Paradise Trinity Day

[Amigoo]

Re: Circular Reasoning design
http://aitnaru.org/psymmetry.html

Rod

[Amigoo]

Re: Open the Box design
"However, one unique isosceles trapezoid exists for a squared circle."

The geometry of this design shows how such a trapezoid should appear
... if Pi is the square of a circle having a diameter of 2.

The midpoint-to-midpoint distance of the sides of the trapezoid must have length equal to the diagonal of a square having sides with length equal to half the square root of Pi. Half of this square is shown as an isosceles right triangle with yellow sides and hypotenuse that is half red and half blue.

Rod

[Amigoo]

Re: Open the Box design
"The midpoint-to-midpoint distance ... "
Why so many halves? Every good Pi should be shared ... and sung:
Sing along, substituting "yellow Pi routine" for "yellow submarine"
in the old Beatles' song "We all live in a yellow submarine."
and "yellow Pi routine" may inspire cool enlightenment!

Rod

[Amigoo]

Of course, "yellow Pi routine" is open for interpretation, but here's a good one
(and it's remotely related to real Pi): http://pining.ning.com/profiles/blogs/pi-s-routine

But what would "yellow" mean? Lotsa sun? Jealousy and envy?
It probably depends on your typical "Pi routine".

This geometry has suddenly waxed so philosophical
that I'll need a nap on a cool yellow pillow.

Rod

[Amigoo]

Re: sCorrespondence (new design)

Obviously, my "Pi routine" for months has been the exploration of the geometry of a squared circle. sCorrespondence (composite of "sibling correspondence" to reflect the two CSC nested squared circles) is the isolation of core geometry.

Not an impressive design (and perhaps chaotic to non-geometers), but the lines and angles of sC will be found in every CSC set of squared circles ... and even in CSCSC.. circles and beyond!

What's next? For this summer, more yellow (lotsa sun) is best to atone for a long winter Pi routine with seemingly 24-7 mouse massage.

Rod

[Amigoo]

Re: sCorrespondence design
http://aitnaru.org/threepoints.html

A simpler version of geometry that easily becomes quite complex
and highlighting the influence of triangles on squared circles.

Rod ... ...

[Amigoo]

Re: Concentric Sees (new design)

Hmmm ... Why did I not recall that "balance" is a universe mandate;
that a "simpler design" would beg a complex design?

Concentric Sees (three concentric circles, all squared,
with "Sees" both allusion to the CSCSC pattern and emphasis on "See!")
visually communicates such balance!

Hmmm ... "Sees" sounds like "seas", perhaps forecasting a new voyage.

Rod

[Amigoo]

Re: Concentric Sees design
When squared circle geometry waxes whimsical.

Every squared circle voyage needs a mascot.
Meet "Sees 'em", the adventure retriever.

Caution: "Sees 'em" has a large appetite for adventure!

Rod

[Amigoo]

Re: Concentric Sees design

Regarding the FAQ, "Is Sees 'em a boy or a girl?"
Sees 'em always answers this question: "young adult - unaffiliated".

Rod

[Amigoo]

Re: Concentric Sees design

Only to share evidence that a Sees 'em voyage may be in progress, a new design is recorded (sPointers Island).

Days after embarcation, Sees 'em was surveying the horizon when a small island was spied, illuminated by the setting sun. With investigatory senses aroused, Sees 'em rushed to the captain to report a squared circles investigatory opportunity on sPointers Island.

"sPointers Island?" guffawed Captain Incredulous. There's no such island on my charts and they're from the international hydrographic center!

Sees 'em was not dismayed - squared circles investigatory senses (SCIS, pronounced "sighs") are most reliable, at least when possessed by a voyage mascot of Sees 'em's renoun. In the day's final scene of this voyage (at the ship's observation post) we see Sees 'em's SCIS in solitude.

To be continued (or not), depending on the quality of evidence at sPointers Island.

Rod

[Amigoo]

Re: sPointers Island design

After a refreshing late afternoon nap, I awakened slowly and contemplated the sPointers Island geometry. I finally understood Sees 'em's SCIS when I modified the geometry with "just a few more lines".

... but first, this unresolved consternation:

Popular myth maintains that the CSC geometry of sPointers Island relates to two two-word sets (not 22-word sets) that are combinations of these four words: Spooners, Schooner, Court, and Port.

Here are the probable combinations (only "Spooners" is plural, for obvious reasons):

Court's Port, Port's Court,
Spooners' Port, Spooners' Court,
Schooner's Port, Schooner's Court,
Schooner's Spooners, Spooners' Schooner.

Say what?!

Having now studied the geometry, I suspect that the two (and only two) mysterious two-words sets identify two locations in close proximity on the coast of the Island. And further, the juxtaposition of these two locations somehow gives precise meaning to the geometry.

What's between "here and there" if "between" were a straight line?
Say what?! A range? A straight line called "between"? Not in my Math 101 books!
Perhaps, a straight line positioned with "here" on one side and "there" on the other side.

Oh, did I mention that the myth narrates that the straight line between "here and there" does not exist in the geometry?
But "between", by definition, does exist!

Just forget it! Any visitor to the Island, with sufficient time and leisure,
can investigate all possible locations identified by combinations of "Spooners, Schooner, Court, and Port".

As for the relationship of the two actual locations, that's between "here and there".

What?! The non-existing straight line?
No problem! Just assess Sees 'em's succinct saying:
"May the (lines of) force be with you".

Rod

[Amigoo]

Re: sPointers Island design

Because of the months-long research leading to discovery of a certain scalene triangle* that squares every circle, I quickly learned that the sPointers geometry presents a visual key for squaring every circle in the CSC range (from larger circle having a diameter of 2 to smaller circle having a diameter equal to the square root of 2).

However, I have no uncertain instruction from Sees 'em to share no more "at this point in time."

"What's the point?" I probed, already knowing a few points about this geometry challenge.

"I Sees 'em, voyage mascot, you geometry junior, Grasshopper.
Now hop along Hopalong and refresh your zest -
sPointers Island is still just a point on the horizon."

* Every scalene triangle of circle-squaring potential contains similar angles (one of them a 45-degree angle), a side with length equal to the side of a square inscribed in the circle, and another side with length equal to a side of the square of the circle.

Rod

[Amigoo]

Re: sPointers Compass (new design)

Since Sees 'em only responded "No comment!" to any discussion of a mysterious "sand dollar" (aka "sea cookie", "snapper biscuit", "pansy shell") reportedly discovered long ago on sPointers Island, I assumed that I could share its geometry.

According to legend, a young geometer from the regional Sanitas Cyclometricus congregation discovered that the object represented a squared circle. However, adding the perimeter of the square did not reveal the purpose of the four straight lines, all having equal length (1.7005552066.. units).

(What happens on sPointers Island, stays on sPointers Island: "Who can tell?' how such precise measurements were made years ago on a remote island. Alien technology?)

Eventually, the object acquired the nickname "sPointers Compass" because of its resemblance to a land explorer's compass. However, every explorer who evaluated this "device" complained that it was only as valuable as a burrowing sea urchin belonging to the order Clypeasteroida.

Once the local Chamber of Commerce claimed that it was a treasure map, its popularity soared amongst island inhabitants and infrequent tourists; facsimiles were printed even faster the regional FRB's island bounty bucks. But this quantitative expansion hastened the demise of this island enterprise - most of sPointers' paper sand dollars were soon dumped into the ocean (not the cause of today's rising seas).

Long story short (or short story long), the sPointers Compass geometry appears to have real value as a guide to circle squaring
... but otherwise has no other value imaginable!

Rod

[Amigoo]

Re: sPointers Compass design

This simple design refused to go to bed, keeping me up hours past another 5:00 AM bedtime.

Only by mid-morning was the reason apparent: this minimal value "sand dollar" possesses unbelievable core geometry. After hours of exploration with many repetitions of "one more line", its hidden geometry was revealed.

If this geometry cannot convince that a certain scalene triangle squares the circle, then no such geometry exists!

Rod

[Amigoo]

Re: sPointers Compass design
(overlapping squares completed)

Who knew?
An SAT (Spoke Adjustment Tool) is available for those
who plan to "square the circle", educationally speaking.
Get a fast ride to school!

Visually, the SAT (as a squared circle tool) appears to ...
1. increase the diameter of the circle and decrease the size of the square (turn SAT clockwise).
2. decrease the diameter of the circle and increase the size of the square (turn SAT counterclockwise).

Thus, the circle is squared when perfectly adjusted with the SAT!
... but a prototype is required to test this conjecture.

Rod

[Amigoo]

Re: Candle or Flame (new design)
"Which is more important, the lamp or the bulb,
the candle or the flame, the herald or the light?"

A previous design focused on an abstract bulb - this design substitutes a candle (and not so abstract).

As often happens, I'm focusing on the "delicious" geometry (a geometer perspective) and don't see the message developing until I'm converting the CAD drawing into a design. In this case, the 27.597.. degree angles (two right triangles forming the sides of the candle) are indicative of precise* squared circle geometry. The outer side of this triangle has length equal to the side of the larger circle's inscribed square; the side of the candle has length equal to the hypotenuse of a right triangle whose sides have length equal to half the square root of Pi (circle's diameter = 2).

Intuition about the message: Future symbolism about squared circle geometry will be finessed by our celestial friends to support their agenda (promoting the brotherhood of man and fatherhood of God). Whatever we humans choose to do with any squared circles successes (regarding math and geometry) is not necessarily important to the planet's march to Light and Life.

* Potentially precise since the effective limit is the CAD software, but visually no inaccuracy can be detected
... and this is why a geometric proof is ultimately required to validate a squared circle.

Rod

[Amigoo]

Re: Candle or Flame?
(updated with "just a few more lines")

Suspecting that this design is advice to me as well (about this "geometry challenge from antiquity"), I'll light the candle and meditate away from the geometry toy box for a while. However, why miss an opportunity to summarize experience?:

Over the recent years, I have learned that few people (even mathematicians) know much about the geometry that accurately* describes squared circles. And the rest keep asking "What's the point? Remember Lindemann, 1882?" Intriguing! Only one point?! ... in a Pi containing trillions of tasty points!

Here's my simple argument for the record:

The area of the square of a circle must be less than the area of a square that encloses the circle and greater than the area of a square inscribed in the circle. Within this range, the circle's perfect square exists with sides that touch the circle at 8 points only (2 points on the circle per side of square). Much geometry exists within the symmetry of this well-ordered Cartesian space! The many designs prove this.

* Earlier discovery of Pi-related trigonometry reveals that precise squared circle geometry must support the existence of 27.5971126356906044517322047523.. degree angles in right triangles whose hypotenuse is the circle's radius.

Disclaimer: Circle-squaring geometry must exist, but only a geometric proof can confirm this reality
... despite the visual confirmation of the many designs.

Rod

[Amigoo]

Re: Scalene Spiral (new design)

When squared circle geometry starts drawing spirals, it's time for me to exit the creativity sandbox.
But this first spiral pattern is worth the effort since it shows direct relationship of circles to their squares
(via the circle and its unique circle-squaring scalene triangle).

In the design, when the smallest scalene triangle is rotated counter-clockwise, the triangle keeps enlarging and maintains direct size relationship to diameters. The 45-degree vertex* of the scalene triangle is observed following a counter-clockwise spiral track (golden).

A bit TMI for my geometry skills, but the obvious pattern is easy enough to draw
and the design valuable for future study of "impossible" squared circles.

* scalene triangle's angles = 45, 62.403.., 72.597.. degrees.

Rod ... ...

[Amigoo]

Re: Scalene Spiral design (added to PDF).

This is one fascinating squared circle pattern! Once the spiral was detected, it's easy to see CSC, CSC pairing in these four concentric circles, all squared (in a CSCSCSC continuum). This may be an "Archimedean Spiral".

The new math supporting this pattern should arrive soon, depending on removal of the Lindemann impasse.

Rod

[Amigoo]

Re: Scalene Spiral design (true project closure?)

Once the spiral was discovered, it was easy to see CSC, CSC pairing in these four concentric circles, all squared (a CSCSCSC* continuum). This pairing pattern was first observed two years ago (see "What's the point?" design in PDF) when the pre-scalene sibling (right triangle, red) was being explored.

The geometry of Scalene Spiral may be an "Archimedean Spiral". The new math supporting this pattern should arrive soon, depending on removal of the Lindemann impasse.

* Circle
within inscribed square
within inscribed circle
within inscribed square
within inscribed circle
within inscribed square
within circle.

Note: three sides of the inscribing square are not displayed for design clarity.


Rod

[Amigoo]

Re: What's the Point? design
"In a Pi of trillions of tasty points?"

The CSC pairing was first observed in this geometry, studied two years ago. Intuitively, the pairing seemed significant then but is emphasized even more with this week's Scalene Spiral geometry.

The Points' CAD conversion to design was redone today with current skills and experience.

Rod

[Amigoo]

Re: Scalene Spiral design (closure? maybe not!)

gJ: "Sees 'em, did you see the neat Scalene Spiral that I discovered?"
Se: "Nice work, Geometry Junior, but what about the two other arms?"
gJ: "Spirals don't have armS - just one arm!"
Se: "Did you not define a scalene triangle that looks like me?"
gJ: "Yes, but the 45-degree vertex is the point in focus."
Se: "Do I not have two other points that also expand outward?"
gJ: "Of course, but I didn't notice them. Are they important?"
Se: "That's why you Geometry Junior, searching for squared circles, but I Sees 'em."
Se: "Take your time - you know that this squared circles toy box is bottomless."

Rod ...

[Amigoo]

Re: Scalene Spiral design ("collector's edition")

Definitely tedious to draw, especially with six squared circles!
But the spiral's three arms are highlighted well!

This geometry would make an impressive crop circle, mesmerizing fireworks,
or get you a free cup of coffee at your favorite bistro (if you leave a large tip).

Rod

[Amigoo]

Re: Discernible Regularity (new design)

"Discernible regularity" is a common expression used in definitions of "pattern". No matter how many times I revisit squared circle geometry, I usually find a certain "discernible regularity" but never find a simple solution to this Greek geometry challenge from antiquity.

I'm forced to accept that which should have been intuitive, especially since no solution to "squaring the circle" has ever been found and most geometers/mathematicians now believe that squaring the circle is impossible. What should be intuitive? That a solution would not be simple!
"I knew that squaring the circle was impossible, but I didn't know that it would be so difficult!"

This is where Discernible Regularity geometry well supports Sanitas Cyclometricus, geometer activity complement to Morbus Cyclometricus:

The three squared circles (CSCSC continuum) in this geometry contain discernible geometric components (especially, isosceles right triangles) that repeat in each circle. In fact, one set of components (including the circle) appears to facilitate generation of the next larger (or smaller) set of components!

Study of squared circles should include reflection on this very discernible but complex regularity!

Rod

[Amigoo]

Re: Discernible Regularity design ("just a few more lines")

Today the DR is in, and the comprehensive evaluation advises:
"If this core geometry of squared circles is not meaningful to the geometer, then squaring the circle will indeed be impossible!"

This geometry of three squared circles (a CSCSC continuum) well highlights the unique circle-squaring scalene triangle inscribed in each circle. Each red isosceles right triangle is part of that circle's scalene triangle: the hypotenuse of the red triangle has length equal to one side of that circle's square; the side of the red triangle when doubled (or halved) has length equal to another circle's square.

Wait! There's more! ... but not today.
Exploration can be self-rewarding in an ambiance of Sanitas Cyclometricus.

Rod