Paradise Trinity Day

[Amigoo]

Re: Pythagorean Pi-ramid design

Bedtime snack (algebraic Pi):
2/(Pi/2) = (2/sqrt(Pi))^2

2 / 1.5707963267948966192313216916398..
= 1.2732395447351626861510701069801..
= (2 / 1.7724538509055160272981674833411..)^2
= (1.1283791670955125738961589031215..)^2

Rod

[Amigoo]

Re: Pythagorean Pi-ramid design

Bedtime snack (more algebraic Pi):
Pi - (Pi/2 + Pi/4 + Pi/8 ...) = 0

0000 3.1415926535897932384626433832795.. Pi

0002 1.5707963267948966192313216916398.. Pi/2
0004 0.7853981633974483096156608458198.. Pi/4
0008 0.3926990816987241548078304229099.. Pi/8
0016 0.1963495408493620774039152114549.. Pi/16
0032 0.0981747704246810387019576057274.. Pi/32
0064 0.0490873852123405193509788028637.. Pi/64
0128 0.0245436926061702596754894014318..
0256 0.0122718463030851298377447007159..
0512 0.0061359231515425649188723503579..
1024 0.0030679615757712824594361751789..
2048 0.0015339807878856412297180875894..
4096 0.0007669903939428206148590437947..
8192 0.0003834951969714103074295218973..

Rod ... ... (queued for the 1st slice)

[Amigoo]

Re: Pythagorean Pi-ramid design
(embellished, suggesting "As above, So below")

When squared circle geometry explains "Who's on first?",
the Cartesian Neighborhood gets mysteriously esoteric
'2' and sqrt(2) unite to illuminate in this "Pi Corral".

Rod

[Amigoo]

Re: Pythagorean Pi-ramid design "final, final"
Updated in: http://aitnaru.org/images/Sqrt_Pi_Ratios.pdf

Rising sqrt(2) decrements to infinity contained by largest circle
and confirming "transcendental Pi" ... within a "Pi Corral".

Geometers' secret:
Pi - (Pi/2 + Pi/4 + Pi/8 ...) = 0
is true only if Pi is transcendental.

Rod ... ...

[Amigoo]

Re: Pythagorean Pi-ramid design

Pi = 3.141592653589793238462643383279502884197169399375105820974944..

Maximum sequence per digits 0-9 in one million decimal digits of Pi:
00000,111111,222222,3333333,444444,555555,666666,777777,888888,999999
(00000 = 5 digits, 3333333 = 7 digits, all others = 6 digits)

100+ decimal digits hints that math disconnects from practical geometry
... since when is atomic-level measurement real-world geometry

Rod

[Amigoo]

Re: Pythagorean Pi

Important sqrt(Pi) ratios:

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

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

= 1.7724538509055160272981674833411.. sqrt(Pi)
x 1.2533141373155002512078826424055..
<-- 1.4142135623730950488016887242097.. sqrt(2)
/ 1.2533141373155002512078826424055..
= 1.1283791670955125738961589031215.. 2/sqrt(Pi)

1.7724538509055160272981674833411.... sqrt(Pi)
/ 1.1283791670955125738961589031215.. 2/sqrt(Pi)
= 1.5707963267948966192313216916398.. Pi/2

1.1283791670955125738961589031215..^2
= 1.2732395447351626861510701069801..
= 2/(Pi/2) = (2/sqrt(Pi))^2

Rod

[Amigoo]

Re: Pi Ratios 1128 design (re: 2/sqrt(Pi))

The algebraic 'x' prefixed to 'ode' (fable or story) ...
or squared circle geometry that "speaks for itself",
or urban "complex and sophisticated algorithm".

Rod ... ... (cruisin' "strophe, antistrophe, epode")

[Amigoo]

Re: Pi Ratios (x+ode) design
(renamed from "Pi Ratios 1128")

Geometers' secret:
This esoteric geometry is all about the "two-tine fork"
where three two-line pairs have the sqrt(Pi) ratio
in that Pythagorean circle-squaring right triangle.

Rod

[Amigoo]

Re: Two-Tine PR design
Answers “What's the point?”

(long sides of circle-squaring right triangle adjoined
by a shorter line creating a sqrt(Pi) relationship;
short side similarly related to its opposite side)

and just in time for this 2018 Pi Day (March 14th)
(Peye Day according to Pythagorean Pi-ramid)

Rod

[Amigoo]

Re: Two By Two design

Who knew that squared circle geometry highlights the obvious:
Pi is directly associated with sqrt(2); as circle's diameter increases by 2,
circle's square (Side of Circle's Square) also increases by 2 ... precisely
... and vice versa.

Rod

[Amigoo]

Re: Two By Two design
"sqrt(Pi) and sqrt(2), united in quadrature"

What's impossible about squared circle geometry
is trying to study it without colored lines, especially
when circles are nesting in the Neighborhood.

Rod ... ...

[Amigoo]

Re: Two By Two design ("final, final")
"via sqrt(2) and seraphic blue"

"Seraphic blue" refers to the overlapping, blue "flutterby" objects
that might be symbolic representation (IMO) of the seraphim pair
(guardian angels) that serve humans. After all, would not
seraphim be associated with this "impossible" geometry ...
and desire recognition on Pi Day (March 14th)?

Rod

[Amigoo]

About Pi Day 2018 (March 14th) ...

A Brief History of Pi (known about 4000 years):
https://www.exploratorium.edu/pi/history-of-pi

Pi Day concept born 1988 at Monterey, California,
becoming an official U.S. national holiday March 2009:
https://www.exploratorium.edu/pi/pi-day-history

Who knew Release the blue butterflies on Pi Day!
(authentic slice is a circle-squaring right triangle)

Rod

[Amigoo]

Re: Two-Tine Enterprise design
(renamed from Two-Tine PR)

Answers "What's the point?"

Long sides of circle-squaring right triangle adjoined by a shorter line,
creating sqrt(Pi) relationship; short side similarly related to opposite side
... and all supported by quadrature-definng ratio 2/sqrt(Pi)
= 1.1283791670955125738961589031215..

Note: Several circle-squaring right triangles exist -
this focus is the large RT outlined with green lines.

Rod ... ...

[Amigoo]

Re: Two-Tine Enterprise design

Intriguing but unintentional abstract symbolism of Hawking in a wheel chair, considering that
this "impossible" geometry correlates with Hawking's "impossible" 55 years of life long after
his 1963 two-years-to-live diagnosis of motor neurone disease.

Re: http://www.dailymail.co.uk/news/article ... ad-76.html

Excellent coverage of Stephen Hawking's life (he departed just "hours before Pi Day").

Re: https://www.amazon.com/Theory-Everythin ... _twi_dvd_2

Hawking was very pleased with Redmayne's portrayal of him,
saying "At times, I thought he was me."

Rod ... ...

[Amigoo]

iGolden Ratio of Squared Circles
('i'= "impossible" and "outside the box")

Re: Golden Ratio: "a + b is to a, as a is to b"

Given: circle-squaring right triangle, where
b = short side, a = long side, c = hypotenuse,
c = circle's diameter, a = Side of Circle's Square (SoCS),
c/a = 1.1283791670955125738961589031215.. = 2/sqrt(Pi)

c = 2.5464790894703253723021402139604.. 2((2/sqrt(Pi)^2)
a = 2.256758334191025147792317806243.. 2(2/sqrt(Pi))
b = 1.1796599400543220225826496809606.. Pythagorean Theorem
a/b = 1.9130583802711007947403078280194.. "iPhi"

iGolden = ((a+b)/b)-1 =
2.256758334191025147792317806243..
+ 1.1796599400543220225826496809606..
= 3.4364182742453471703749674872036..
/ 1.1796599400543220225826496809606..
= 2.9130583802711007947403078280194..
-1 = 1.9130583802711007947403078280194..

a/b = 1.9130583802711007947403078280194..

Rod (and math needs verification)

[Amigoo]

iGolden Ratio of Squared Circles

The math is confirmed (it's just algebraic magic):
((a+b)/b)-1 = (a/b)+(b/b ) -1 = a/b

Translation: Go back to the electronic abacus.
(but a/b is a still a special ratio)

Rod

[Amigoo]

Re: Two-Tine Enterprise design (Squared Circles 101)

Given: circle-squaring right triangle where c = 2, a = sqrt(Pi),
b = sqrt(c^2 - a^2) from Pythagorean Theorem

4 - 3.1415926535897932384626433832795..
= 0.8584073464102067615373566167205.. = b^2
sqrt( = 0.92650275035220848584275966758914.. = b

Ratio of long side to short side = a/b
= 1.7724538509055160272981674833411..
/ 0.92650275035220848584275966758914..
= 1.9130583802711007947403078280204.. = a/b

Ratio of hypotenuse to long side = 2/sqrt(Pi)
= c/a = 1.1283791670955125738961589031215..

Rod ... ...

[Amigoo]

Re: Squared Circles 101 design

Geometry that highlights the consistent squared circles ratios.
Line lengths of golden "cross" over trapezoid have 2:1 ratio.

Coincidental angle of cross identifies with Easter celebration.

Rod

[Amigoo]

Re: Trapezoidal Inclination design (renamed from Squared Circles 101)
Now added to http://aitnaru.org/images/Sqrt_Pi_Ratios.pdf

Focus on ratio 1.9130583802711007947403078280203..
= long-side-to-short-side in circle-squaring right triangle.

Geometers' secret: Bottom and top lines of trapezoid
have length ratio 1.913.. as well as long-side-to-short-side
of circle-squaring right triangle (red); triangle and trapezoid
are geometrically related ... obviously.

Rod ... ...

[Amigoo]

Re: Trapezoidal Inclination design (recap)

Squared circle geometry inclined to speak for itself,
with focus on ratio 1.9130583802711007947403078280203..
= long-side-to-short-side in circle-squaring right triangle.

Geometers' secrets:

1. Bottom and top lines of trapezoid have length ratio 1.913..
and long-side-to-short-side of circle-squaring right triangle (red);
triangle and trapezoid are geometrically related ... obviously.

2. Vertical and horizontal lines of "cross" have 2:1 length ratio.

3. Hypotenuse-to-long-side = sqrt(Pi)/(Pi/2) = 2(sqrt(1/Pi))
= 2/sqrt(Pi) = 1.1283791670955125738961589031215..

Rod

[Amigoo]

Re: Quadraturial Quiescence design

Evidence that only 2 points on the circle's circumference are required to "square the circle",
2 points that create a chord permitting construction of 3 unique circle-squaring objects:
right triangle, isosceles trapezoid, and scalene triangle.

Metaphysical translation: The human mind and guiding light within.

"What are those two points?"
The chord that exists between the circle's diameter
and the chord created by diameter/sqrt(2).

Rod

[Amigoo]

Re: Quadraturial Quiescence design

Geometers' secret: Diagonal and side midpoint-to-midpoint lines
of trapezoid (green) have sqrt(2) relationship, begging the question (again):

"Whence transcendental Pi?" since trapezoid is constructed of 2 overlapping,
circle-squaring scalene triangles!

Rod ... ...

[Amigoo]

Re: Quadraturial Quiescence design

Squared circle geometry that speaks for itself with a riddle:
What is the significance of the red number '7'

Number of lines in an isosceles trapezoid constructed of
two overlapping, circle-squaring scalene triangles?

Seventh day of rest after "impossible" quadrature?

Or maybe it's two circle-squaring right triangles.

Rod

[Amigoo]

Re: Quadraturial Quiescence design (The List)

Squared circle geometry that speaks for itself with a riddle:
What is the significance of the red number '7'

Number of lines in an isosceles trapezoid constructed of
two overlapping, circle-squaring scalene triangles?

Seventh day of rest after "impossible" quadrature?

Or maybe it's two circle-squaring right triangles,
nested in 2:1~1:2 line length relationship!

Or does it signal esoteric completeness, by adding number
3 (triangle, equilibrium, trinity, divinity) and the number
4 (square, material & terrestrial world).

Or Easter Shopping List for a Remembrance Meal:
"five barley loaves and two dried fishes" (152:2.7)

Rod