Paradise Trinity Day

[Amigoo]

Re: Discernible Regularity design

The suggestion that men prefer less lavender in their squared circles was accepted by the design committee.
But who knew that men and women communicate on different wave lengths?
(lavender is in 400-450 nanometer range)
"Momma, is a rainbow when men and women yell at each other or when they cuddle?"
"No dear, a rainbow is when children are born."

Rod

[Amigoo]

Re: Discernible Regularity design

In Cartesian workspace, addition of three more circles (CSCSCSCSCSC continuum) confirmed the repeating patterns.

Each red isosceles right triangle is part of that circle's unique scalene triangle:
- the hypotenuse of the isosceles triangle has length equal to one side of the circle's square.
- the side of this triangle, when doubled, has length equal to the side of the next larger circle's square.
- the hypotenuse of this triangle, when halved, has length equal to the side of the next smaller circle's isosceles triangle.

In other words, there is backward/forward correspondence of these CSC.. circles and their isosceles triangles.
Does this discernible regularity show how to square a circle or prove that it's squared?

I don't know ... but the apparent existence of squared circle geometry is encouraging!

Rod

[Amigoo]

Re: Scalene Points (new design)

As the turtle "rushes" to the finish line, all participants discover that this was just the qualifying race. That Greek challenge issued centuries ago is now reclaimed in the name of Sanitas Cyclometricus!

The Scalene Points design shows why squared circle geometry exists and why a unique scalene triangle is the trademark of squared circles.
Yet, this geometry neither shows how to square the circle nor prove that it's squared (necessary disclaimer).

That's the reissued challenge: Squared circle geometry exists,
so discover how to draw it (according to the Greek rules) and create a geometric proof.

Scalene Points, created to show the scalene triangle, was drawn with these simple steps:

1. Isosceles right triangle (green) with hypotenuse length equal to square root of Pi.
2. Circle (lavender) with diameter equal to 2.
3. Circle's inscribed square (dark blue).
4. Isosceles right triangle (red), main portion of scalene triangle.
5. Vertical dark blue line to complete scalene triangle.

Rod ... ...

[Amigoo]

Re: sPointers Treasure (new design)

For several weeks, I've reached research plateaus where I'm convinced that it's time for closure! Not necessarily "the goal", but the point beyond which there is no trail and little hope for progress. And then, after a refreshing nap or brief distraction, the Cartesian workspace entices yet again.

Unlike the movies, this treasure box will not be raised from the dunes, but will be stuffed with more treasure (how this happens without raising the box will have to be imagined). The "treasure" is today's discovery that sets of geometric components (circle, circle's square, inscribed scalene triangles, trapezoid) are interrelated - proof of a circle's square may exist within this CSCSC continuum.

As in life itself, not all treasure is finally discovered or found to be "treasure". This is why the treasure box is stuffed rather than opened;
tasty squared circle morsels awaiting future exploration. The design could drift upon the dunes in a few days.

Rod

[Amigoo]

Re: sPointers Treasure (new design)

Reviewing the geometry for future development of this new design, I searched for correspondence of the inscribed trapezoids in the CSCSC circles. After observing that the length of the diagonal of the trapezoid in the larger circle was twice the length of the diagonal of the trapezoid in the smallest circle, I glanced at the clock: 9:54 PM.

Why did this time appear to be a Midwayer prompt?! I suspected a line length correspondence but already knew the lengths of both diagonals (smaller diagonal is 954225.. units long). Corresponding lengths, but I knew this! So, I explored again and discovered a possible reason for the prompt:

The smaller diagonal had half the length of the larger diagonal as well as being equal to the length of the horizontal midpoint-to-midpoint line in the middle circle's trapezoid. Another clue about the geometric correspondence of CSCSC squared circles and their inscribed trapezoids!

Say what?!

This complex geometry is often "Greek to me!" but I focus on patterns and think twice before ignoring prompts that might be clues.
Did I mention that each trapezoid is composed of two similar scalene triangles?!

Rod

[Amigoo]

Re: sPointers Treasure (new design)

Oops! A bit TMI for such limited Cartesian workspace.
(design clarity quickly degrades with so many geometric objects)

So, a less ambitious version might be attempted next week
but will still include the touted trapezoids.

Rod

[Amigoo]

Re: sPointers Treasure (first draft)

"I know what you're doing!" lectured Sees 'em.

"You're pre-planning a convenient excuse to take breaks from all that yard work and home maintenance that you've postponed all winter. Post a draft version of this Geometry 101 concept and take your breaks when you earn them."

Aye! Aye! Sees 'em ... Arrrrgh!

Rod

[Amigoo]

Re: sPointers Treasure design

This was going to be an updated draft, but now qualifies as completed design (without diagonal lines that identify scalene triangles).

This design is an excellent example of how patterns promote squared circle geometry, how these unique trapezoids (two similar, overlapping scalene triangles) identify the circle's square, and how the CSCSCSC continuum (four circles) directly relate to the entire composition!

Rod

[Amigoo]

Re: sPointers Treasure design
http://aitnaru.org/psymmetry.html

When one scalene triangle per circle in a CSCSCSC continuum (four circles)
is not enough to convince that squared circle geometry exists,
consider two per circle (overlapping in a trapezoid).

Rod

[Sandy]

That's an intriguing design, Rod! I keep finding I am turning my head from side to side looking for something I've missed... new patterns that jump out at me.
XXSandy

[Amigoo]

Re: Scalene Transition (new design)

Sandy,

Me too ("looking for something I've missed") but in the geometry not in the designs. And the conclusion has been repeating for a month: this scalene triangle (and related trapezoid) is the core geometry of a squared circle. End of story!

While "Scalene Transition" was a title selected today for the clockwise (or counterclockwise) decreasing (or increasing) size of the scalene triangles, relative to the diameters, I now suspect that the title also refers to this project - there is no need to keep searching. Many more designs can be created but the core geometry will not change ... if the circle is truly squared.

A summer break is at hand ... and I like to review products to post comments online. I could focus on sunscreen lotions or cold beer ... or both! Even sketch a few squared circles in the sand while enjoying a cool ocean breeze.

Life is good!

Rod ... ...

[Amigoo]

Re: Scalene, VP (new design)

The first time for this ...
I was not enthusiastic about creating the Scalene Transition design ("Been there - Done that" ... several times), but reviewed the geometry and even created an acceptable design. I was thinking "These scalene triangles are impressive, but what else can they do?"

Yawn! I decided to take advantage of a decent bedtime (1:30 AM) and postpone the PDF update until morning.

I went to bed. Twenty minutes later, I was back at the computer investigating something that I had casually observed an hour before: one of the "vertical" scalene triangles effectively defined both the circle and the square of the next larger circle ... and created a vesica piscis with the larger circle!

Scalene, VP is this design.

Rod

[Amigoo]

Re: Scalene, VP design

In this geometry, the large yellow circle was drawn last,
guided by potentials of the scalene triangle in the smallest circle,
creating a vesica piscis (VP) with the largest light blue circle.

A circle is squared, creating a square that is circled
... and then confirmed by a vesica piscis
... and authenticated by a scalene (red).

Nice performance, Vesica and Scalene!

Rod

[Amigoo]

Re: Scalene, VP design
There's nothing more thrilling (figuratively speaking)
than to claim so much and know so little:

After experiencing a certain scalene triangle, I would say:
“Even complex math cannot yet describe squared circles
… and the derivation of Pi may be the reason.”

Rod

[Amigoo]

Re: Scalene, VP design ("one more line")

When rushing to square a circle ...
If you don't know where you're going,
at least know where you've been ...
or you'll be running in squared circles.

The "Texas T" prompt was added to acknowledge its motivation and guidance for many months. Its position in the geometry now appears salient, perhaps identifying the very core of squared circle geometry!

Rod

[Amigoo]

Re: Scalene Fource (new design)

"Circle-squaring magic of this scalene triangle,
not once, twice, or thrice, but fource!"

When a certain scalene triangle is present, once you've seen a squared circle (geometrically speaking) you've seen them all.
This set of four nested circles, a CSCSCSC continuum, is a good example.

Rod

[Amigoo]

Re: Scalene Fource design ("one more line")

Sees 'em commented "Nice work, but may I suggest 'one more line'?"
"What?!" I protested. "I'm ready to wrap! Seriously!"
He continued, "Please include: 'May the Fource be with you.'"

Say what?! But I added the line (actually, moved part of a line).
The red scalene triangle now explains this "Fource".

Rod

[Amigoo]

Re: iScalene (new design) - http://aitnaru.org/threepoints.html
If the Scalene Fource geometry seems esoteric, evaluate the simplicity* of this iScalene design,
the basic anatomy of a squared circle.

diameter = 2, side of blue triangle = half the square root of Pi.

* Simplicity as geometry but profound symbolism:
A squared circle is identified by a pair of overlapping scalene triangles (a trapezoid). For centuries, a squared circle has been considered the metaphysical union of heaven and earth. The pairing of these unique triangles in an "impossible" geometric construction easily symbolizes a pair of seraphim, man's attending angels.

Since these "angels of time and space are endowed with the ability to discern both spirit and material beings" (44:0.17), the union of heaven and earth is within our mindal grasp. Thus, spiritually advancing mortals have potential awareness of the already existing "union" of heaven and earth even now as human residents of our current abode in time and space: Urantia.

Rod

[Amigoo]

Re: iScalene design

This basic anatomy of a squared circle provides opportunity to revisit Pi, an approximation. "Aproximation"?! Of course, or the world's fastest computers wouldn't be calculating more decimal digits of Pi every year. While this scalene-triangle-focused geometry neither shows how to square the circle nor prove that it's squared, it has this potential.

But how?

What seems certain is that squared circle geometry exists that parallels the accuracy of Pi: the more precise Pi, the more accurate the geometry. When the final digit of Pi is determined, we will know the final squared circle geometry.

And if we reflect on the seraphim symbolism, with Pi having now been calculated to billions of decimal digits, the impressive current accuracy of Pi suggests an equally impressive presence of man's attending angels (plus other celestial beings)!

Rod

[Amigoo]

Re: iTrapezoid (new design)

The geometry of recent weeks is more documentation than design, but "design" also indicates conversion from the simple lines of the CAD drawing. This iTrapezoid geometry, a CSCSCSC continuum (four circles), highlights the four integrated isosceles trapezoids (two scalene triangles per trapezoid; one inscribed trapezoid per circle).

Interesting consistencies of the trapezoids:
0. the largest circle's diameter = 2; side of circle's square = square root of Pi.
1. the many sets of angles in these nested objects are similar - even visually redundant!
2. the longest line of the trapezoid always has length equal to the side of that circle's square.
3. the mid-point to mid-point line of the non-parallel lines is also the diagonal of the next smaller circle's trapezoid.

So, "What's the point?"
If repeating patterns in nested geometric objects, all integrated within a CSC.. continuum, cannot confirm that squared circle geometry exists, then time and space are an illusion (but keep paying taxes until this is proven).

Rod

[Amigoo]

Considering local "invitations" to other opportunities, this seems timely:
Note to self: The project has ended - enough with "documentation!"

Rod ... ...

[Sandy]

it's ended? I hope not, Rod...but if so...please do consider this space your own and feel free to document all the other wonders you discover from other opportunities. It has been encouraging and comforting for me to come here every day and read about what you have been up too in your geometric search. I did not always understand what you are talking about but is a blessing all the same.
Love,
Sandy

[Amigoo]

Sandy,

Now I'm ready to compromise and say that the geometry reached a solid plateau,
providing a break for renewing creative energies and time to attend to
postponed home maintenance and domestic chores.

Midwayer prompting humor about this?: x:07 followed by x:08
(x:07 relates to the geometry, x:08 is part of my house number)
The x:07,x:08 display has been a frequent visitor recently.

Rod

[Sandy]

You can't argue with the numbers.
Yes, those home maintenance chores are of some import if we wish to keep a roof over our heads.
Enjoy!
Love,
Sandy

[Amigoo]

Re: Son-iT (new design, son of iTrapezoid)*

How is it possible? Can one complex pattern replicate another?
(all lines deleted except those for two circles, a CSC continuum)

The geometry in this subset of iTrapezoid appears to be significant.
I'll review this in the days ahead while attending to chores.

* not related to Son-aB, truly impossible geometry.

Rod