Paradise Trinity Day

[Amigoo]

Re: 8th Note of Pi design where
"Pi is meat in a geometric sandwich"

"Geometry Junior, there's a lot of wiggle room
for transcendental Pi between sqrt(2) and 2.0 "

The equal and precise areas of a circle and its square
when diameter = 4.0 suggest that the 8 points on a circle
(whereupon rests the circle's square) create Pythagorean
relationships that effectively corral the touted Pi.

"Interesting recipe, Geometry Junior! We should explore
the concept ... after the recipe is awarded a blue ribbon
... elsewhere in the mathematical universe."

Rod

[Amigoo]

Re: Square1 design
Added to: http://aitnaru.org/images/Sqrt_Pi_Ratios.pdf

"Square1", as in "Back to Square1", is much better than "Second Base"
because Square1 can suggest that you've "Been there! Done that!"
at every base! and are now ready for another mile so high.

Rod ... ... (been to Square 3.14)

[Amigoo]

Re: Square One design

Now crafted as an "impossible geometry" compass showing how to
navigate to the center of the universe (if you know the point).

Regarding "What's the point?" ...

According to this geometry:
- squared circles must exist.
- the circle cannot be squared.
- only patterns prove a circle's square.
- so savor such salient samples!

Rod ... ...

[Amigoo]

Re: Point One UC design
(pronounced "you see")

Speaking of "center of the universe" ... this CSC geometry proves
that squared circles can be linked relative to diameter and SOCS.
In this linkage, a "Universe Center" (UC) appears, showing that
all circles have a common center (albeit not concentricity).

CSC = Circle inscribed in Square inscribed in Circle ...
SOCS = Side of Circle's Square

Regarding "What's the point?" ... The UC, you see!
Regarding "Point One" ... There are others; at least 8!
Regarding "transcendental Pi" ... hosted by sqrt(2)
(maybe this is the real "Point One")

Rod

[Amigoo]

Re: Point One UC design (pronounced "you see")
This research hints that Pi has a silent sibling - even "nemesis" ...

Pi itself, the mathematical creation called "constant" is transcendental ... apparently
But the actual ratio of Circumference to Diameter is not transcendental ... apparently
... unless sqrt(2) is transcendental. ... "you see"

Rod

[Amigoo]

Re: Point One UC design

hints that Pi has a silent sibling - even "nemesis"

Better display of the Point One "flutterby", geometry proving sqrt(2) domination
of this Cartesian Neighborhood of 4 circles squared. Notice how the colored patterns
suggest Russian nesting dolls: https://en.wikipedia.org/wiki/Matryoshka_doll

Rod

[Amigoo]

Re: Point One UC design

Point Two: (sqrt(Pi)/sqrt(2))^2 = Pi/2

Simple, intuitive algebra evidenced by this geometry ...
despite sqrt(Pi) being transcendental and sqrt(2) not.

Rod ... ...

[Amigoo]

Re: Point One UC design

Point Two: (sqrt(Pi)/sqrt(2))^2 = Pi/2

About analyzing this geometry, Hadron protested:
"If I ever see an impossible quark, I'll let you know
... until then, mine your own Ps and Qs
(Pythagorean Quadrature)"

Rod

[Amigoo]

Re: Point One UC design

Point Two: (sqrt(Pi)/sqrt(2))^2 = Pi/2

Slightly recolored to highlight isosceles right triangles* that control dimensions
of the linked circle-squaring right triangles, right triangles where all three sides
have sqrt(2) length relationship to similar lines in preceding/following triangles
... with sqrt(Pi) one of those line lengths.

Tip: IRTs are geometrically synonymous with sqrt(2).

IMO: What's the difference between quarks and quadrature

1. quarks have arks; Pythagorean Quadrature smells a rat.
2. quarks are not directly observed or found in isolation whereas
quadrature is not directly observed or found outside of patterns.
3. quarks are studied inside the box; quadrature outside.

Rod

[Amigoo]

Re: Point One UC design

With this geometry speaking for itself so eloquently,
this geometers' tip is effectively "Note To Self":

Short side of the linked circle-squaring right triangles
is a chord of the circle, a chord that creates the angle
between the hypotenuse and long side. This angle
propagates in the sqrt(2)-linked right triangles.

What' are the always-present angles?
27.597112635690604451732204752339..
62.402887364309395548267795247661..
... and of course, 90 degrees

Rod (Note To Self: stop trying to explain
geometry that "speaks for itself")

[Amigoo]

Re: Square One design

This geometry must be the master pattern of a squared circle,
a universe pattern proving that heaven and earth align.

Not only does this pattern include circle-squaring scalene
and Pythagorean right triangles, but also isosceles trapezoids
and squared-circle-defining line length ratios and angles.

Whenever I try to isolate a geometric circle-squaring object
(Pythagorean right triangle, for example) as THE foundation of a
squared circle, other objects quickly appear (or seem necessary)
to support circle-squaring functionality in the Neighborhood).

"only patterns prove a circle's square" ... apparently

Rod

[Amigoo]

Re: IAM UC design

"G. T. S. F. I." ... with patterns, of course.

Rod ... ...

[Amigoo]

Re: IAM Quadrature design
(renamed from IAM UC, aka "IAM Universe Center")

G. T. S. F. I. ... eloquently!

Rod

[Amigoo]

Re: IAM Quadrature design
("final, final")

If any geometric object can prove that the circle is squared,
it must be this "two-tine fork" with its convincing display
of circle-squaring, line-length ratios sqrt(Pi) and 2/sqrt(Pi).

sqrt(Pi) = 1.7724538509055160272981674833411..
2/sqrt(Pi) = 1.1283791670955125738961589031215..
sqrt(2) = 1.4142135623730950488016887242097..

Rod

[Amigoo]

Re: Pythagorean Palette design

More evidence that circle-squaring, line-length ratios
of sqrt(Pi) and 2/sqrt(Pi) easily create G. T. S. F. I.

Next on your geometer's palette: "impossible" Pi
or a popular, inside-the-box, mental wedgie

Rod ... ... (still cruisin' for clues about this concept):
Is "thong" a wordless wedgie or "wedgie" a noteless thong?

[Amigoo]

Re: Pythagorean Palette design

According to short-lived myth, Salvador Dalí was going to channel
Pythagorean squared circle geometry and had created this sketch ...
but decided that "surreal" he could do - not "impossible".

Rod

[Amigoo]

Re: Pythagorean Palette design
“Outside the box”, but surreal or “impossible”?

Salvador Dalí ... decided that "surreal" he could do - not "impossible".

... or maybe he was taken aback by “outside the box”.
... or maybe “outside the box” then became his style.

Rod ... ...

[Amigoo]

Re: Pythagorean Palette design
“Outside the box”, but surreal or “impossible”?
... or transcendental?

"Pi Corral" becomes more meaningful when an arc (1/4 of circle),
representing a finite portion of Pi is sandwiched between
the two sides and hypotenuse of an isosceles right triangle
having sides with length 2,2 and hypotenuse 2(sqrt(2)).

In Pythagorean Palette, this IRT appears as a golden hat
in the upper right quadrant of the design.

The line length sandwich?
4.0, 3.1415926535897932384626433832795..,
2.8284271247461900976033774484194..

Rod ... ...

[Amigoo]

Re: Pythagorean Palette design
“Outside the box”, but surreal or “impossible”?
... or transcendental?

New insight: Don't go fishin' without your Pi Sandwich.
(arc sandwiched between two isosceles right triangles
in the "golden hat" represents a Pi/4 increment)

Caveat: You have to be "outside the box" to catch a box,
and know that the best fishin' aligns with sqrt(2) angles.

Rod ... ... (oft fishin' at all angles)

[Amigoo]

Re: Pythagorean Palette design

Re: https://en.wikipedia.org/wiki/The_Adven ... Tom_Sawyer

Tom Sawyer wanted Becky to go with him to the fishin' hole
to show her how he could catch an elusive box of "impossibles"
(and she was soon enthralled by his square root).

At the fishin' hole, Tom was visualizing a much delighted Becky
when he caught her "impossible" box that he had forecast.

And that reminded him about his COT (Constraint Of Pi):
two geometrically similar arcs have opposite points with the
distance between points at the two ends precisely zero

The Constraint? Becky's smile (length of IRT's hypotenuse).
Say what? Just gaze upon Tom's Get Lucky! "golden hat"
with the green frog face.

Rod

[Amigoo]

Re: Pythagorean Palette design (at the fishin' hole)

And that reminded him about his COT (Constraint Of Pi)

Note: "Constraint Of Pi" is reference to "Constraint Of Transcendental"
since the length of Becky's smile is a finite sqrt(2) value via the IRT.

Rod ... ...

[Amigoo]

Re: Pythagorean Pi-ramid design
(further study of the Pythagorean Palette geometry
after catchin' the "ïmpossible" box at the fishin' hole)

Pre-sunrise observation (via LED illumination) ...

With length of circle's diameter associating with arc Pi/2
and length of side of inscribed square associating with arc Pi/4,
a tall isosceles triangle can be constructed having precise
angles: 67.5, 67,5, and 45 degrees:

0.39269908169872415480783042290995.. Pi/8
0.5553603672697957808769851237576..
0.78539816339744830961566084581989.. Pi/4
1.1107207345395915617539702475152..
1.5707963267948966192313216916398.. Pi/2

Not only does this "impossible"* Pi-ramid reflect absolute
sqrt(2) domination of squared circle geometry, but also hints
that this Pi-ramid effectively corrals transcendental Pi.

* re: sqrt(2) and the "impossible" squared circle.

Rod

[Amigoo]

Re: Pythagorean Pi-ramid design

"Sqrt(2) beacon of Cartesian enlightenment
in local Neighborhoods of circles squared"

An intuitive* geometry visual that ...
"either sqrt(2) is transcendental or Pi is not"
(since 1/4 of the circle associates with Pi/4
as well as a side of circle's inscribed square)

* apparent ratio of circumference to diameter -
not necessarily long-popular constant called "Pi".

Rod ... ...

[Amigoo]

Re: Pythagorean Pi-ramid design

Re-colored two horizontal lines (diameter, side of inscribed square)
in the same circle to reveal a geometers' secret of this Pi Corral:

When each line is associated with its arc (line is a chord),
the difference between the length differences of these two sets
(line and arc) is sqrt(2) ... which begs the question (again):

Whence transcendental Pi? (Pi is represented by the arcs)

Rod

[Amigoo]

Re: Pythagorean Pi-ramid design (the numbers)
Calculate Ratio of Ratios arc-Diameter, arc-InscribedSquare

Given: Circle's Diameter = 2.0,
SoIS = Side of InscribedSquare = sqrt(2),
Pi = 3.1415926535897932384626433832795..,
Circumference = Pi x D
= 6.283185307179586476925286766559..

Calculate arc-Diameter ratio arc-Dr:
arc-D = (Pi x D) / 2 (= length of arc-D)
= (3.1415926535897932384626433832795..(2.0)) / 2
= 3.1415926535897932384626433832795.. arc-D
arc-Dr = arc-D / D (= ratio of arc-D to Diameter)
= 3.1415926535897932384626433832795.. / 2.0
= 1.5707963267948966192313216916398.. arc-Dr

Calculate arc-InscribedSquare ratio arc-ISr:
arc-IS = (Pi x D) / 4 (= length of arc-IS)
= 3.1415926535897932384626433832795..(2.0) / 4
= 1.5707963267948966192313216916398.. arc-IS
arc-ISr = arc-IS / SoIS (= ratio of arc-IS to SoIS)
= 1.5707963267948966192313216916398.. / sqrt(2)
= 1.1107207345395915617539702475152.. arc-ISr

Calculate Ratio of Ratios arc-Diameter, arc-InscribedSquare:
RoR = arc-Dr / arc-ISr
= 1.5707963267948966192313216916398..
/ 1.1107207345395915617539702475152..
= 1.4142135623730950488016887242097.. sqrt(2)

Say what And whence transcendental Pi
Relax and consult the arc-Dr in the morning.

Rod ... ...