadraturial "lock".

Geometers' secret (about golden 'X' shape lock):
Top and bottom lines have sqrt(Pi) length relationship;
diagonal lines have 2/sqrt(Pi) length relationship.

sqrt(Pi) = 1.7724538509055160272981674833411..
2/sqrt(Pi) = 1.1283791670955125738961589031215..

Rod ... ...

[Amigoo]

Re: Three Squares, Pi II design

Now, with a Quadraturial "lock".

Geometers' secret:
Astute geometers have noticed that this Neighborhood
is similar to those earlier that host a "Pi Fork".

Rod

[Amigoo]

Re: Three Squares, Pi II design

"Three squares?! There are more than three!
... there are squares from here to there!"

The beginning 3 confirm line length (and angle) ratios
that propagate Quadrature all over this Neighborhood.

Rod

[Amigoo]

Re: Three Squares, Pi II design

this Neighborhood is similar to those earlier that host a "Pi Fork"

About the Pi Fork ...
All lines participate in Quadrature-defining
line length ratios: 2, sqrt(Pi), 2/sqrt(Pi),
within influence of the sqrt(2) host.

Rod

[Amigoo]

Re: Three Squares, Pi II design

this Neighborhood is similar to those earlier that host a "Pi Fork"

Geometers' secret extraordinaire ...
sqrt(Pi)/2 = 0.88622692545275801364908374167057..

Angles in all circle-squaring right triangles
90.0 degrees
62.402887364309395548267795247661..
27.597112635690604451732204752339.. acos(sqrt(Pi)/2)

Rod

[Amigoo]

Re: Plane2C design

“Squared circle geometry that speaks for itself”
… and it's Plane2C!

Two-dimensional labryrith with three-dimensional ambience,
featuring "Lines! and Triangles! and Squares!, Oh Pi!" ...
not to mention trapezoids, parallelograms, and objects
that exist in "impossible" circles all squared.

Rod

[Amigoo]

Re: Plane2C design

“Squared circle geometry that speaks for itself”
… and it's Plane2C!

Well, it would make sense ...
that Quadrature identifies with an Einstein–Rosen bridge!
Enter at your own risk (of believing in the "impossible")

Rod

[Amigoo]

Re: Plane2C design

“Squared circle geometry that speaks for itself”
… and it's Plane2C!

Who knew?! This geometry defines a new polyhedron,
a "quadrahedron" if Quadrature is acknowledged.

Rod

[Amigoo]

Re: Plane2C_iQ design ("impossible" Quadrature)

“Squared circle geometry that speaks for itself”
and it's Plane2C ... a quadrahedron!

Not all lines of interest are highlighted,
but enough to start visualizing the "impossible"
... and counting the geometric faces.

Rod ... ...

[Amigoo]

Re: Plane2C_iQ design ("impossible" Quadrature)

“Squared circle geometry that speaks for itself”
and it's Plane2C ... a quadrahedron!

Geometers' secret ...
Two fan-shape objects overlap with each object having three adjoined triangles
and each triangle containing a 45-degree angle. Perimeter of this juxtaposition
contains six connecting lines having equal length.

... juxtapostional complexity expected o'er an E-R Bridge.

Rod

[Amigoo]

Re: Plane2C_iQ design ("impossible" Quadrature)

“Squared circle geometry that speaks for itself”
and it's Plane2C ... a quadrahedron!

Apparently, this geometry defines a dodecahedron: 12 faces, 6 per side
with each side having five 45-degree angles and one 135-degree angle.

Who knew?! The first starfish o'er the ER-Bridge!

Rod

[Amigoo]

Re: Plane2C_iQ design ("impossible" Quadrature)

Apparently, this geometry defines a dodecahedron

Re: http://www.dodecahedron.us/Dodecahedron.html

"The Dodecahedron has been a source of metaphysical interest for at least 2000 years. Like a crystal or gem, its facets and symmetries compel our eyes and hearts to observe life more deeply. Some have believed that the Dodecahedron represents an idealized form of Divine thought, will, or idea. To contemplate this symbol was to engage in meditation upon the Divine."

Rod