Paradise Trinity Day

[Amigoo]

Re: Transcendence of Vesica design, “a real pinch of Pi”

Geometers' tip:
The two circles forming this "Vesica Piscis" are derived from the geometry
of two overlapping scalene triangles (which square the largest circle),
as well as all the circles with sqrt(2) relationship.

Who knew?!
Hopscotch was subtle intro to sqrt(2) (re: length of diagonal of a square)
See also: https://en.wikipedia.org/wiki/Hopscotch

Rod ... ... (shoppin' for butterscotch, Hop's cousin)

[Amigoo]

Re: Transcendence of Vesica design, “a real pinch of Pi”
(final tweakin' & embellishment) Design updated in:
http://aitnaru.org/images/Squarely_Entwined.pdf

When squared circle geometry prefers to speak for itself,
especially about a greater Vesica Piscis, everyone listens
... at least those who "have a mind to".

Rod

[Amigoo]

Re: Harmony of Circle, Square,Triangle design

Considering popular symbolism that the circle represents heaven (the spiritual world)
and the square represents planet earth (the material world), of what representation
is the circle-squaring triangle (a "miraculous" right triangle)?

Who knew?! This right triangle whose hypotenuse (circle's diameter) and long side
(side of circle's square) have the 2(sqrt(1/Pi)) ratio both defines a circle and its square
as well as confirms the squaring of the circle. Is it possible the triangle reflects trinity
existence before heaven and earth? If so, this may be a "paradisiacal trinity"!

Rod ... ...

[Amigoo]

Re: Harmony of Circle, Square,Triangle design

New PDF just for these designs:
http://www.aitnaru.org/images/Harmony_of_CST.pdf

Rod

[Amigoo]

Re: C'est La Vie design
aka, CST INspiraling (or OUTspiraling)

Such influence of sqrt(2) that we ponder:
Is sqrt(Pi) evenly divisible by sqrt(2) ?!

And who knew
That sqrt(Pi)/2 defines a circle-squaring right triangle
... including the circle (D = 2.0) and its square!

Rod C'est La Vie ...
when running in squared circles.

Re: https://www.youtube.com/watch?v=H7rhMqTQ4WI
"We'll find love, we'll dance, yes this is life!"

Not necessarily related to squared circle geometry,
but LIVELY! Look for "Garlic Chicken" on the menu
at 2:22 into the video.

[Amigoo]

Re: C'est La Vie design

Hmmm ...
sqrt(Pi)/sqrt(2) defines a transcendental triangle spiral
((sqrt(Pi)/sqrt(2))/sqrt(2))/sqrt(2) ...

Rod ... ... (on TTS Highway,
ridin' toward the "go inward" exit)

[Amigoo]

Re: C'est La Vie design

sqrt(Pi)/sqrt(2) defines a transcendental triangle spiral
((sqrt(Pi)/sqrt(2))/sqrt(2))/sqrt(2) ...

Numerical pattern of Transcendental Triangle Spiral
('/n=' refers to these first two increments of Pi):

1.7724538509055160272981674833411.. sqrt(Pi)
1.2533141373155002512078826424055.. sqrt(Pi)/sqrt(2)
/2=
0.88622692545275801364908374167057..
0.62665706865775012560394132120276..
/4=
0.44311346272637900682454187083529..
0.31332853432887506280197066060138..
/8=
0.22155673136318950341227093541764..
0.15666426716443753140098533030069..
/16=
0.11077836568159475170613546770882..
0.07833213358221876570049266515035..
/32=
0.05538918284079737585306773385441..
0.03916606679110938285024633257517..
/64=
0.02769459142039868792653386692721..
0.01958303339555469142512316628759..
/128=
0.0138472957101993439632669334636..
0.00979151669777734571256158314379..
/256=
0.0069236478550996719816334667318..
0.0048957583488886728562807915719..

Rod

[Amigoo]

Re: C'est La Vie design (re: Wiggly Numbers)
Re: http://aitnaru.org/images/Squarely_Entwined.pdf

Note: C'est La Vie is technical version of Squarely Entwined.

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

/8= (marks 360 degrees of spiral)
0.22155673136318950341227093541764..
0.15666426716443753140098533030069..

Hmmm ...
What happens to "transcendental" at the next 360-degrees?

Rod ... ...

[Amigoo]

Re: C'est La Vie! design

I've observed many times that two circle-squaring triangles exist:
the primary c-s right triangle and the secondary scalene triangle.
Since the right triangle seems sufficient for circle squaring,
of what value is the scalene

Years ago in this project, I conjectured that circle-squaring
geometry must show obvious association of sqrt(2) and sqrt(Pi).
The scalene triangle does this well: one side represents sqrt(2)
and another side represents sqrt(Pi).

Of what value is side three? Besides joining the two sides,
completing the symbolic association of sqrt(2) and sqrt(Pi),
in C'est La Vie! geometry this third side is shown to decrease
by increments of sqrt(2) in the scalenes of smaller circles.

C'est La Vie! "Yes, this is life!" in squared circle geometry.

Rod

[Amigoo]

Re: C'est La Vie! design
Regarding 360 degrees of triangle spiral ...

1.7724538509055160272981674833411.. sqrt(Pi)
/ 8 = 0.22155673136318950341227093541764..

1.4142135623730950488016887242097.. sqrt(2)
/ 8 = 0.17677669529663688110021109052621..

0.22155673136318950341227093541764..
/ 0.17677669529663688110021109052621..
= 1.2533141373155002512078826424055.. sqrt(Pi)/sqrt(2)

Analysis: Sqrt(2) is master of the Pi Corral.

Rod

[Amigoo]

Re: C'est La Vie! design
(a transcendental triangle spiral)

Now, more complex, but ...
would a squared circles spiral not be boldly esoteric

Rod

[Amigoo]

Re: C'est La Vie! design
(a transcendental triangle spiral)

In 360 degrees of a squared circles spiral,
sqrt(2) maintains direct relationship to sqrt(Pi):

Regarding inscribed squares of circles ...

1.4142135623730950488016887242097..
^2 = 2

1.4142135623730950488016887242097..
/ 8 = 0.17677669529663688110021109052621..
^2 = 0.03125

2 / 0.03125 = 64

Regarding area squares of circles ...

1.7724538509055160272981674833411..
^2 = 3.1415926535897932384626433832795..

1.7724538509055160272981674833411..
/ 8 = 0.22155673136318950341227093541764..
^2 = 0.04908738521234051935097880286374..

3.1415926535897932384626433832795..
/ 0.04908738521234051935097880286374..
= 64

Obviously , sqrt(2) is transcendental
... or sqrt(Pi) is not.

Or is this a mathematical "As above, So below"
where Pi is the only "transcendental" ?

Rod ... ...

[Amigoo]

Re: Los Dieciséis design
"180 degrees of the Pi Corral"

Intuitively, the number 16 relates to the practical limit of Pi's decimal digits
... and NASA is quite happy with 15 decimal digits (3.141592653589793):
https://www.jpl.nasa.gov/edu/news/2016/ ... ally-need/

Is that '7777' symbolism in this Cartesian neighborhood ?

Rod ... ... (off to the bakery to get 180 degrees of a Pi)

[Amigoo]

Re: Los Dieciséis design
"180 degrees of the Pi Corral"

Considering the well-accepted 'Circumference = Pi x Diameter' (C = Pi x D),
for a circle having Diameter = 2 .0, length of half the Circumference = Pi.

Since the area square of a circle where D = 2.0 has a side length of sqrt(Pi),
this shows mathematical association of half the Circumference (an arc; length = Pi)
with a side of the circle's square (a straight line; length = sqrt(Pi)).

Not mathematical revelation , but a feature of the conjectured "Pi Corral":
The length of the Diameter is fixed (2.0) and, by definition, Pi defines
both the length of the Circumference and the side of the circle's square.

Yet, the Diameter length, a straight line surrounded by circle and its square,
is not a transcendental value! (and neither is the ubiquitous sqrt(2)).

Rod

[Amigoo]

Re: Los Dieciséis design
"180 degrees of the Pi Corral"

More wiggly numbers:

1.1283791670955125738961589031215.. 2(sqrt(1/Pi))
= 2.0 / 1.7724538509055160272981674833411.. sqrt(Pi)
= 1.7724538509055160272981674833411.. sqrt(Pi)
/ 1.5707963267948966192313216916398.. Pi/2

2.0 / 1.5707963267948966192313216916398.. 2.0 / Pi/2
= 1.2732395447351626861510701069801.. (2(sqrt(1/Pi)))^2

1.1283791670955125738961589031215.. 2(sqrt(1/Pi))
/ 1.4142135623730950488016887242097.. sqrt(2)
= 0.79788456080286535587989211986876..

1.1283791670955125738961589031215.. 2(sqrt(1/Pi))
x 1.4142135623730950488016887242097.. sqrt(2)
= 1.5957691216057307117597842397375..

Next design ("1, 2, Pi", aka "1 to Pi"), exploration of C'est La Vie!,
will better display this consistent 2(sqrt(1/Pi)) influence.

Rod

[Amigoo]

Re: 1, 2, Pi design
D1 = 2.0, D2 = 2(sqrt(1/Pi))

Simple squared circles triangulation.

"Lines and triangles and squares! Oh, Pi!"

Rod

[Amigoo]

Re: 1, 2, Pi design
Updated in http://aitnaru.org/images/Squarely_Entwined.pdf

The squared circles toy box promotes "impossible" creativity:
Lengths of parallel light blue lines = 1, 2, Pi (sqrt(Pi))

From whence this "creativity"?
I knew that "1, 2, Pi" was the perfect design name
before I discovered the third line (sqrt(Pi)) today.

Rod ... ... (still chasin' the flutterby)

[Amigoo]

Re: 1, 2, Pi design

still chasin' the flutterby

I discovered the type of net required to catch this flutterby:
Model 1:2.Pi - handcrafted by those who "think outside the box".

While having no such net, I recolored the design to highlight
that the diameter of the largest circle has twice the length of a side
of the smaller circle's square where D = 2(sqrt(1/Pi)).

Rod ... ... (caught in the "impossible" net)

[Amigoo]

Re: 1, 2, Pi design

This may be the year to inaugurate the Squared Circles Soirée
on Leftover Pi Day, November 10th (Julian Day 314 this year).

Rod

[Amigoo]

Re: "Four and Twenty" design

Indirect reference to the "sixteen permanent members" of the Four and Twenty Counselors (UB 45:4.1) ... perhaps; more to follow ... perhaps. This geometry is so unlike all previous designs that it might even be "a word from the sponsor", revealing a Paradise Trinity "Tree of Life"

Even more mysterious are the five symmetric points of a triangle (golden circles in this design) that appeared on my thigh five days ago, assumed to be insect bites, folliculitus, or sleep time branding by a celestial cowboy. This did not occur in my home (details are censored).

And if it were not for the unusual symmetry of the geometry, I would have assumed that I had been sleep walking (sans chaps, spurs, and western bolo) in a North Texas briar patch after a wild night at the Pi Corral ... and that there was nothing personal to share (in this venue).

The BB-size circles are healing quickly after daily application of raw apple cider vinegar
and the geometry easily expanded to include 16 esoteric points of overlapping circles.

Rod

[Amigoo]

Re: "Four and Twenty" design

the five symmetric points of a triangle

Hmmm ... a glaring error (later); there are six points - not five!
(and I took a revealing photo for lasting evidence)

"Who can tell?" why the largest circle's diameter needed to be 3333333.333..
and the two smaller concentric circles (all light green) have diameters
of 16666666.6665.. and 8333333.33325.. (D/2 and D/4), but they support
the subsequent creative and intriguing geometry.

Rod

[Amigoo]

Re: "Four and Twenty" design ("final, final")

When the number 3 (a trinity) refers to the "sixteen permanent members"
of our planet's Four and Twenty Counselors ... with "sacred geometry".

Rod

[Amigoo]

Re: "Four and Twenty" design ("final ^2")

Considering that the geometry might represent seating for the Triennial Assembly
of the Four and Twenty Counselors, seats for the non-permanent 8 were added
... effectively highlighting possible interactions of groups of members ...
as well as suggesting a central podium for the deliberations.

Rod

[Amigoo]

Re: "Four and Twenty" design

After many thousands of hours of study of squared circles,
and with so many circles in Four and Twenty Counselors
(with none squared), a message seems delivered:

Pi, an invention of man (a math constant), did not exist
prior to the material universe. And still, post-Trinity,
squared circles may exist but cannot be created
... but might be proven!

At least, Pi proportions* in the next design, Los Dos (objects),
hint a proof may be just around a squared circle corner.
* Pi/2 : sqrt(Pi) ~ sqrt(Pi) : 2.0

Rod ... ...

[Amigoo]

Re: Los Dos design

"Pi/2 : sqrt(Pi) ~ sqrt(Pi) : 2.0 = 2(sqrt(1/Pi))
= 1.1283791670955125738961589031215.."

It seems a mystery to which geometric objects "Los Dos" refers
... and some sets of objects are valid "Los Dos". But the salient "Dos"
(relative to squared circles) are squared-circle-defining pairs of lines:

In the smallest right triangle (red lines), exist four pairs of lines
having sqrt(Pi) or 2(sqrt(1/Pi)) line length relationship. Of course,
squared circle geometers know which lines are these.

That four pairs of lines exist in a right triangle seems "impossible"
... and perhaps transcendental (by ethereal definition)!

But these sets of ratios are also an impressive "Los Dos":
Pi/2 : sqrt(Pi) ~ sqrt(Pi) : 2.0

Whichever ... Viva Los Dos!

Rod