Paradise Trinity Day

[Amigoo]

Sandy,

A mind checker would also help!
I read your "sun and lake is beckoning", then my mind substituted a word:
"... I will need a bathing attachment to my push mower."

Hmmm .... what's "bathing attachment"? A water sprinkler?

Rod

[Sandy]

Hi Rod,

A mind checker would also help!
I read your "sun and lake is beckoning", then my mind substituted a word:
"... I will need a bathing attachment to my push mower."

Hmmm .... what's "bathing attachment"? A water sprinkler?

Now you're talking!!! It is very hot here today and so a "bathing attachment" to my mower would be just the ticket! if only there was such a miracle device!! Hey, we'd make millions! and people would be lining up to "cut their neighbor's grass."

xxSandy

[Amigoo]

Re: Transcendentally (new design concept)
“Anticipating one trillion ... patterns.”

Expansion of the CSC geometry to CSCSC presents new opportunities for complex patterns, all confirming the probable existence of squared circles (at least, in the universe). This Transcendentally geometry highlights very obvious patterns and posits that transcendental Pi cannot exist in an infinitely repeating geometric landscape dominated by sqrt(2).

... or perhaps there is indeed an irrational/transcendental mathematical boundary where the decimal digits of Pi stop being irrational (only) and become transcendental (in all definitions of this word). "Who can tell?" The universe patterns know and often disclose the "unknowable".

Rod

[Amigoo]

Re: Transcendentally design
“Whereupon the border of irrational and transcendental ...”

Then what happens? Some will enter therein and explore,
but might return unchanged. Others will stand at the gate
and remain unchanged. Whence derives the adventure?

Rod ... ...

[Amigoo]

Re: Transcendentally design
“Whereupon the border of irrational and transcendental ...”

With the lengths of the bottom line of each of two isosceles trapezoids (light blue; inscribed in the inner and outer circles of the set of three) equal to 1 and 2, respectively, and those lines representing a side of the square of each circle, the transcendental digits of Pi are effectively transferred to the diameters of the two circles ... apparently.

Since sqrt(2) defines the relative difference in the lengths of the two diameters, whereupon this Cartesian border of irrational and transcendental does this inspired geometric perspective wax and wane?

Rod ... ... R 'n R (near the border ... apparently)

[Amigoo]

Re: iTrapezoid, iSpiral designs

“Whereupon the border of irrational and transcendental”
sqrt(2) can oft be found, nigh rational - not incidental.

Hmmm ...
if the bottom line of a trapezoid is a side of the circle's square,
the sides of the trapezoid the two sides of inscribed squares,
the ratio of the diagonal to the mid-point line equals sqrt(2)
... how irrationally transcendent is that?!

About iSpiral ...

The smallest squared circle (green square) has area
equal to 1/16 of the area of the largest squared circle
and created in a decreasing spiral!

Rod

[Amigoo]

Re: iTriangle, iMetamorphosis designs
“Been there! Done that!”

iTriangle, with its right triangle simplicity, suggested project closure
... until iMetamorphosis appeared in the Cartesian neighborhood, as both
contrast to iTriangle's simplicity and forecast of pending metamorphosis.

Obviously, a sqrt(2) perspective of squared circle geometry.

Rod ... ...

[Amigoo]

Re: iMetamorphosis design
“Been there! Done that!”

Squared circle geometry speaks with eloquence
... when its message is "transition time"!

The portfolio: http://aitnaru.org/images/Tripartite_Soul.pdf

Rod

[Amigoo]

Re: Scalene Fource (new design concept)

Introduction ... ...

What?! There is only "once, twice, thrice" ... but no more!
No whey! The recipe pattern shouts that "fource" is next.

Consider: "on ce, tw o ice, thr ee ice, four __"
is this pattern: "once, twice, thrice, fource"

- maintains one-syllable words, ending with 's' sound
- each uses beginning letters of word (number) with 1+ sequence after "once" (2:tw-, 3:thr-, 4:four-)
- "fource" sounds like "force" which emphasizes "not once, twice, or thrice, but fource!"

What comes after "fource"? "Fice" might suffice ... within a few centuries.

Say what?

When squared circle geometry teases with a vesica duo (two inscribed squares and two squared circles) upon a trio of circles having the same diameter, this geometric familiarity is promoted by a certain "scalene fource" ... not once, twice, or thrice but fource in this Cartesian neighborhood.

"But there are five squared circles ", you protest.
Sorry ... "Fice" cannot exist until we go with the "fource".

Rod

[Amigoo]

Re: Scalene Fource design

Good enough for the "proof of concept".
(two vesica piscis, each centered on a side of the
circle-squaring scalene triangle, is a fource to behold).

Rod

[Amigoo]

Re: Scalene Fource design
About vesicae pisces (plural of "vesica piscis") ...

I've long been a bit casual (math teachers and good students would insist "ignorant" ) about the term, vesica piscis, applying it to any overlapping of two circles having the same diameter* length. Besides, the inherent inscribed squares of two such overlapping circles (this "vesica" vertical line is a shared side of the two squares) and the similarly shared sides of the two squares of such circles is significant geometry in a squared circles Cartesian neighborhood (IMO).

About the real vesica piscis ...

Re: https://en.wikipedia.org/wiki/Vesica_piscis

"The vesica piscis is a shape that is the intersection of two disks with the same radius, intersecting in such a way that the center of each disk lies on the perimeter of the other."

"Mathematically, the vesica piscis is a special case of a lens, the shape formed by the intersection of two disks. The mathematical ratio of the height of the vesica piscis to the width across its center is the square root of 3 " (1.7320508075688772935274463415059..)

Obviously, this self-described "geometer" proffers that these "vesicae pisces" in a squared circles Cartesian neighborhood are quite significant, geometrically speaking (but only in serious conversations about the "impossible" squared circles). At least, the Scalene Fource geometry tempts squared circle beliebers ( confusing the 'b' and 'v' sounds from foreign language study) to consider this controversial math reality (IMO).

* Pop Quiz (answer is Given ... somewhere):
When two identical circles have the same diameter, how do you know that there are two circles?

Rod

[Amigoo]

Re: Scalene Fource design (re: vesicae pisces)
About the foundational scalene triangle (red) ...

What's the "vesica" perspective of three circles
having the same length diameter, with two circles
having adjoined inscribed squares and two having
overlapping squares (of those two circles)?

sqrt(Pi) = 1.7724538509055160272981674833411..
sqrt(2) = 1.4142135623730950488016887242097..

1.7724538509055160272981674833411..
/ 1.4142135623730950488016887242097..
= 1.253314137315500251207882642406..

Ratio of the square of the circle (side length in one "vesica")
to the shared side of adjoined inscribed squares = sqrt(Pi)/sqrt(2)
= 1.253314137315500251207882642406..

Conjecture: sqrt(3) ratio in the vesica piscis "lens" is less significant,
geometrically, than the ratio of these two sides of the circle-squaring
scalene triangle; the circle is not squared until this scalene triangle
with its own "dual vesica" ratio is present in the geometry.

Rod ... ...

[Amigoo]

Re: Scalene Fource design (a few more lines)
"Lines and triangles and squares! Oh my!"

Of course, a Cartesian, squared-circles jungle (with redefined vesicae pisces)
would be intimidating! After all, for centuries few have dared to explore therein
... not to mention that its existence was "proven" impossible in the 1800's.

What's the "four"? The four points of the circle-inscribed quadrilateral,
which includes the circle-squaring scalene triangle. This quadrilateral contains
three triangles: a right, isoceles right, and scalene. Hmmm ... three = four!

Ready to explore? May the Scalene Fource be with you.

Rod

[Amigoo]

Re: Inscribed Alignment design

Apparently, my continuing curiosity about the 'X' shape geometric object
(golden lines) must have been "good & sufficient" request for clarification.
This design was constructed with such minimal planning and effort that
it qualifies as an Addendum to this geometry research.

These golden lines show the geometric relationship of one side
of two inscribed squares where D = 2 and the smaller D = sqrt(2)
... with both circles displaying their area squares.

Rod ... ...

[Amigoo]

Re: Inscribed Alignment design

If an Addendum is "a thing to be added", this geometry is quite a thing to be added!
The design "constructed with such minimal planning and effort" was just a first draft.
When redrawn, it expanded to include two 'X' shapes, one in each circle.

Lesson learned! And curiosity satisfied ... totally!!
And to guarantee "Lesson learned!", I was forced to keep searching for the X & Y axis,
if only for familiar guidance in this "impossible", X-rated Cartesian neighborhood.

Rod

[Amigoo]

Re: Inscribed Alignment design
(see in: http://aitnaru.org/images/Tripartite_Soul.pdf )

If all of this squared circles research can be expressed in one Cartesian display,
this is that expression! Even the symbolism (including the mini x/y at the bottom)
shouts that squared circles reflect geometric balance of lines, angles, and objects.
D = 2 and D = sqrt(2); side of square = sqrt(Pi), sqrt(Pi)/sqrt(2)

Got belief, faith, and balance? Tour the neighborhood at your leisure.

Rod ... ...

[Amigoo]

Re: Inscribed Alignment design
Who knew? There was another Pi in the oven (D = 1).

For Diameter = 2, sqrt(2), and 1,
side of circle's square = sqrt(Pi), sqrt(Pi)/sqrt(2), sqrt(Pi)/2.

Rod

[Amigoo]

Re: Inscribed Alignment design (updated again)
This geometry keeps nagging for "minor" improvements.
(probably related to its value as a squared circles reference)

With the recoloring of the scalene triangles (red, one in each circle),
a Pop Quiz seems timely to emphasize the geometric integration:

Are the three overlapping scalene triangles geometrically similar?
How do you know?

Rod

[Amigoo]

Re: Inscribed Alignment design (updated yet again)

Apparently, when you discover that this geometry supports a Vesica Trinity
(on the sides of inscribed squares), it's time to accept that you may have climbed
to the squared circles mountain top. Sample the precise air for future reference
before returning to the popular valleys of the "impossible".

Rod ... ...

[Amigoo]

Re: Inscribed Alignment design

Caution: Those who comprehend this geometry may believe that they can
walk on troubled waters, but Scalene Sailing is more difficult than it looks.

At least, align with planetary x/y coordinates before you Ro, Ro, Ro your boat.

Rod

[Amigoo]

Re: Inscribed Alignment design

Since the largest circle (light blue) was not Vesica-paired because of increased design size, a few more lines were added to identify that circle's center. Once these lines were included, another squared-circle-identifying quadrilateral appeared! And this justifies restatement of the Caution:

"Those who comprehend this geometry may believe that they can walk on troubled waters, but Scalene Sailing in the universe is more difficult than it looks. Align with this planet's spiritual coordinates before attempting such a voyage."

Rod

[Amigoo]

Re: Inscribed Alignment design

Oy Vey! Even more lines! (but better geometric balance)
Not to worry - the x/y axis supports all "stabilized neural energies"
... even when stabilization is still in progress.

Rod

[Amigoo]

Re: Familial Wormholes (new design concept)
(see also: https://en.wikipedia.org/wiki/Wormhole )

While "researchers have no observational evidence for wormholes" in space,
one researcher has observational evidence of wormholes in Cartesian space
occupied by nested, juxtaposed, or otherwise familial squared circles.

Say what? I might be concerned about being labeled "the geometer
who channels squared circles" if the beginning steps were intentional. Yet,
those steps always start as late-night (usually) wandering in Cartesian space.

This Familial Wormholes concept began as curiosity about a Vesica pair of
squared circles, aligned on a shared side of their inscribed squares. Having
observed that this Vesica object was shadowed by a large scalene triangle
of squared circles reputation, the wandering became focused (IMO).

I'm discovering that other interesting possibilities appear when I don't
force the evolving geometry to first align on the x/y axis. How the Vesica pair
first aligned at 27.597.. degrees* is a mystery! But such may be the essence
of squared circle channeling when it's not that intentional tapping.

* http://aitnaru.org/images/Alright_Triangles.pdf
This PDF seems to contain salient references to the 27.597.. angle, but I closed
that file immediately after noticing that it represents a different wormhole:
where much of my time was spent in recent years.

Rod ... ...

[Amigoo]

Re: Familial Wormholes
"Cartesian points in squared circles spacetime."

"Been there! Done that!" .... sans agave larvae.
Fortunately, the Marnier was geometrically inspirational,
but now I'm all cracked ice and sans Marnier.

Rod

[Amigoo]

Re: Familial Wormhole
".emitecaps selcric derauqs ni stniop naisetraC"

“!taht enoD !ereht neeB”

Everything seems the same (maybe twisted slightly)
... after a test voyage through this wormhole.

Rod ... ...