Paradise Trinity Day

[Amigoo]

Re: http://aitnaru.org/thekingdom.html
The updated design, now with "esoteric balance".

Michele & Sandy,

I sense that many of these geometry designs do not communicate information as much as they cause one's mind to be attuned for the reception of information "from above". In this respect, that which is received is that which the receiving mind best comprehends and appreciates.

If this is true, Crown Squared will be quite a cosmic portal for many seekers of truth, beauty and goodness!

A mesmerizing artistic display for the future Temple of Light.

Rod

[Sandy]

Hey Rod....

I was struck by your latest design this time tears forming in my eyes. It seemed as I viewed it it changed and moved as different aspects of existence moved forward to attention... The pure beauty of it struck me and I felt drawn into it as first a seeker and then finding myself a part of all that is...
This is so true....

I sense that many of these geometry designs do not communicate information as much as they cause one's mind to be attuned for the reception of information "from above". In this respect, that which is received is that which the receiving mind best comprehends and appreciates.

Like a true work of art, your geometry and your designs move us into those higher realms. I am thrilled to realize that I do not have to understand how they came about. Understanding and divine purpose can come through to all of us as we view them!
Thank you Rod!
Love,
Sandy

[11light11]

Here, here, Sandy!

This is beautiful . . . I see you've added the all-seeing eye at the top. I agree with Sandy; you feel this sense of recognition, maybe, when you look at these images. She and I are in the same mess as far as not understanding the mathematical theories that created these designs . . but they still resonate with us! Isn't that just magical? It's something you can feel. We're maybe good test subjects for you, since you know we're paying so much attention and we're not exactly mathematically-savvy. It just goes to show how you do land upon inherent 'truths' with your work -- the resonance tells the story.

And you wrote at 11:22!

Also, your work reminds me of something that has been tugging at me for years . . .that there's no true distinction between art and science. Your work straddles both spheres and truly illustrates this principle!

[Sandy]

Hey Michele,
You wrote:
Also, your work reminds me of something that has been tugging at me for years . . .that there's no true distinction between art and science. Your work straddles both spheres and truly illustrates this principle!
That's it exactly! It does and provides "a bridge" for all of us.

Sandy

[Amigoo]

I'm finally convinced that it's time to vacate the geometry "studio" if only to allow time for cleaning. Yet, during the preliminary survey of the creativity debris, one more pattern emerged: "Incremental Roots of 2" (in the Parallel Dimensions PDF file).

This design provides a satisfying (geometrically speaking) plateau since it highlights the presence of this unique scalene triangle in the four squared circles. Those diameters, displaying increments of the square root of 2, from the smallest to the largest: 0707106.781.., 1000000.000.., 1414213.562.., 2000000.000..

Pythagoras would comment: "A coloful scalene doodle, but many of my Pythagorean triangles still lurk within."

Rod ...

[11light11]

That's awesome Rod! And you've written at :33 again!

I wonder if taking a little break will cause insights to flood in . . . that happens to me whenever I take a break from concentrating on something that's had me stumped (though usually you keep making strides , so I wouldn't exactly call that being stumped . . .). I was reading about the unconscious aspects of ourselves that continue to work on problems we're not consciously focused upon. They used an attorney as the example. For those who simply glance at the case materials once, then set them aside and go about other tasks for weeks . . . when they return to the case at hand they are ready with large ideas, and they tackle the complexities with ease. For those who go into court blind and glance at the case materials just prior to speaking to the court, they fumble around. Though both sets of attorneys did no research whatsoever, the first group has been 'working on it' in the background. Wild huh? We're more powerful than we think. So I wonder if that's what your break'll wind up feeling like, to you.

Enjoy it! Michele

[Amigoo]

I protested that I'm vacating the studio, but "they" insisted on one more revision of the current design.
So, "Incremental Squares of 2" became that revision:
One less circle (now 3 squared) for better focus on the mysterious dark blues lines.
since these two lines may unlock the secrets of squared circle geometry.

The name, "Incremental Squares of 2", now refers to this two-line set,
to the square-root-of-2 increments in the circles' diameters,
and to the squares of those circles.

The geometry highlights the presence of this scalene triangle in the 3 squared circles:
their diameters from smallest to largest (increments of the square root of 2):
1000000.000.., 1414213.562.., 2000000.000..

Rod ... ...

[Amigoo]

Nice confirmation (apparently, the revison of the design is acceptable) ...
I walked away from the computer, searched for a sweater, then glanced at the clock: 12:34 PM.

Rod

[Amigoo]

Somehow, that design revision was still not satisfying to me - some other symbolism needed to be included. And the more I contemplated this, the more I sensed that this design would represent "closure" for this journey of convincingly suggesting that "squaring the circle" was indeed possible. I gradually learned that this mysterious journey was not about proving the concept (and presenting a solution) but simply resurrecting the belief that such was possible.

The overlapping squares in this geometry from several years ago ( http://aitnaru.org/homepage/freewill.html ) was the conceptual foundation for the research that continued to this day. It's sudden presence (via inspiration) was unexpected - the symbolism is powerful and convinces me that closure is at hand. The goal has been achieved ("their" goal of restoration of belief - not my sometimes ego-driven goal of "squaring the circle").

So, the "Incremental Squares of 2" design in the Parallel Dimensions file (click on the design here to view: http://aitnaru.org/providence.html ) now includes a symbolic "tip of the hat" to Pythagoras (each of the 4 right triangles in the center square that green circle) and is surrounded by the two overlapping red squares (which, along with the green parallelogram, square the small golden circle in the center).

I seem to be well-advised by this symbolism that any further creativity from this geometry sandbox will only keep me running in squared circles.

But 5 circles, all squared, remind me of another geometric object that needs investigation: a pentagon.
Maybe later in the year when the studio has been cleaned and organized for other research.


Rod ... ...

[overmind]

I added 9 squares and got a circle, each is about 10 degrees apart. Does that count?

[Amigoo]

Overmind,

I knew that one of the points of those 9 squares would have to be at 62.4028873643093955482677952.. degrees (not achievable without a computer). But the focus and timing of your comment was perfect! As I reviewed the two red squares in the design, considering how 9 squares could be drawn to complement the geometry, I noticed that an important geometric object was missing: the inner scalane triangle!

So, your comment becomes the equivalent of reminding the pyramid builders that the capstone had not been placed on top ... and then the engineers discovering that the capstone fit perfectly (notice how the scalene triangle nestles perfectly within the two overlapping red squares).

The pharaoh will be pleased!

Rod ... ...

[11light11]

Wow, what a great string of updates . . . we've got :22, 12:34, :44, :53 and :39! Am I missing anything? I find it remarkable that you tried taking a break and vacating the studio . . "and they pulled me back in!" to quote Michael Corleone. I really enjoyed your updates -- and Overmind, that design is beautiful! And it seems to have jogged something during the "no work" time, LOL!

It reminds me of the old joke:
"How do you make God laugh? . . . You make a plan."

[overmind]

If I made the picture using 90 squares, the outside would be circular as well. Each point on the square would be a degree.

[Amigoo]

"A tipped hat, indeed!" complained Pythagoras. "All 4 Pythagorean triangles in the center need to be visible."
So, the design was revised. But now the scalene triangle protests: "What am I, swiss cheese?"
Anticipating protests from other geometric objects, I proffer: "Stick it in the suggestion box!"
Now the straight lines complain: "Well! Since when did we become sticks?"
Oy vey! What a noisy geometry toy box this week!
Every object is running in squared circles.

Rod ... ...

[Amigoo]

Overmind,

You're moving in the right direction (Pythagoras would love the pun), but what about the ".4028873643093955482677952.. degrees"?

However, your suggestion highlights the more complicated math reasoning why "Squaring the circle is impossible." The precision required to draw a squared circle directly relates to Pi - over 10 trillion decimal digits now and they're still counting!

Since even a simplified version of the analysis is overwhelming to me, I discarded that expertise and went back to the basics: the geometry only needs to show equal comparison between objects (or angles, etc.). As long as both objects (or points of comparison) are equal, proof of a squared circle might be possible.
The many geometry designs that I've shared attempt to show geometric objects that have potential for proving the square (perhaps, even drawing the square according to the Greek rules). Of course, the challenge is more complicated than my simple explanations, but the designs tend to be visually pleasing - even mind stimulating!

Rod

[11light11]

Rod, you posted at 12:22! I can never get over how often that happens.

Now the straight lines complain: "Well! Since when did we become sticks?"

This reminds me of a quote from a river architectural tour, where I used to work as a bartender. The docent quoted a famous architect (Bertrand Goldberg?!) as having said "There are no straight lines in nature." This is why he created buildings that always curved, such as those seen here. https://www.google.com/search?q=twin+co ... 66&bih=644
(Apparently these buildings are called "Marina City" . . I never knew that, because in Chicago we call them the Twin Corncobs! )

I'm trying to fit this idea into your research, Rod. Any thoughts?

[Amigoo]

Michele,

This squared circle geometry suggests a theme of contrasts and inclusions: the circle and the square and all other geometric objects. When the geometry "works", the visual message becomes powerful ... even symbolic of the entire universe.

Re: http://www.richardcassaro.com/tag/squar ... day-rimmer

Ultimately, it becomes a "spiritual reference to man’s instinctive quest to harmonize our physical and spiritual natures."
"These are not just squares and circles. These are squares and circles in balance, united in harmonious rhythm, a reunion of opposites."

And the inclusion of other geometric objects greatly expands this circle/square symbolism. The discovery of so much geometry (potentially) in squared circles was unexpected - the reason I started referring to "geometry toy box" (so much hidden within, awaiting discovery).


Rod

[Amigoo]

Here I go again ... back into the toy box (driven by more curiosity: what else is in there?)

The Incremental Squares of 2 design now displays a new expression of geometric balance:
the center of the design appears to be balanced as a gyroscope with one end touching the point of the large red triangle.
Squared Circle Geometry - a bottomless toy box!

Incidentally, I've been seeing x:03 and x:37 recently, but have not identified their associations.


Rod ... ...

[overmind]

Here's another idea. A circle is a square existing at all angles. Put differently, if any shape points in every direction, you are going to get a circle or a sphere. A circle becomes a point of infinite movement and possibility. It's just like the field of an electron. How do you discover the actual location of the electron? Observe or measure it. If there are 8 electrons, they can form a cube. However, there is still a sphere/field of possibility. Maybe this is too abstract.

Actually, when I think about it, the field might be a torus, but you know what I mean.

[Amigoo]

"So, you think the universe totters like a gyroscope?", teased Pythagoras. "Not so fast, green grasshopper!"

"After you arrive on Mansonia One, attend one of my introductory lectures on the real geometry of the local universe (I'm still learning about nearby universes). But sit in front of the class - even there you'll need super-vision for a while."

Translation: the other half of the large red triangle was discovered in the geometry toy box and was added to the design.

Rod

[Sandy]

Here I go again ... back into the toy box (driven by more curiosity: what else is in there?)

I'm glad to hear!...I enjoy hearing about your latest endeavors even if I don't understand it...I certainly appreciate the "fruits of your mathematical labors."
Keep on rolling, Rod!
XX Sandy

[Amigoo]

Overmind,

"I think I'm thinking"* about your explanation.

But I understand that the "squaring the circle" challenge relates more to the two-dimensional Cartesian plane; esentially, what can be solved with pencil, paper, compass and straightedge. My inexpensive CAD software helps me explore this geometry (and image editor creates the designs), but I keep in mind that the ancient Greek rules must be followed for any proposed solution and proof.

I'll need to refer to these articles again to see if they encompass your perspective:
http://en.wikipedia.org/wiki/Squaring_the_circle
http://en.wikipedia.org/wiki/Cartesian_ ... ate_system
http://en.wikipedia.org/wiki/Euclidean_space

* A friend, now a resident of the mansion worlds, often responded with this comment when prodded by her bridge partners to play a card.

Rod

[overmind]

Maybe you can look at it this way:



When both shapes have the same area, each end of the circle has to match the area of each corner of the square. You can also measure the area of one of the smaller squares, so the corner is going to be a little less than half of that. If you look at the middle 4 squares, you can see that the bigger square is the squared version of the smaller one (2x2, 4x4). You can also draw a circle around the middle 2x2 square the same way the first one was drawn and the outer circle will become the squared version of the smaller circle. So it looks like you can create a squared circle if you work backwards.

[overmind]

After doing some reading, it looks like I'm trying to solve the wrong problem. I was thinking of how someone can literally square a circle on paper, not draw a circle with an area that matches the square.

[Amigoo]

Overmind,

Your diagrams are similar to my early analysis of how to approach this challenge by "thinking outside the box". But once I discovered that a 62.4028873643093955482677952.. degree angle (the angle of a radius) must be present in a solution, I realized that a solution would not be easy (most of the math experts say "impossible").

Eventually, I discovered the trigonometry (click on the image here to view the PDF file: http://aitnaru.org/homepage/freewill.html ) which proves that a precise Pythagorean triangle effectively squares a circle. The problem - equally as intimidating as Pi - is how to draw that precise triangle by the Greek rules. However, the trigonometry samples in this PDF file convincingly suggest that the cosine angle can effectively match Pi digit for digit! And this well-supports the new Pythagorean concept.

The majority of the designs that I've been sharing attempt to reveal the various geometric lines, angles and objects which have potential for supporting this Pythagorean approach to squaring the circle. My recent Incremental Squares of 2 geometry is especially intriguing because it displays 5 squared circles, 4 of which have diameters that are increments of the square root of 2 (in units: 1000000.0, 1414213.562373095.., 2000000.0).

These interdependent lines, angles, and objects should promote future squared circle discoveries ... perhaps, including solutions and proofs for this Greek challenge. Until then, this new concept of Pi should at least stimulate new interest in the study of geometry.

Rod