Paradise Trinity Day

[Amigoo]

Re: Squarely Entwined design

Today's squared circles meditation (short & sweet) ...

It should be no surprise that sqrt(2) is associated with Pi since sqrt(2) is the essence
of a circle squared (diameter is the hypotenuse of an isosceles right triangle).

And if this isosceles right triangle is symbolically associated with this 1111Angels'
Current Events topic (Paradise Trinity Day), then the question is answered:
Which came first? The circle or the triangle? (not circle or square?)

How? The circle is the symbolic source of all reality, the triangle (a trinity)
follows from that beginning that never happened (always was).

Sqrt(2) and Pi, squarely entwined ... from the post-Trinity era.

Rod

[Amigoo]

Re: Squarely Entwined design
(the Math Of Pi, a squared circles MOP")

Speaking of "trinity", Pi itself defines a trinity,
a transition of adjoined right triangles: Pi/2, sqrt(Pi), 2
(each value is both hypotenuse and long side):

1.7724538509055160272981674833411.. sqrt(Pi)
/ 1.5707963267948966192313216916398.. Pi/2
= 1.1283791670955125738961589031215.. sqrt(Pi)/(Pi/2)

1.7724538509055160272981674833411.. sqrt(Pi)
x 1.1283791670955125738961589031215.. aka 2(sqrt(1/Pi))
= 2 circle's diameter

Thus, sqrt(Pi) defines the side of a square for D = 2
and 2(sqrt(1/Pi)) defines circle-squaring geometry
... for every circle! (between here and there)

Rod

[Amigoo]

Re: Squarely Entwined design
Who knew?!

That a heavy, natural-stone object that I've owned over 30 years,
usually covered by weeds, vines, and leaves and never named,
now is "Pi~nosaurus" (relating to stone's mature age and shape).

Pi~no the no-dino was elevated (literally) after I dragged across the yard
(with rope around butt, pulling backward inch-by-inch) three even heavier
landscaping stones (also covered by weeds, vines, leaves, ... and dirt).

How does "Pi~no" relate to Squarely Entwined, et. al.?
Dunno! Transcendental inspiration?

Rod (needed pyramid building skills)

[Amigoo]

Re: Pi's in the Corner design

Two Pi's in the corner of the Pi Corral ...
obviously, chaperoned by sqrt(2).

Rod ... ...

[Amigoo]

Re: Upside Down 'n Backwards design

Decades ago, our computer school instructor described how an impact printer worked:
"Imagine, inside the printer, a monkey sitting upside down and typing backwards."

Imagination wasted on we who tried to comprehend how a monkey could do this.

Fast forward decades and it's the first concept that came to mind when I comprehended
the significance of the contrasting geometric objects in two circles squared, one having
half the diameter of the other.

In this "upside down 'n backwards" contrast, proof of circles squared may exist.

Rod

[Amigoo]

Re: Upside Down 'n Backwards design
Geometers' tip from this monkey sitting up, typing forward:

Despite the camouflage of "upside down and backwards",
it's easy to see two sets of overlapping scalene triangles
(both sets part of geometrically similar trapezoids).

The point (re: "What's the point?") is that the small angle tips
of each scalene (in a set) defines a point-to-point line that
represents one side of that circle's inscribed square.

Also, many of the lines in the contrasted objects are parallel
and have 1:2 line length relationship.

Rod (just another day in the neighborhood)

[Amigoo]

Re: Pin Point Precision design
and recent conjecture that "the portal is open" ...

Pin Point Precision suggests that scarafinesse is a new skill:
Geometers' coaxing of designs from squared circle geometry.

The several, consistent geometric objects of squared circles
provide mysterious opportunity for such creative coaxing!

scarafinesse = scarab + finesse
re: this finesse of a beetle (scarab)

Rod

[Amigoo]

Re: Starring Pi design

"Lines and triangles and squares! Oh, Pi!"
featuring 8 overlapping, circle-squaring scalenes

Relationship of inner circle to outer circle
(of Diameter = Pi/4 to Diameter = 4):

4/(Pi/4) = (2(sqrt(Pi)/(Pi/2)))^2
= 5.0929581789406507446042804279205..

Calculation confirmation:

sqrt(5.0929581789406507446042804279205..)
= 2.2567583341910251477923178062431..
/ 2 = 1.1283791670955125738961589031215..
= sqrt(Pi)/(Pi/2)
= 1.1283791670955125738961589031215..
* 2 = 2.2567583341910251477923178062431..
^2 = 5.0929581789406507446042804279205..

Rod ... ...

[Amigoo]

Re: Starring Pi design
"The other 'transcendental'"

Some lines removed for better presentation of this "kaleidoscope"
(but geometric points remain for design's impressive scalenity).

Current collection: http://aitnaru.org/images/Squarely_Entwined.pdf

Rod

[Amigoo]

Re: Starring Pi design

“The other 'transcendental',
'impossible' scalenity of triangles”

Sequence with circles-squaring significance:
(increments of 2(sqrt(1/Pi)) )

0.78539816339744830961566084581988.. Pi/4
0.88622692545275801364908374167057.. sqrt(Pi)/2
1.0000000000000000000000000000000.. 1
1.1283791670955125738961589031215.. sqrt(Pi)/(Pi/2)
1.2732395447351626861510701069801.. (sqrt(Pi)/(Pi/2)^2

Rod

[Amigoo]

Re: Starring Pi design

“The other 'transcendental',
'impossible' scalenity of triangles”

Intriguing groups of 5 circles squared,
surrounding D=4.0, D=2.0, D=1.0 sequence:

5.0929581789406507446042804279205.. (2(sqrt(Pi)/(Pi/2)))^2
4.5135166683820502955846356124862.. 4(sqrt(Pi)/(Pi/2))
4.0000000000000000000000000000000.. 4
3.5449077018110320545963349666823.. 2(sqrt(Pi))
3.1415926535897932384626433832795.. Pi

2.5464790894703253723021402139602.. 2((sqrt(Pi)/(Pi/2))^2)
2.2567583341910251477923178062431.. 2(sqrt(Pi)/(Pi/2))
2.0000000000000000000000000000000.. 2
1.7724538509055160272981674833411.. sqrt(Pi)
1.5707963267948966192313216916398.. Pi/2

1.2732395447351626861510701069801.. (sqrt(Pi)/(Pi/2))^2
1.1283791670955125738961589031215.. sqrt(Pi)/(Pi/2)
1.0000000000000000000000000000000.. 1
0.88622692545275801364908374167057.. sqrt(Pi)/2
0.78539816339744830961566084581988.. Pi/4

Note: These circles not displayed in Starring Pi

Rod

[Amigoo]

Re: Trinity of Concentricities design

Geometric explanation for the 5-circle concentricities
in the 4,2,1 wiggly numbers sequence of Starring Pi.

"Say what?" This geometry speaks for itself ...
in a mysterious trinity of concentricities.

What's outside the 4,2,1 concentric circuits?
Undisclosed outer space levels?

Transcendental Pi extraordinaire!

Rod

[Amigoo]

Re: A Circle Squared design

"Geometers' tip: Red, yellow, and green
line pairs have sqrt(Pi) line length ratio."

Speaking of "extraordinaire!" ...
Finally! Squared circle simplicity!

Rod ... ... (off to see red, yellow, and green high above)
hint: http://www.todayifoundout.com/index.php ... ic-lights/

[Amigoo]

Re: A Circle Squared design ("new & improved")

Enhanced to highlight another ratio (hypotenuse of right triangle to long side):
2(sqrt(1/Pi)) = 1.1283791670955125738961589031215.. , aka "rPi".

When reviewing this geometry, "robust Pi" came to mind since from this constant
(and a known diameter), geometry of a squared circle can be derived. Incidentally,
"rPi" was proffered for this constant several years ago, referring to "radial Pi"
... and sounding like "our Pi" (for the new millennium).

Rod ... ... (stoppin' & goin' ... with interim caution)

[Amigoo]

Re: Credible Closing design
"Evidence of the long journey post voir dire"

aka "Been there! Done that!"

Rod

[Amigoo]

Re: ACS Redux design
(ACS = A Circle Squared)

More "evidence of the long journey post voir dire" ...

and indication that this geometer might need therapy to stop assembling
these Cartesian neighborhoods. Such is the ease of composition construction
once squared-circle building blocks (geometric objects) are known.

Geometers' tip: If a squared circle does not have the Trademark Angle*
(a shared point on a circle where rests its square), the circle is not squared

* 90 - 27.597112635690604451732204752339.. (a cosine angle)
= 62.402887364309395548267795247661.. rPi TAngle
(reduced to 62.403 for preliminary construction)

Rod

[Amigoo]

Re: CC Redux design
(renamed from ACS Redux since it's not similar)

"Geometric building blocks of circles squared,
featuring the foundational 62.40288.. rPi TAngle"

... albeit, an ACS Redux might "round" the pages to 11
if other meaningful (and related) geometry is found.

Current 10: http://aitnaru.org/images/Squarely_Entwined.pdf

Rod

[Sandy]

That's a beautiful design , Rod. It reminds me of one of my mourning dove friends.
xxSandy

[Sandy]

That's a beautiful design , Rod. It reminds me of one of my mourning dove friends.
xxSandy

[Amigoo]

Re: BIONics design
Believe It Or Not (in circular scalenity)

Somehow the number 3 comes to mind when thinking of "scalenity".
This number must be related to triangles ... somehow.

This complex geometry (that can speak for itself ... quite well),
begs the question: "Is the circle squared?"

Well ... Does a bear sit in the woods? 3 times a day?

Rod

[Amigoo]

Re: BIONics design
Believe It Or Not (in circular scalenity)

Somehow the number 3 comes to mind when thinking of "scalenity".
This number must be related to triangles ... somehow.

This complex geometry (that can speak for itself ... quite well),
begs the question: "Is the circle squared ... yet?"

Does a bear sit in the woods? 3 times a day?

Note: Comments for several designs created before BIONics
(but after the PHP burp) will be reposted later.

Rod

[Amigoo]

Re: BIONics II design

Both simple and complex, highlighting the consistent geometric objects
of circles squared: scalene triangle, right triangle, isosceles right triangle,
and isosceles trapezoid ... all integrated with a certain quadrilateral
containing a 90-degree angle.

Rod

[Amigoo]

Re: aSign design

As for circle-squaring Scalenity in the local universe
... let there be aSign.

Rod ... ...

[Amigoo]

Re: aSign design

As for circle-squaring Scalenity in the local universe
... let there be aSign.

There is a sign, but not the expected "circular Scalenity"
"Spiral of Circular Scalenity" is the apparent sign.

SoCS has been used to designate "Side of Circle's Square"
in descriptions of the geometry of circles squared.

While Pi - even as line length ratios - is well-highlighted,
sqrt(2) is the foundation of this scalenity spiral.

Rod

[Amigoo]

Re: aSign design

As for circle-squaring Scalenity in the local universe
... let there be aSign.

Geometers' tip: This geometry proves that Pi is directly linked to sqrt(2).
For every 45 degrees (clockwise) of the spiral, the Pi value (as line length)
decreases by sqrt(2) ... and so do the circles' diameters.

Rod