Paradise Trinity Day

[Amigoo]

Re: Pi are Square (design in progress)

This design highlights the pattern created by a squared circle
having a diameter = 2(sqrt of Pi) and side of square = Pi.

This pattern identifies the two squares of a CSC set
having diameters of 4 and 2(sqrt of 2).

Rod

[Amigoo]

Re: Pi are Square design (completed)

Center symbolism added for future reference.

Rod ... ...

[Amigoo]

Re: Pi are Square design

Observation ...

One large right triangle (red; two are shown with adjoining hytpotenuses) is sufficient to construct this entire geometric composition, including the Urantia Pearl symbolism in the lower portion of the center. The length of each triangle's hypotenuse = 2; length of long side = sqrt of Pi.

The Pearl's geometry of 4,3,2,1 relative diameters begins with the largest circle which has half the diameter of the large golden circle; the bright green circle symbolizes Terra Firma, our current planetary residence.

Analysis: Pi are Square! Who knew? Pythagoras?

Rod

[Amigoo]

Re: Pi are Square design

Caveat ...

This geometry is useful for studying* the still-increasing decimal digits of Pi,
but cannot show perfectly squared circles until all digits are known
... and the experts say fuhgeddaboudit.

* long side of largest right triangle lengthens with more Pi digits.
How to see changes in the geometry? Fuhgeddaboudit!
(unless using CAD software on a supercomputer)

Rod

[Amigoo]

Re: Pi are Square design
Manifestation of double 'T' alignment.
(new insight and geometric symbolism):

"When pivoting in squared circles,
the pendulum's equilibrium may liberate
the champions of the impossible."

Rod

[Amigoo]

Re: Pi are Square design
Manifestation of double 'arrow' alignment.
(more insight and geometric symbolism):

Regarding the Pythagorean triangle that defines a circle and its square ...

Small arrow shows why the 'Pi/2, sqrt of Pi' line set
defines the square of a circle having a diameter of 2
and not (for example) the '2/2, sqrt of 2' set.

Say what? Only Pi are square.

Rod

[Amigoo]

Re: Pinnacle of Pythagoras (new design)
The geometry is certain; conversion from CAD drawing is in progress.

Conceptual symmetry is what I should have been searching for
and not geometric symmetry since this POP effectively correlates the
circle-squaring of that right triangle with its 27.597 degree vertex.

Speaking of POP, MOM was conceived at the same time! (MOM refers
to a new minced onion and millet recipe, posted as a product comment).

Rod

[Amigoo]

Re: Pinnacle of Pythagoras design
At least, a "draft" quality presentation.

Of the two inner circles (green, both squared), the juxtaposition of the smaller having a diameter of 1 and the larger having a square with side length of 1 highlights the circle-squaring magic of the Pythagorean triangle with a vertex of 27.597.. degrees.

The two outer circles? Both are squared by a similar triangle.

Rod ... ...

[Amigoo]

Re: Pinnacle of Pythagoras design
"Cast doubt and ascend the Pinnacle of Pythagoras!"

Final version, highlighting the circle-squaring triangle
(vertex of 27.597112635690604451732204752339.. degrees
between two sides having lengths of Pi/2 and sqrt of Pi).

About those Midwayer prompts ...
x:01 has been a personal prompt for several years,
indicating "progress". I've been seeing x:02 lately.
Time to wander along a new path?

Rod

[Sandy]

Hello Awesome Rod,

About those Midwayer prompts ...
x:01 has been a personal prompt for several years,
indicating "progress". I've been seeing x:02 lately.
Time to wander along a new path?

Maybe you are making double the progress!
hugs,
Sandy

[Amigoo]

Sandy,

"Double the progress" may be on target:

The lower yellow line (length = sqrt of Pi) forms an
obtuse triangle where the lengths of the two shorter sides = 1
(the radii of the inner largest circle).

This triangle perfectly correlates the two smaller squared circles
where the length of one diameter = 1 and the side length of the
other circle's square = 1.

And, now I suspect that this yellow line forms the top of the
Texas 'T' that had been a prompt as I drove on that tollway
several years ago.

Rod ... ...

[Amigoo]

Re: Pinnacle of Pythagoras design
'T' time (closure for this geometric prompt)

For a circle having a diameter of 2,
the Texas 'T' is part of an isosceles trapezoid
with the top of the 'T' = sqrt of Pi and the
two adjoining sides = sqrt of 2.

Rod

[Amigoo]

Re: Pinnacle of Pythagoras design

Finally, a design worthy of "pinnacle".

Is it possible to get there from here?
Of "there" now familiar to Pythagoras,
he would say: "Come and see!"

Rod

[Amigoo]

Re: Pinnacle of Pythagoras design
Definite prompting guidance (long story short) ...

I was uncertain that the juxtaposition of the two smaller squared circles (length of one diameter = 1; side length of the other circle's square = 1) was definitive enough. A recent comment on the dianasenechal site referred to a "diagonal" and, later, a 19:08 prompt appeared on my computer's clock (a diagonal in the geometry has a line length of 1.9084486..). This was sufficient inspiration to keep exploring, whereupon a better position was found for the smaller of the two circles.

The significance of this repositioning is that, now, two identical Pythagorean triangles participate in the correlation of these squared circles. And this geometry also provides resolution for the long-mysterious x:52/x:53 prompts (one immediately followed by the other, with x:52 alluding to two line lengths representing half of the square root of Pi and x:53 representing the square root of Pi*). Neither that diagonal nor all of the lines of this isosceles right triangle are shown to allow better presentation of the design.

* From the chart of prompt associations (x:52 = two x:26):
x:26 = 0.8862 26 9254.., x:53 = 1.7724 53 8509..

More prompting guidance ...
7:37 appeared on a kitchen clock as I poured a caffeine refill before converting the updated CAD drawing to design.

Rod

[Amigoo]

7:37 PM again as I awakened naturally from an afternoon nap (some "afternoon" naps explain the 3:00+ AM bedtimes ).

When I suspect that I should finally be finished exploring the foundational geometry of squared circles, another lure appears. Now, the geometric squared-circle-pairing* of the circles in the Pinnacle of Pythagoras design are enticing another drawing. "Juxtapositional Pi" is the probable title since it refers to this intriguing pairing.

Maybe before Christmas.

* In Pinnacle of Pythagoras, a large circle's radius defines a smaller circle and both of the two-circle pairs are geometrically interconnected by the circle-squaring Pythagorean triangles (and by the incremental Pi line lengths of those triangles). Say what? The geometric patterns are obvious and telling.

Rod ... ...

[Amigoo]

Re: Juxtapositional Patterns of Pi (in development)

Expecting simplicity, I found complexity.
But these integrated patterns of 8 squared circles in sets of 2 creates fascinating squared-circle geometry.
And the 2-sets-of-4 pattern enhances the circle squaring magic of these similar Pythagorean triangles!

This geometry may wish to speak for itself
... at least, when first presented.

Rod

[Amigoo]

Re: Juxtapositional Patterns of Pi

Once started, conversion of this complex CAD drawing to design was easy!
The obvious patterns helped ... and color selection seemed intuitive.

That "this geometry may wish to speak for itself" is confirmed,
a "long winter's nap" is now anticipated".

Rod ... ...

[Amigoo]

Re: Juxtapositional Patterns of Pi

For numerical reference:

Diameter of largest green circle = 2.

Diameter of 2nd largest green circle
= 1.5707963267948966.. = half of Pi.

Hypotenuse (red) of right triangle
(whose long side = half of Pi)
= 1.7724538509055160.. = sqrt of Pi.

Lower vertex of right triangle
= 27.5971126356906044.. degrees.

Angle of hypotenuse (red)
= 62.4028873643093955.. degrees.

Rod

[Amigoo]

Re: Juxtapositional Patterns of Pi

"Y" not display the geometric foundation of a vesica piscis,
inherent in this squared circles geometry

When the vesica piscis is drawn,
the circles' center-to-center length = 1.

One what? What "Y"?
A design worthy of "pinnacle";
the big red one.
One what? "'Y' not"?
Y(es); Yes.

Rod

[Amigoo]

Re: Juxtapositional Patterns of Pi

Regarding the invisible (but not imaginary) vesica piscis,
the side length of each circle's square = 1.

One what?
One - number.
Which number?

Rod

[Amigoo]

Re: Juxtapositional Patterns of Pi
"What 'Y'"? indeed! (now replaced with a double 'T')

But not just any "T' - the Texas 'T' relating to the x:52/x:53 prompts.
And not just one 'T' but two 'T's! (lengths of vertical lines equal sqrt of Pi/2).

What a ride! literally:
As I drove under a set of "Texas 'T's (tollway logo) tonight, I was thinking:
"Those were the days!" (when these 'T's appeared as prompts - but no more).
However, after tonight's geometry exploration, I should have been thinking:
"These are the days!" (the adventure continues).

I was convinced that geometric closure for the 'T's had already appeared
several weeks ago ... but now a double set arises! Not only a double set
but accompanied by the intriguing 52/53 prompt (as 3:52/3:53 AM).

This design now integrates these two lines having length equal to sqrt of Pi/2
and one line (hypotenuse of red triangle) having length equal to sqrt of Pi;
an impressive confirmation of the long-mysterious x:52/x:53!

How enlightening! a vesica piscis hosts this double set of 'T's!

Rod ... ...

[Amigoo]

Re: NNI (new design)

Two thoughts came to mind on how to describe this design:
"Network-to-Network Interface" or "Needs No Introduction".

And the squared-circle-identifying "Texas 'T's" in all three
may herald future significance of the geometry's symbolism.

Apparently, another example of ...
"When squared circle geometry speaks for itself."

Rod

[Amigoo]

Re: Duality of Vesicae Piscis (new design)
"When squared circle geometry speaks for itself."

Captured essence of divine mitosis in an ever-expanding universe
... or simply vesicae piscis in squared circles correlation.

Rod ... ...

[Amigoo]

Re: Pivotal Duality (new design)
"When squared circle geometry speaks for itself."

This appears to be a good and sufficient plateau, tempting a "long winter's nap". In particular, the geometry presents (finally) a juxtapositional concept by which a squared circle might be proven vis-à-vis a set of circle-squaring Pythagorean triangles.

Say what? The triangles must be positioned on a Cartesian plane such that their similarity is destroyed if the length or angle of a shared line, for example, is changed. Otherwise (re: the irrational, transcendental Pi), proof will remain elusive ... or "impossible".

That a circle-squaring Pythagorean triangle must exist seems confirmed by these geometry explorations. But proof remains just beyond the horizon ... though the horizon now appears closer than ever.

"I knew that squaring the circle was impossible, but did not know that it would be so difficult." At least, the portfolio of designs presents convincing proof of Sanitas Cyclometricus - the journey still has value. "Morbis" seems reflective of premature opinion - the mathematical models of Pi may need to be updated (IMO).

Rod

[Amigoo]

Re: PD Addendum (new design)
"When squared circle geometry speaks for itself."

This design is actually more "Note to Self", a quick & easy perspective
on the conjectured set of juxtapositioned Pythagorean triangles.

The lower squared circle and its circle-squaring Pythagorean
are used to create the larger squared circle above, wherein
the lower circle reflects its geometric similarity.

Say what?
Geometric balance is the essence of an intelligent universe,
wherein "As above, So below." is the mortal invitation.

Rod