Paradise Trinity Day

[Amigoo]

Re: SoSL IV design
(IV as in "intravenous")

With all this SoSL Slush (4 designs), an intravenous boost of imagination
might be required if the "consistent components" of squared circles are
not yet obvious. SoSL IV will be a worthy "final" to this series since
the flutterby pattern is very prominent!

I pronounce SoSL as "'sah sill", but "social" makes a better tongue twister
... and "social slush" reflects on the controversy of squared circles.

Rod ... ... (off to find scintillating SoSL Slush galoshes)

[Amigoo]

Re: SoSL IV design
(IV as in "intravenous")

"Flutterby butterfly near infinite roam.
Butterfly flutterby on the way home."

Rod

[Amigoo]

Re: SoSL IV design
"Flutterby butterfly, near infinite roam.
Butterfly flutterby, on the way home."

When overwhelmed by advanced math and scientific proofs, consider the patterns in these five integrated circles, for example. These patterns suggest that, while the circle "cannot be squared", squared circles may indeed exist in our geometric universe.

About the small blue "lightning bolt" ...
Line lengths increase/decrease by sqrt(2);
side length of largest circle's square = sqrt(2).

Rod

[Amigoo]

Re: SoSL IV design
"Flutterby butterfly, near infinite roam.
Butterfly flutterby, on the way home."

Yikes! Follow a blue "lightning bolt" (after a good nap)
and you'll find a SoSL aka with similar pronunciation: SoCL

About the 'C' ...
Smooth sailing ahead, perhaps "on the way home",
from one corner of the universe to the other.

Rod

[Amigoo]

Re: SoSL IV design

About "SoCL" ...
"Singularity of Corner Linkage" comes to mind.

Rod ... ... (off to buy a singular ity for supper;
maybe a big bowl of angel hair spaghetti)

[Amigoo]

Re: SoSL IV design

While crusin' (and crusin' (and crusin' ... ))) in this geometric composition,
I observe but one "singularity": sqrt(2) - the host of squared circles
(many pairs of line lengths and angles maintain this sqrt(2) ratio)

Perhaps, a Pi corral begin with a circle's inscribed square,
common carrier of 1.4142135623730950488016887242097..
and "perfect" square root of 2 (best reference diameter).

Is not the Pythagorean Theorem the theme song?

Rod

[Amigoo]

Re: SoSL IV design

"Out on the limb" bold conclusion:
(aka "Geometer Junior Analysis")
(aka "4th of July Declaration")

The formulas that calculate Pi have a transcendental bias,
but the real geometric ratio (C to D) has not this essence.

Rod

[Amigoo]

Re: SoSL IV design
(aka "Geometer Junior Analysis")

Given: three diameters, increasing by sqrt(2)
D = diameter, SoCS = Side of Circle's Square

D = 1.1283791670955125738961589031215.. = 2(sqrt(1/Pi))
SoCS = 1

D = 1.5957691216057307117597842397375..
SoCS = 1.4142135623730950488016887242097..

D = 2.2567583341910251477923178062431..
SoCS = 2

Analysis: How does Pi, a transcendental number,
increase by sqrt(2), a non-transcendental number,
yet remain transcendental while sqrt(2) is not?

Note: sqrt(2) increment relates to inscribed squares.

Rod ... ... (still looking for matching SoCS)

[Amigoo]

Re: Lines of Lasso design (LOL)

Addendum to the SoSL IV geometry, highlighting its symbolic "Pi lasso".
(if SoSL IV is a "Pi Corral", LOL is carpe diem of its capturing chords.)

Rod

[Amigoo]

Re: Lines of Lasso design (LOL)

Current gEotP.pdf (click on image to view):
http://aitnaru.org/threepoints.html
"Lines of Lasso in a Pi Corral dominated by sqrt(2)"

"Three Points" refers to the circle-squaring scalene triangle
(minimum number of points required to square a circle).

Rod ... ... (crusin' from point to point)

[Amigoo]

Re: Lines of Lasso design (LOL)

Better beetle (more lines dominated by sqrt(2))
What?! There are two circle-squaring scalene triangles?
Cheeky geometer!

Rod

[Amigoo]

Re: Lines of Lasso design (LOL)
"Lines of Lasso in a Pi Corral dominated by sqrt(2)"

More exploration ... but same song - next verse:
Geometry proven “impossible” must needs be esoteric.

Intriguing arrival of the red parallelogram!

"I knew that squaring the circle was impossible,
but I didn't know that it would be so difficult."

Rod

[Amigoo]

Re: Lines of Lasso design (LOL)
"Lines of Lasso in a Pi Corral dominated by sqrt(2)"

Entrepreneur Alert! (more art than science)

An HDTV antenna, crafted from copper tubing (a round circle with an inscribed scalene triangle),
is now in development as a prototype. I'll wire it as if expecting the inspired device to actually
receive HDTV signals. At least, I can claim that I finally got a copper "round toit".

Rod

[Amigoo]

Re: Lines of Lasso design (LOL)
"Lines of Lasso in a Pi Corral dominated by sqrt(2)"

HCIT! An isosceles right triangle, somewhat camouflaged,
has one side identfying a side of the square of one circle,
and hypotenuse identifying a side of square of larger circle.

Obvious geometry: lengths of sides have sqrt(2) ratio.
Say what?! There are two Pi - one larger by sqrt(2).

Rod ... ... (off to find coupons for Two Pi Specials)

[Amigoo]

Re: Lines of Lasso design (LOL)
"Lines of Lasso in a Pi Corral dominated by sqrt(2)"

While sqrt(2) may dominate, defining the relationships
of the right triangles now mentioned, 2(sqrt(1/Pi))
(= 1.1283791670955125738961589031215..)
is the signature of these right triangles*
... which also square the circle.

* ratio of hypotenuse to long side = 2(sqrt(1/Pi))

Rod

[Amigoo]

Re: Lines of Lasso design (LOL)
"Lines of Lasso in a Pi Corral dominated by sqrt(2)"

Metamorphosis x2! HCIT!

A new flutterby butterfly helps identify the isosceles right triangle
whose hypotenuse, long side, and dividing line each define a side
of the square of their respective circle - a sqrt(2) spiral!

Rod ... ... (chasing another flutterby
... in my rear view mirror)

[Amigoo]

Re: Lines of Lasso design (LOL)
"Lines of Lasso in a Pi Corral dominated by sqrt(2)"

isosceles right triangle whose hypotenuse, long side, and dividing line
each define a side of the square of their respective circle

If these three lines represent the sides of squares,
with c = hypotenuse of the isoceles right triangle,
and a = side of this triangle, and b = dividing line,
the algebra declares that a^2 + 2(b^2) = c^2
therefore, 1 + 1 = 2.

How was "transcendental" absorbed?

Rod

[Amigoo]

Re: Lines of Lasso design (LOL)
"Lines of Lasso in a Pi Corral dominated by sqrt(2)"

With one more line, this geometry goes from "esoteric" to "Esoteric",
for this one line (and another recolored) now completes the identification
of three similar trapezoids, having similar colors: top/bottom line green,
one side yellow, one side dark blue. HCIT!

While it's "impossible" to square the circle, it's nearly "impossible"
to color code similar trapezoids in tightly integrated geometry.

Rod ... ... (off to decode "ciretosE")

[Amigoo]

Re: Lines of Lasso design (LOL)

Would make an impressive imaginary constellation!
... named ciretosE (pronounced 'seer ah tose)

Rod

[Amigoo]

Re: Lines of Lasso design (LOL)

More perspective on sqrt(2) and Pi
(calculate backward/forward from Pi)

= 2.2214414690791831235079404950303..
/ 1.4142135623730950488016887242097..

3.1415926535897932384626433832795.. Pi

x 1.4142135623730950488016887242097..
= 4.4428829381583662470158809900607..

Therefore ...
4.4428829381583662470158809900607..
- 2.2214414690791831235079404950303..
= 2.0

Analysis: In CSCSC concentric circles ...

If Pi is the length of a side of the middle circle's square,
the length of the side of the outer circle's square is twice
the length of the side of the inner circle's square
... precisely

CSCSC = Circle inscribed in Square inscribed in Circle
inscribed in Square inscribed in Circle

Apparently, "transcendental" exists only
between Here and There in a CSCSC Pi Corral.

Rod

[Amigoo]

Re: Square Pegs - Round Holes design
(exploration of Lines of Lasso geometry)

New insight about the brain arrived with the Square Pegs concept:
With continuing exploration of a complex design, I often see new geometric objects ... but only after a good night's sleep.

I theorize that the brain accumulates many bits of information (patterns, angles, line relationships, etc.) during the day's exploration, but all this stuff of temporary value is erased (or archived?) during sleep. Then (especially in alpha mode) when awakening, new patterns become visible. These Square Pegs - Round Holes are today's "new patterns". Does this not suggest that a daily nap during working hours, especially for mind laborers, would be beneficial ?!

Note: "pegs" are oblong - not square, but have square corners ... and the OPRH acronym would allude to other media.

Rod

[Amigoo]

Re: Pi Are Square II design
Simplicity today!

Rod

[Amigoo]

Re: Pi Are Square II design
aka "Geometry Of Interest" (GOI)

Highlighting recurring circle-squaring objects:
right triangle, scalene triangle, trapezoid.

Ratio of hypotenuse to long side:
2.0 / 1.7724538509055160272981674833411..
= 1.1283791670955125738961589031215..
= 2(sqrt(1/Pi)) = length of red line

Rod ... ... (monitoring consistent drift to complexity)

[Amigoo]

Re: Pi Are Square II design

monitoring consistent drift to complexity

aka "Simplicity Today - Complexity Tomorrow"
(after a late evening nap before bedtime)

Now, the Texas 'T' (D = 2(sqrt(1/Pi)))
"What goes around, comes around."

Rod

[Amigoo]

Re: Pi Are Square II design
"What goes around, comes around."

Anticipating clamoring from squared circle aficionados ...
Yes, the Texas 'T' circle is also squared!

Rod