Paradise Trinity Day

[Amigoo]

Re: Objet d'Pi design
"Pythagorean cSquare (prototype)"

At first, it seemed odd that a Pythagorean cSquare* would be among the final designs in this research, but I soon realized that no one would have known what unique tool would be needed for an "impossible" Cartesian construction. While a satisfactory prototype might be available some day, a precision cSquare will have to await determination of the final decimal digit of Pi.

* proposed geometer's tool (red) for squaring the circle.

BTW: There should be a good market for this tool as a mathematician's gag gift.
(contract for autographed copies might be negotiated)

Rod ... ...

[Amigoo]

Re: Objet d'Pi design
"Pythagorean cSquare (prototype)"

This might be a good example (Rorschach Test?) of one's ability
to "think outside the box" (if you comprehend this email):

"When invited to go to a Nordic sauna for a round of tongue wrestling,
totally timid foreign tourists should inquire 'public or private?':
Public TW is known as debate; Private TW is a different sport.
... according to an unemployed tour guide."

The explanation follows in the next post.

Rod

[Amigoo]

Re: Objet d'Pi design
"Pythagorean cSquare (prototype)"

The Nordic sauna cultural "fact" was presented as humor and focused on "tongue wrestling"
as a term meaning verbal debate (we speak with our tongues). The rest is just setup to
associate French kissing as "a different sport" in a Nordic sauna.

Apparently, the humor is too remote to perceive (or is "out of the box" thinking),
but a good example of my mind's "creativity" when squaring the circle late at night.

Also,"out of the box" thinking (with extensive pattern recognition/analysis) is
my only claim to fame (LOL) in this geometry research.

Rod " "

[Amigoo]

Re: Caja de Objet d'Pi (new design concept)

There must be a few lessons in this about a box and thinking outside
of it, such as "keep exploring and you'll see all that's outside said caja" ...
within a few lifetimes or more.

I'll compromise one more time and complete this design, but will be
less inquisitive about what else exists in squared circle geometry since
that larger box (the one outside) is bottomless ... apparently.

Some "outside the box" humor ...
I know how to pronounce "objet d'art", so
"objet d'Pi" might be pronounced as in this dialogue:

"Baxter, please bring me my obb zhay de pee."
"Sir, your chamber pot is being scrubbed."

Rod

[Amigoo]

Re: Caja de Objet d'Pi design
"Perchance, 'outside the box'"

At least, the geometry has good presentation
for an expedited draft version.

Rod

[Amigoo]

Re: Caja de Objet d'Pi design
"Perchance, 'outside the box'"

To recap the box adventure ...

People understand that "thinking outside the box" is a metaphor,
but few know that the box is real ... in Cartesian space.

In the Caja de Objet d'Pi design, if you see a box and it's open,
you might be TOTB! (if you guessed Too Old To Bother,
you're not TOTB, geometrically speaking)

A proposed geometer's tool for squaring the circle (on paper,
of course) is shown as the red object in the Objet d'Pi design.
BTW: all 5 circles are squared (the smaller 3, effectively).

TIP*: if a 62.403 degree (rounded) radius can be found,
the circle is squared. "But there's no 62.403 degree angle
on the Pythagorean cSquare!" you insist triumphantly.
TIP*: identify the circle-squaring trapezoid.

* no, not Totally Impossible Perspicacity
Verify the geometry with your cSquare.

Rod ... ...

[Amigoo]

Re: Objet d'Pi design
"Pythagorean cSquare (prototype)"

Geometer's Perspicacious Secret (GPS) ...
In the center of Objet d'Pi is an arc for a circle
that encloses the unique scalene triangle
... that squares this circle.

Diameter of this circle? sqrt(Pi)
(same as the similar three, creating four
similar triangles within the largest scalene)

Rod

[Amigoo]

Re: Sqrt(Pi) (new design concept)

Diameter of this circle? sqrt(Pi)

After posting this quote ...
and happily humming the "Closure at last!" mantra,
sqrt(Pi) kept insisting "Show me! Show me!"

So, Sqrt(Pi) gets to be the final design.
Simple, with scalene sponsorship, of course
and including sqrt(2) ... of course.

Rod

[Amigoo]

Re: Sqrt(Pi) (renamed to Pythagorean Pi)

The facts of Pi (this recipe):
- Sides of red square = sqrt(Pi).
- Radius of golden circle = sqrt(Pi).
- Left diagonal of green scalene triangle
= sqrt(Pi) x sqrt(2) = 2(sqrt(Pi/2)).
- Sides of square of golden circle = Pi.

Studying the geometry of Pi
... is like reading a recipe (Lemon Meringue):
1 1/4 cups flour, 1/4 tsp salt, 1/2 cup butter,
1/4 cup ice water, 3/4 cup white sugar, 2 tbsp flour,
3 tbsp cornstarch, 1/2 tsp salt, 1 1/2 cups water,
2 lemons, juiced, 2 tbsp butter, 5 egg yolks, beaten,
5 egg whites, whipped, 6 tbsp white sugar.

... but feasting on out-of-the-box Pi
is a heavenly experience.

Rod

[Amigoo]

Re: Sqrt(Pi) design

Geometer's TIP:
The length of a line drawn between the midpoints of the diagonal sides
of the circle-squaring trapezoid in the large circle is equal to the length
of the diagonals of a similar trapezoid in the smaller circle.

"But there is no such trapezoid in the smaller circle!" you insist.
Of course not - "some assembly required".

"That midpoint diagonal does not have the same length!" you discover.
Well, who said "midpoint diagonal" in the similar trapezoid?

Rod ... ...

[Amigoo]

Re: Sqrt(Pi) design (renamed to Correlation of Pythagorean Pi)

More complexity added to the Pythagorean Pi design ...
but the red square (side length = sqrt(Pi)) now correlates the geometry of
three squared circles, facilitating identification of geometric proportions.

Say what?! This explains the joke (why the objects were drinking):

A square, triangle, and circle are drinking wine at their table for four.
The waiter asks "Will you have another round before dinner?"
The circle responds first "Yes, then bring us all a round."
Why does the waiter bring four glasses of wine? When?

Rod " "

[Amigoo]

Re: Correlation of Pythagorean Pi design

Another circle was added, creating such integration of squared circle geometry,
all hosted by the perfect partnership of sqrt(Pi) and sqrt(2), that gazing upon this
Cartesian complexity too long might cause one's mind to depart the universe
in a chariot of fire.

Rod

[Amigoo]

Re: Correlation of Pythagorean Pi design

Of course, add a circle and another line wants to be added.
But this line proves geometrically that the small red square
is one-fourth the size of the large red square.
What line sizes? sqrt(Pi)/2 and sqrt(Pi)

More elaboration on The Table with Three-Fourths,
logically amusing entertainment before dinner:

A square and triangle, an odd couple, join a circle at a Table for Four
and sit around the table. While waiting for four, a waiter serves them
complementary wine. Later, the waiter asks the circle "Will you have
another round before dinner?" The circle responds "Yes, we're early
but please bring us a round when we're all around the table."

Why does the waiter bring four glasses of wine?
At what time? AM or PM?

Waiter's tip: A full bottle of wine is often consumed
by three-fourths while they savor this dinner logic.
Will this wine be around for the extra round?

BTW: What is the name of the restaurant?
Is this the only restaurant with the name?

Extra credit:
1. According to squared circle geometry, which guest
is sitting at the side of the table that represents sqrt(2)?
2. Another side of the table is closest to sqrt(Pi); is that
guest sitting opposite or next to the sqrt(2) guest?

Rod ... ...

[Amigoo]

Re: Correlation of Pythagorean Pi design

More elaboration on The Table with Three-Fourths,
logically amusing entertainment before dinner:

Re: Extra credit ( third question was missing)

1. According to squared circle geometry, which guest
is sitting at the side of the table that represents sqrt(2)?
2. Another side of the table is closest to sqrt(Pi); is that
guest sitting opposite or next to the sqrt(2) guest?
3. When did you learn that the table is square?

Rod

[Amigoo]

Re: Correlation of Pythagorean Pi design

After claiming geometric "chariot of fire" integration, I was persuaded
to return to the Cartesian whiteboard and produce truth-in-advertising.

This geometry prefers to speak for itself and is a solid reference
for discussing the merits and demerits of circles and triangles
in a heavenly union of sqrt(Pi) and sqrt(2).

I was concerned that afternoon rumbling outside was the sound of chariot wheels
rolling a mi casa, but that weather disturbance passed (apparently) while I was napping.
At least, everything still looks the same in this familiar Cartesian neighborhood.

Rod

[Amigoo]

Re: Smile of Pythagoras design

Project closure with a smile (or at least a grin)
... and relative definition of "closure".

Study of the circle-squaring scalene triangle hints that
if a geometric solution to "squaring the circle" does exist,
it will be found within the Smile of Pythagoras ...
if you believe Acts, upon this promise,
will bear such Cartesian fruit.

Rod

[Amigoo]

Re: Smile of Pythagoras design

Such a critic! But upon reaching the other side,
one can claim that they're on the path to perfection.
(Pythagorean isosceles right triangle is displayed)

Rod ... ...

[Amigoo]

Re: Smile of Pythagoras design

Whoa! This design wants to be more esoteric!

While trying to decide if the symbolism represented
the "smile of Pythagoras" or the "eye of Pythagoras" (maybe both),
I noticed that the Greek upsilon* letter was present.

* Pythagoras's emblem of the path of virtue or vice,
(along the path to squared circles, a point precise).

Rod

[Amigoo]

Re: Smile of Pythagoras design

Many hours later and still no dominant symbolism.
Obviously, the design is meant to be esoteric with
layers of symbolism, ranked in this order:

Eye of Pythagoras, Upsilon of Pythagoras ...
nesting within Cartesian merits and demerits of circles and triangles
in a heavenly union of sqrt(Pi) and sqrt(2), and all blessed with
the Smile of Pythagoras.

Re: http://plato.stanford.edu/entries/pythagoras/

"... while Pythagoras was famous in his own day and even 150 years later in the time of Plato and Aristotle, it was not mathematics or science upon which his fame rested. Pythagoras was famous (1) as an expert on the fate of the soul after death, who thought that the soul was immortal and went through a series of reincarnations ..."

This history suggests that the geometry challenge of "squaring the circle"
was not necessarily the true focus of the Eye of Pythagoras and, perhaps,
explains the design's "mandated" complex symbolism.

This enlightened perspective reveals why the geometry
seems to display an eye peering into a larger eye;
the Eye of God? symbolic essence of "soul"?

Rod

[Amigoo]

Re: Smile of Pythagoras design

Difficult but fascinating evolution of this design.
Today's version has a good chance of being "final"
... especially since the "eye" now appears to look
both inward and outward.

Of greatest geometric significance is the blue
butterfly shape whose sides (one green, one golden)
effectively associate sqrt(2) and sqrt(Pi).

Hmmm ... looking to the left of the geometry,
I "see" replication and balance - the symbolic Eye of God?
It's time for another visit to this Cartesian neighbor.

Rod

[Amigoo]

Re: Smile of Pythagoras design

It's time for another visit to this Cartesian neighbor.

"Been there! - Done that!"

First impression of this extended design:
symbolic doorway between the secular and the sacred;
open passage from the material to the spiritual;
linkage of the mortal and cosmic minds.

Rod ... ...

[Amigoo]

Re: Smile of Pythagoras design

So, there I was ... back in the Cartesian neighborhood
... all serious and metaphysical ... and inspired to
embellish the design even more ... when it took
a sudden turn to the whimsical!

Oy Vey! Pythagoras is really smiling!

Rod

[Amigoo]

Re: Smile of Pythagoras design
"Anticipating the merits and demerits of circles
and triangles in a union of sqrt(Pi) and sqrt(2)."

Oy Vey! Pythagoras is really smiling!

... or laughing on the way to Morbus Cyclometricus.
(peering through the doorway is harmless)

Rod " "

[Amigoo]

Re: Smile of Pythagoras design
"Anticipating the merits and demerits of circles
and triangles in a union of sqrt(Pi) and sqrt(2)."

Recap of the 24-hour path of exploration ...

A difficult but fascinating evolution of this design. Today's version has a good chance of being "final" ... especially since the "eye" now appears to look both inward and outward. Of greatest geometric significance is the blue butterfly shape whose sides (one green, one golden) effectively associate sqrt(2) and sqrt(Pi).

Hmmm ... looking to the left of the geometry, I "see" replication and balance - the symbolic Eye of God? It's time for another visit to this Cartesian neighbor.

Later ... "Been there! - Done that!"

First impression of this extended design:
~ symbolic doorway between the secular and the sacred;
~ open passage from the material to the spiritual;
~ linkage of the mortal and cosmic minds.

Even later ... "Rested and ready!"

So, there I was ... back in the Cartesian neighborhood ... all serious and metaphysical ... and inspired to embellish the design even more ... when it took a sudden turn onto Whimsical Way. Oy Vey! Pythagoras is really smiling now! ... or laughing on our way to Morbus Cyclometricus. Apparently, peering through the windows of the bus is harmless, nevertheless keep asking the tour guide "Are we there yet?"

Rod ... ...

[Amigoo]

Re: Scalene Correlating Redundancy (SCoRe; new design concept)

So, there I was ... back in the Cartesian neighborhood ... all jovial and lighthearted ... and inspired to explore with less Pi precision ... when it took a sudden turn onto Whiplash Way. Oy Vey! Pythagoras is more serious than ever now! ... but chuckling on our way to Morbus Cyclometricus? Peering through the windows of the bus still seems harmless, but I'm persuaded to keep asking the tour guide "Are we there yet?"

Naturally, I'm doubting the "bus" metaphor - this is a roller coaster!
... and have a new question: "What's the SCoRe"
(aka, "Are we there yet?")

Rod