Paradise Trinity Day

[Amigoo]

Re: As Above, So Below design

While many geometers and non-geometers may ask "What's the point?",
usually only geometers want to know "What's the angle?".

Anticipating the second question, I revised the As Above design to show
how the angles are divided in this Cartesian workspace of a circle squared.

Rod

[Amigoo]

Re: As Above, So Below design

Qualified for its own page:
http://aitnaru.org/threepoints.html

A squared circle color wheel for this millennium!

Rod

[Amigoo]

Re: As Above, So Below design

After visitation by prompts 3:33 and 4:44 today, and with few geometry-related prompts in recent days, As Above seems to be the last design needed to promote the geometry of a squared circle. With summer break quickly approaching, "just a few more lines" were added to the design.

As Above, So Below is a bit more esoteric now, geometrically speaking, but may have future value for when
"squaring the circle" is not an impossibility but a way of life.

Rod ... ...

[Amigoo]

Re: As Above, So Below design
"New & Improved!"

And gaining reputation as the 2014 Pi Chart of squared circles.

Rod

[Amigoo]

Re: If ___, Then ___ (new design; a project footnote)

If Pi is the square of a circle having a diameter of 2,
then this geometry visually describes that squared circle.

If I would attempt humorous reflection on this total research effort,
now continuing over three years, I would say:

"I was permitted to explore the geometry of a squared circle
... and lived to tell about it."

Rod

[Amigoo]

Re: If ___, Then ___, and more. (embellishment of If ___, Then ___. design)

I should have known better - crafting an esoteric "envelope" in the geometry was good and sufficient closure.

But then I had to investigate the envelope:
Could it be more symmetric? Does it really integrate with the symbolism of the geometry? Will it "move" in synchrony with the other objects?

Long story short (envelope was left unopened):
The envelope was discovered to be the foundation of another impressive squared circle pattern. And this discovery becomes the ultimate lesson about squared circles. They're less about the equality of area measurement of a circle and its square and much more about universe patterns:
complex, symmetric, expanding, mysterious, inviting, artistic, inspirational, ...

Until I'm ready to fill a new portfolio with colorful, squared circle patterns, this will be the final temptation to improve a "completed" design.

Rod

[Amigoo]

Re: Raison D'être design (just the geometric facts)

Apparently, the full circle has been traversed in this research and an important lesson learned: it's all about patterns
Raison D'être presents the geometry that motivated this continuing research for so many months:

Theorizing that the square of a circle must be no smaller than a square inscribed in the circle and no larger than a square enclosing the circle, I created this guide to remind me that a geometric transition exists - the geometry of a squared circle is real ... without a doubt! But "getting there from here" is a bit difficult; some argue "Impossible!".

Yet, consider a squared circle an unmarked exit along a well-mapped highway ... ...
Somewhere along this highway, having very precise length, the squared circle exit should exist. The fact that it has not yet been marked, does not diminish the reality. Advanced math suggests that such an exit is a black hole (worse than pot hole) along a very real highway.

However, "black hole" may be the better description when researchers are labeled with the conjectured disease of morbus cyclometricus. But the Raison D'être geometry counters that sanitas cyclometricus is the true affliction - just don't drive so fast or you'll always pass the unmarked exit.

And while enjoying a more scenic drive, you're certain to view the many patterns indicating the presence of a squared circle. Who knows? An intensity of complex patterns may be the only possible evidence of an exit to the impossibly precise land of squared circles.

Rod

[Amigoo]

Re: Raison D'être design

There may be prescience (especially, regarding world events) in the symbolism of this geometry. For signs or persons may appear, saying "It is here!" or "It is there!", when in fact "it" is a mirage or supported by similar evidence.

This symbolism hints that one's inner awareness (with the influence of other universe forces) will remain the more certain guidance.

Rod

[Amigoo]

Re: If __, Then __, but wait! (new design)

Perhaps knowing that I'm trying to abandon the project without being completely convinced about the geometry, the forces of inspiration and motivation issued yet another "Wait! There's more!".

This next design contains a very convincing integration of geometric objects, all well supported by the squared circle on this Cartesian palette.
This pattern and too many recent 5:00 AM bed times should finally persuade me to feast - not nibble - on the evidence.

Rod

[Amigoo]

Re: If __, Then __, but wait! design

Even upon a 6:00 AM bedtime, this design seemed to be incomplete.

So, one more round of exploration later finally revealed the message of this geometry:
We have arrived at an intensely fertile moment in our planet's spirituality.

Rod ... ...

[Amigoo]

Re: If __, Then __, but wait! design

Balance seems to be an eternal quest ...
(design name changed to Fecundity to complement message)

Rod

[Amigoo]

Re: Incredible Palette (new design)

What is more incredible is the motivating power of a kitchen full of dirty dishes!
(I can linger for hours at the computer, in a distant room of the house)

Incredible Palette reflects continuing exploration of the If __, Then __ geometry,
but is just an extreme example of this squared circle pattern.

Unless I plan to permanently bond with paper plates and plastic utensils,
further postponing domestic priorities will be impossibly incredible!

Rod

[Amigoo]

Re: Raison También (new design)
(a mixed-languages title from the creativity sandbox)

This design should be another good reference for future study of squared circles. The geometry that integrates two squared circles (the larger circle and its csc* sibling) is colored to promote this more technical but very revealing pattern.

csc - circle within inscribed square within circle.

Rod

[Amigoo]

Re: Raison También design ( http://aitnaru.org/threepoints.html )

Everyone knows that "squaring the circle" requires thinking outside the box!
But who knew to open the box?

Rod ... ...

[Amigoo]

Re: Raison También design

Squared circle geometry waxes so loquacious and erudite that I have to consult a comprehensive online dictionary.

To complement this erudition, RT was enhanced to show three squared circles with CSCSC integration.

[ CSCSC - circle within inscribed square within circle within inscribed square within circle ]

Rod

[Amigoo]

Re: Raison También design
The online page featured this design until its early morning cloning to "Open the Box!"

Apparently, the 12:34 AM prompt confirmed that opening the box halfway was perfect.

But halfway relative to what? The geometry's upper red lines seem to satisfy this halfway specification
... and they allow the isosceles trapezoid to observe its destiny.

Rod

[Amigoo]

Re: Open the Box design

The 1:2 ratio of the height of the trapezoid to the separation of two corners of the "boxes" is quite consistent as these integrated objects are replicated outward. A draft version of three squared circles looks very good, geometrically, but I'm not going to convert this into design now.

I'm convinced that this geometry well-describes a squared circle ... and many more in CSCSC relationship!

Next task: Create a geometric proof. "Not my job!"

Rod

[Sandy]

awesome Rod!

[Amigoo]

Sandy,

Thanks for the feedback! It will be awesome if I can ever climb out of this geometry toy box.
I rationalize now that I'm just completing visual documentation of the adventure.

Today's insight (re: Open the Box design):

There's always room for one more box on a squared circles Cartesian palette.
This one gives better definition to the isosceles trapezoid.

Previous insight (re: Raison También design):

Everyone knows that "squaring the circle" requires thinking outside the box!
But who knew to open the box? How far to open the box? Best guess: halfway.

The two upper red lines define "halfway" and this 1:2 ratio repeats for these lines
as the geometric objects are replicated outward, maintaining CSC relationship.

CSC - inscribed circle within inscribed square within circle.

Rod

[Amigoo]

Re: Open the Box design

"What?!" I complained.
"I just posted comments about two upper red lines and now I have to add two more red lines!"

"May the (lines of) force be with you." was the only response.

Well, they make the geometry more dynamic and slightly esoteric, I have to admit.

Rod

[Amigoo]

Re: Open the Box design (one line removed)

Why Pi has been so elusive ...

The design now presents an abstract view of Pi hustlin' down the road,
spinnin' squared circles along the byways of our ordered universe.

For the next 3.14 Pi Day celebration ...

The problem with "squaring the circle" is that all digits of Pi do not have to be considered - just 8 (or as this geometry shows, at least 3). This free spirit even believes that 2 points are sufficient (center of circle, midpoint of side of square) ... if the geometry is painted with a Cartesian palette.

Rod ... ...

[Amigoo]

Re: "all digits of Pi"

What's that nonsense? Points on the circle should have been the focus.

Obviously, "I think I'm thinking" cannot be claimed before the day's first cup of coffee.
But "wiggly numbers" (many decimal digits) are the reason I prefer patterns.

Rod

[Amigoo]

Re: Open the Box design
Today's lesson: "Not so fast, geometer genius!"

The 1:2 ratio is intriguing, but it exists for various positions of the two halves of the "box".
However, one unique isosceles trapezoid exists for a squared circle.

Current analysis: the goal is between here and there.
But how many decimal digits exist between 0 and 1?

Rod

[Amigoo]

Re: Circular Reasoning (new design)
Yesterday's lesson, Part II: "We repeat, it's all about patterns,
often the communication of celestial guidance."

Although the 1:2 ratio was a thrilling (but short) roller coaster ride, the Circular Reasoning pattern is soothing to this geometer's soul. And it incorporates a center CSC cameo which confirms the importance of this pattern (CSC) in the research.

Rod

[Amigoo]

New insight about "patterns" ...

While pattern recognition may be evolutionary, what universe intelligence planned this
pattern recognition for later use in celestial communication and guidance?

Rod