Paradise Trinity Day

[Amigoo]

Re: PPoSC design
Now four circles, all squared (one completely, three effectively).

The green circle on the far left with its green circle overlay at the top uniquely describes a squared circle set.
Within the overlay, exists the "Texas 'T'" (not completely drawn).

This PPoSC is now as complex as it can be and still show as much integrated, squared circle geometry as possible.

Rod ... ...

[Amigoo]

Re: PPoSC design (updated)
Within the green circle overlay, the "Texas 'T'" is colored blue.

The right triangle that squares this circle would be formed by the left and top red lines
plus an hypotenuse (upward extended vertical blue line of the 'T').

Rod

[Amigoo]

Re: 'T' Ring (new design)
Finally, convincing geometric explanation of the "Texas 'T'".

A set of concentric circles, with crowning circles half the diameters of the larger set,
form a precise circle-squaring "ring"; another very convincing pattern of Pi (cPoP).

As sweet as an Apple-Walnut Tea Ring! Move over Pi and cake ...
'T' Rings will soon appear at every Pi Day celebration.

Diameters in this design, from largest to smallest:

2.0
1.772453850905516027298167483341 - sqrt(Pi)
1.570796326794896619231321691639 - Pi/2

1.128379167095512573896158903121 - 2(sqrt(1/Pi))
1.0
0.886226925452758013649083741670 - (sqrt(Pi))/2

Rod

[Amigoo]

Re: Infini-T Ring (design embellished and renamed)

This geometry prefers to anticipate infinite expansion,
light years into the sky above the Texas 'T'.

Rod

[Amigoo]

Re: Infini-T Rings (design embellished and renamed ... again)

When infinity rings (calls) during our life on this planet,
how do we respond?

Ring Ring!
"Infinity calling - will you accept the invitation?"

Rod

[Amigoo]

Re: Infini-T Rings geometry
How do you know that Infinity is calling?

The carrier wave of the circles' diameters is related to
1.1283791670955125738961589031215.. (increments of 2(sqrt(1/Pi))

Of course, Caller ID will truncate a few decimal digits.

Rod ... ...

[Amigoo]

Re: Infini-T Rings geometry

Intuitively (and with limited CAD measurement for support), the Pythagorean triangles at the top of this geometry are all directly associated with a squared circle. If one triangle squares a circle (any circle, full size or half its diameter), then all triangles square their respective circles ... gracias a trigonometría (the vertex angle remains constant for all of these Pythagorean triangles.

So, "What's the point?"
Only three points are required to square a circle:
two points on the circumference and the center of the circle.

But the vertical line of a Texas 'T' (the two points at either end of this line) begs an out-of-the box(body?) perspective:
Only two points are required to square a circle! If this is true, this line might be the Cartesian complement of an astral line,
the umbilical silver cord!

Rod ... ...

[Amigoo]

Re: Quadrilateral Quiescence (new design)
A good example of the juxtaposition of geometric lines, angles, and objects
that will be required to define the finite boundaries of a Pi Corral.

Say what?

"In this Pi Corral, the PPoSC rules ... diagonally!"

Say what?

"What."

Rod

[Amigoo]

Re: Quadrilateral Quiescence (elevated symbolism)

So, I inquired: "What, exactly, is Sanitas Cyclometricus"?"

The response: "Sanitas Cyclometricus is a state of mind
... a delicate, nurturing balance of complementary forces."

(The resulting symbolism in this geometry speaks for itself.)

Ring Ring!
"Infinity calling - will you accept the invitation?"

Rod

[Amigoo]

Re: http://aitnaru.org/psymmetry.html

Fascinating! This repositioned image (Quadilateral Quiescence) entices a smile.

Rod

[Amigoo]

Re: http://aitnaru.org/symmetry.html
http://aitnaru.org/baktun14.html (click on design)

Dr.-Ing. Helmut Sander discusses the same quadrilateral geometry here
but as two adjoining right triangles and without reference to trigonometry:
http://www.emis.de/journals/NNJ/Sander.html

BTW (re: top of Sander's web page) ...

"Engineer Helmut Sander reflects on the impossibility of squaring the circle and the irrational number pi,
and provides a geometric construction that approximates the operation."

The trigonometry discovered later suggests that Sander's "approximates" is a bit modest
... at least, in this geometric description of a squared circle.

Rod ... ...

[Amigoo]

Re: http://www.emis.de/journals/NNJ/Sander.html
and comment previously posted to this topic:

Dr.-Ing. Helmut Sander discusses the same quadrilateral geometry here
but as two adjoining right triangles and without reference to trigonometry

Sander's angle A in his right triangle ABC is directly related to (sqrt(Pi)/2):
cosine (27.597112635690604451732204752339..) = 0.88622692545275801364908374167057..

Therefore, with triangle ABC defined as sqrt(4-Pi), sqrt(Pi), 2,
Sander's geometry (as definition of a squared circle) is always
as accurate as the known digits of Pi.

... making his "approximates" (re: top of web page) a bit modest.

Rod

[Amigoo]

Re: Spiral of Quadrilaterals (design in development)

If a spiral is "a representation of process and growth", this composition of integrated quadrilaterals
is such a representation ... especially as reflection of ongoing research of squared circles.

But I digress ...

This geometry's purpose is to highlight line lengths from right angle to right angle in the quadrilaterals.
These lengths increase (and decrease) by increments of the square root of 2:

From largest to smallest (*, to 3 decimal places):
19084489.882, 13494778.849, 9542252.574, 6747390.237
Respective diameters = 2, sqrt(2), 1, sqrt(2)/2.

Geometric noise to non-geometers, but valuable technical notes of the journey
... and new perspective on "irrational" and "transcendental".

* lengths may change slightly when draft geometry is redrawn.

Rod

[Amigoo]

Re: Spiral of Quadrilaterals design

This design had to be rushed to completion -
the mouse was aflame, the keyboard crunching,
the monitor melting, and the operator quickly
drifting "out to lunch".

Say what?
"Geometric noise" in this design is understatement,
but the geometric objects are all distinct.

The revised numbers for the dark blue lines,
right angle to right angle line lengths that are
increments of sqrt(2):

1.90844931129.., 1.34947744957..,
0.95422465565.., 0.67473872478..
Diameters = 2, sqrt(2), 1, sqrt(2)/2

Rod

[Amigoo]

Re: Spiral of Quadrilaterals design (now five circles)
Design nickname: Rational Spaghetti (long story short):

Oh, the power of suggestion ... even at midnight! The "out to lunch" humor sent me to the kitchen for a bedtime snack - nothing to snack on. So, what's a logical next step? Make spaghetti (logical when you're "out to lunch" at midnight). After a generous bowl of spaghetti, bedtime had to be delayed to promote better digestion. So, what's a logical next step? Go back to the computer and review the latest geometry.

End of story: one more squared circle replicated (next larger circle in the progression).

Rod (one for increased design complexity, one for too much spaghetti)

[Sandy]

I am enjoying your sense of humour, Rod...and yum, a bowl of spaghetti would taste good right now!
hugs,
Sandy

[Amigoo]

Re: Texas~T (new design)
"Riding the range of possibilities."

I've been wondering how many pages will be in the final cPoP 5253 PDF file.

The symbolism of this latest geometry hints that 21 is the number: this geometry provides nice closure for this current research, contrasting two squared circles (one with a diameter of 1, the other with a square whose side length =1) with visually obvious geometric correlation ... and both properly chaperoned by a "Texas 'T'".

Rod ... ...

[Amigoo]

Re: Texas~T design
"Riding the range of possibilities."

About the dark blue circle ...
- diameter = sqrt(2), large circle's diameter = 2
- blue arc is embellishment to create a lasso:
http://www.dreamstime.com/illustration/lasso.html

Get a 4-legged ride and explore this free-range concept!
and bring your lasso to capture geometric insight.
Or ride the Texas~T with real wheels.

Rod ... ...

[Amigoo]

Re: Texas~T design
"Riding the range of possibilities."

About the blue scalene triangle ...
This circle-squaring geometric object seems to be
the important correlation of squared circles.

When riding the Texas~T range of possibilities,
if you lasso one of these unique scalene triangles,
you lasso them all (they're in all squared circles).

Rod (riding the range)

[Amigoo]

Re: Blue Cheese, Aligned (new design)
"Fresh cheese - Stale crackers"

I've been wondering how many pages will be in the final cPoP 5253 PDF file.
The symbolism of this latest geometry hints that 21 is the number:
this geometry provides nice closure for this current research ...

On the other hand, 22 is also a good number. Say what?

In a dream two nights ago, a woman approached and asked "Are you finished?" When I failed to respond with a firm "Yes!", she walked away, saying "Take your time, but other projects await." Considering that the blue cheese has long been aligned (scalene triangle was discovered and allowed to roam all over the Cartesian range), it's time to sweep the stale crackers out the door and discover tastier crunchies. I suspect that if I keep munching on stale crackers, I will soon be asking "Who moved my cheese?"

Interestingly, the person who approached in the dream might be symbolic
of the mysterious x:06 midwayer prompt that visits often.

Talk about closure! x:46 just said "Hello" and "Goodbye" ... once more.

BTW: What are "crackers"? Best guess: neurons (re: brain and mind),
since a friend used to refer to her ex-boyfriend as "Crackers".

Rod ... ...

[Amigoo]

Re: Blue Cheese, Aligned design
Finally, an acceptable version of the geometry ...

Small vertex of the similar, circle-squaring red and dark blue Pythagorean triangles
is the same: 27.597112635690604451732204752.. degrees.

"It's time to move along Hopalong" (a different cowboy):
https://www.youtube.com/watch?v=zkbhadcVwWo

About 4 minutes into this black & white movie,
a barrel of Krispy Crackers appears (in a country store).
Hmmm ... I may need a Topper for the next project.

Rod

Postscript: I went back to the design to see how it might be improved,
then glanced at the clock: 12:46

[Sandy]

I'm enjoying this ole western classic. Been many years since I even heard the name Hopalong Cassidy... The acting styles are different in these earlier periods but still fun to watch. Thanks Rod!
xxSandy

[Amigoo]

Re: Blue Cheese, Aligned design
An observation (while munching on cheese and crackers ) ...

Given: diameter of large green circle = 4.

The large red Pythagorean triangle (two triangles) at the top right
defines the algebra proportion: sqrt(Pi) / 2 = 2 / x where x = 4/sqrt(Pi)

Rod

[Amigoo]

Re: Pairs of Squares (new design)
"Every squared circle has a soul mate."

Interesting "closure" ... that keeps on closing!

I now suspect that "closure" means that this toy box is bottomless -
there will be no final geometry ... except the last one drawn.

So ... page 22 (of 23) is currently the last one drawn.

Rod

[Amigoo]

Re: Pairs of Squares design
"Every squared circle has a soul mate."

About the dark blue Texas 'T' (both symbolism and possible clue to "a circle, squared") ...

A man was preparing to cross a broad, remote section of the range and stopped at a small country store to buy supplies for himself and his 4-legged ride. As he was about to leave the store, he turned and asked the proprietor for a small bag of tea. "Tea?" inquired the proprietor. "You have plenty of coffee ... and why a small bag of tea?"

The traveler reached into his pocket and pulled out a large thimble with a flat bottom. "You sew! ... out on the range!" laughed the proprietor. "Maybe 'sow', verbally speaking." the traveler winked. "When someone is attentive and listening. But I enjoy the tea often after a good day's journey, when the stars shine brightly that night."

"I drink from my coffee cup - the thimble is a symbolic cup for my traveling companion, my 'spirit within'. A simple ceremony - not even ritual - but those quiet moments of reflection enliven the journey and complement it with assurance of safe and certain guidance."

"A pair we are, indeed!" he grinned. "But my companion always seems to be two steps ahead of me and in precise alignment with my plans ... or should I say 'our plans'. Our quiet tea time is capped with a symbolic two-for-one toast: my companion's two steps for every one of mine.

Rod ... ...