Paradise Trinity Day

[Amigoo]

Re: Replication en Trio design

Perhaps, intended to be accompained by a Trio Sonata,
but convincing evidence that squared circle geometry
is simply complex ... and all about replicating patterns.

WYSIWYG ... depending on your belief and vision.

Rod

[Amigoo]

Re: Replication en Trio design (final, final, final)

The only thing difficult about squared circle geometry
is scheduling extra naps needed for brain overload.

Obvious pattern replication:

Circle 1: “Been here! Done that!”
Circle 2: “Been here! Done that!”
Circle 3: “Been here! Done that!”

Looks like a broadcast antenna:

Circle 3: “CYHMN?”
Circle 2: “CYHMN?”
Circle 1: “CYHMN?”

Rod ... ... (?NMHYC)

[Amigoo]

Re: Circular Triunity design

Well ... "final, final, final" did not prevent another improvement of Replication en Trio, but I'm quite satisified with the f,f,f,f version!

Yet, this is one more instance of creativity transition to another design immediately after using this 'F' word to describe a completed design. Obviously, I'm emotionally challenged by "final" and subconsciously (at first) plan to prove that it's not a controlling word.

However, many days are now passing where I don't roll the stone (front door) away from my man cave - with the vines growing so rapidly in Spring, this is not wise. There will have to be serious quiescence - maybe sabbatical - after posting Circular Triunity. Besides, the "butterfly" shape is positive reinforcement that "abode" should not be synonymous with "incarceration".

Rod ... ... (off to find examples of "sunshine" and "fresh air")

[Amigoo]

Re: Circular Triunity design
Now esoteric and open for interpretation. Here's one:

Golden chord of connectivity to a living universe of things, beings,
and unimaginable forces will soon be propelled into our Milky Way.

Rod ... ...

[Amigoo]

Re: Paradigmal Scalene Integration (PSI) design

Also "esoteric and open for interpretation".
(the 321,123 circular patterns inspire "paradigmal")

Diameters increase/decrease by sqrt(2) ...
(= 1.4142135623730950488016887242097..)

creating incremental diameters of 2(sqrt(1/Pi)).
(= 1.1283791670955125738961589031215..)

Rod ... ... (off to integrate line colors ...
with purposeful paradigmal persuasion).

[Amigoo]

Re: Paradigmal Scalene Integration design:
http://www.aitnaru.org/images/gEotP.pdf

If you've seen one squared circle, you've seen them all,
but those red flutterbys cocoonify within the boundaries
of infinity (paradigmally speaking).

Rod

[Amigoo]

Re: Paradigmal Scalene Integration design

(red flutterbys cocoonify within the boundaries of infinity)

Now, this squared circle geometry is "paradigmal"
(the vertical golden line has length = sqrt(Pi)/2 and
diagonal line (hypotenuse of right triangle) has length = 1)


1.0 / sqrt(Pi)/2 = 2(sqrt(1/Pi))

1.0 / 0.88622692545275801364908374167057..
= 1.1283791670955125738961589031215..

Rod

[Amigoo]

Re: Paradigmal Scalene Integration design

Another upgrade! Now, visual "all-you-can-eat-day"
in the Cartesian Neighborhood. But it's just a study of
the geometry associated with the lines of the large
circle-squaring scalene triangle* with golden lines.

* this triangle exists in every squared circle,
as well as the similar trapezoids which include the
scalene, right triangle, and isosceles right triangle
(all similar geometric objects in the Neighborhood).

Rod

[Amigoo]

Re: Paradigmal Scalene Integration design

as well as the similar trapezoids which include the
scalene, right triangle, and isosceles right triangle
(all similar geometric objects in the Neighborhood)

Re-colored to highlight perimeter of trapezoids.

Rod ... ...

[Amigoo]

Re: Paradigmal Scalene Integration design

So, where's the scalene triangle within the trapezoid?

It's defined by the shorter corner-to-opposite_corner line (not drawn)
and identifies a side of the squared circle's inscribed square
... with a consistent 45-degree angle in the scalene.

Rod

[Amigoo]

Re: Paradigmal Scalene Integration design
(also online: http://aitnaru.org/lighttruth.html )

"Integrated geometric objects in an 'impossible' Cartesian neighborhood,
highlighting similar trapezoids that all include a right triangle, isosceles right,
and circle-squaring scalene" ... all geometrically familial and similar.

Is the circle squared? "Who can tell?!"
It's an "impossbile" Cartesian neighborhood.

Rod

[Amigoo]

Re: Paradigmal Scalene Integration design

The two longer sides of the large golden scalene are intriguing
because they are the hypotenuse of an isoceles right triangle ... and
the four points provided (largest red flutterby) identify the diameter
of largest dark blue circle as well as side of its inscribed square:

Squared-circle geometric integration extraordinaire!
... and uniquely mathematically paradigmal!

Rod

[Amigoo]

Re: Paradigmal Scalene Integration design

My petite computer mouse began to slink off the desk
after the PSI Caliper (light blue) arrived, shaking
and mumbling that "They're here!!"

"Nonsense " I insisted. "They've always been here!
... even before Neighborhood hardware stores!"
... or circles with radials!"

The angle measured?
27.597112635690604451732204752339.. (approx.)
Hmmm ... that looks familiar!

Rod ... ...

[Amigoo]

Re: Paradigmal Scalene Integration design

27.597112635690604451732204752339.. (approx.)

I was reading "upside down and backwards"*
But the actual measurement relates to this value.

* During my first computer course (last millennium ),
the instructor explained how an impact printer works:
"a monkey sitting upside down and typing backwards".

Rod ... Well, I'll be an uncle's monkey!
... to remember the early 20s (mine)

[Amigoo]

Re: PSI 'n Focus
(Paradigmal Scalene Integration and Focus)

Curiosity about the flutterby shape motivated exploration
of this PSI geometry. Upon realizing that sqrt(2) has significant
presence associated with the flutterby, I noticed the obvious
sqrt(2) geometric signature: an isoceles right triangle (IRT).

PSI 'n Focus highlights the dominating IRT of the flutterby.

Rod ... ... (off to explore the Dominating IRT;
maybe my new pinpointer will help. )

[Amigoo]

Re: PSI 'n Focus design

Who knew? that so many lines and objects could be displayed
in this "impossible" Cartesian neighborhood!

The obvious theme is the confluence of scalene triangles,
but the isosceles right triangle (lower right, two green sides,
hypotenuse half green half light blue) is host for two similar
scalenes, both with lines colored red, yellow, and green.

"Squared-circle geometry that speaks for itself"
is serious understatement in this composition.

Rod ... ...

[Amigoo]

Re: Simply Scalene design

Finally, respite from the neuron-numbing complexity!
(but welcome excuse for multiple daily naps)

The two dissimilar, overlapping isosceles trapezoids
that share one side* while inscribed in a circle ( HCIT )
inquire of the scalene siblings: "Who's your daddy?"

* length = side of circle's inscribed square

Rod

[Amigoo]

Re: Scalene Soirée design
(renamed from Simply Scalene)

"Simply" was simply impossible - the geometry wants to be complex ... with its own "simplicity". But the repeating patterns stand out (especially in concentric circles) and now mandate a new geometry shape: TriScalene Pentagon, a similar object in every squared circle

... and composed of three similar, circle-squaring scalene triangles, a circle-squaring right triangle (close cousin of the circle-squaring scalenes), two dissimilar isosceles trapezoids, and an isosceles right triangle.

What's the point? All of these integrated objects derive from the five points of the TriScalene Pentagon.

Rod ... ... (off to navigate sharp turns in the Neighborhood)

[Amigoo]

Re: Scalene Soirée design

Proffer a new geometry shape with impressive name
and the geometry wants to be even more complex!
Now, hidden amongst the scalene siblings are two
right triangles, each squaring its respective circle.

Most noteworthy are the similar ratios of the lengths
of the hypotenuses and the lengths of long sides: sqrt(2)
(= 1.4142135623730950488016887242097..)

Replicating TriScalene Pentagon geometry!
What's next? A Diversey Parkway address?

Rod

[Amigoo]

Re: Scalene Soirée design

"final, final, final"

I'm certain because I tried to insert an inner circle (with it's squared-circle objects) and this Cartesian composition protested most convincingly: "You complain about complexity, then try to add your own! Take your extra tsp to a different recipe - this one is just right with 2 tsp."
( the little green circle has its own tsp, but not all golden lines )

All is well, racing along, on Terra Firma 000000 6 000000 ... ...

Rod

[Amigoo]

Re: Scalene Soirée design

"final, final, final"

... until the next "final"

Whereas insertion of a circle was frowned upon, an enclosing circle was not
(probably because it would not interfere with the central integration).
Circle-squaring TriScalene Pentagons in concentricity

Rod ... ... (off to price TSP trifocals)

[Amigoo]

Re: Scalene Soirée design
(re: http://www.aitnaru.org/images/gEotP.pdf )

"final, final, final ... final"
I'm certain because I just purchased a paperback* to read
that will stop the boisterous running in squared circles.

* There is a River by Thomas Sugrue (1942, 1973)
(an old story but well-written and informative)

Of particular geometric interest in Scalene Soirée
are the similar TriScalene Pentagons (largest and
its sibling within), similarly positioned.

Methinks that Pi cannot be transcendental
with this geometric similarity.

Rod

[Amigoo]

Re: Scalene Soirée design

Best guess about the concentric TriScalene Pentagons:

Geometrically prove that the length of one side of the area square of the larger circle
is precisely twice the length of one side of the area square of the smaller circle
... and suggesting that this Pi has no transcendental aroma.

Rod

[Sandy]

Re: Scalene Soirée design

it's a very beautiful design, Rod.
xxSandy

[Amigoo]

Re: Scalene Soirée design

Believing that the Scalene Soirée design was finished
and remembering that the central portion was "sacred",
I was curious about the recent x:28 midwayer prompts*.

This morning exploration helped me understand why
the central portion was such Scalene Soirée - not "sacred"
but reserved for specific geometry ... once discovered.

* x:28 relates to 2(sqrt(1/Pi))
= 1.1283791670955125738961589031215..

Geometers' clue about this lively center:

Diameter of smallest concentric circle (green)
= sqrt(2) x sqrt(1/Pi)
= 1.4142135623730950488016887242097..
x 0.56418958354775628694807945156077..
= 0.79788456080286535587989211986876..

BTW: 5:28 AM appeared as I was typing this.

Rod ... ...