Paradise Trinity Day

[Amigoo]

Re: I-Lambda V design
(even more esoteric)

"Pivotal evidence
of a Pi Corral crock
in design esoteric?"
queried sage Locke.

"What's in the bucket?"
"Perpendicular tease?"
Said No One to Lindemann:
"Opine if you please."

Rod

[Amigoo]

Re: I-Lambda V design
Riddle poem revised:

"Pivotal evidence
of Pi Corral crock
in design esoteric?"
queried yon Locke.

"What's in the bucket?
Perpendicular tease?"
Said No One to Lindemann,
"Opine if you please

or channel a message
your faithful will hear
regarding these tease
if squaring this year.

Magisterial sunrise
will follow - Prepare! -
the blessed uniting
of circle and square."

Rod

[Amigoo]

Re: I-Lambda V design

While attempting to confirm the geometry in this design, I found too many "wiggly numbers" (unmatched angles and line lengths at the .xxx decimal level). Visually, the geometry still looked great, but 72.597 is not equal to 72.593 (or whatever).

Re-drawing every line in the geometry with verified accuracy would have been an all-nighter (of geometric persuasion). Not a practical choice!

So, Plan B became operative: find self-confirming geometry. Within a few hours, an abstract anvil appeared (see center of design). Although the color selected for this anvil required much work to "layer it" into the background, at least it enhances the colorful presentation!

This "anvil of necessity complete with hammer of anguish" (re: 23:2.12) complements the design's esoteric motif. Consider: the length of the base of the anvil has 1/4 the length of the top of the anvil. These two lengths represent the distance between the sides of two "buckets", the smallest one in the center and the largest sibling, positioned upside down.

So, what's in the bucket now (besides the geometric "tease")? A large anvil.
Who knew? A bucket is made for carrying anvils, other buckets, and perpendicular tease.
Obviously, just a "drop in the bucket!" (if there's room).

Rod

[Amigoo]

Re: I-Lambda V design
(Now with .xxx accuracy)

In squared circle geometry,
what's the difference between XXX and .xxx?

XXX is what you say every time you fail to achieve
.xxx accuracy for each line and angle.

Note: XXX includes texting phrases.
(LOL was not one of the exclamations)

Rod

[Amigoo]

Re: I-Lambda V design
(redrawn completely to verify .xxx accuracy)

Just in time for the 10-10 anniversary!
So, here's a geometer's secret for the celebration:

The "perpendicular tease" overlap and pivot independently.
Consider how the extended sides of the "bucket" (dark blue lines)
confirm the geometry when the sides are positioned as shown.

Rod

[Amigoo]

Re: I-Lambda V design

Another secret: "perpendicular tease" is also whimsical:
confirmation is limited to .xxx (true accuracy is unknown).

Rod

[Amigoo]

Re: I-Lambda V design
(redrawn again without bucket extensions)

"Perpendicular tease" was more self-advisory than I realized when crafting this expression - the bucket side extensions (dark blue lines) were geometrically close but not close enough. So, the lines were removed - the geometry remaining is solid; the isosceles trapezoids all have identical angles and the "wiggly numbers" are confirmed at .xxx accuracy.

Nice research plateau, but it's time to climb out of the bucket and kick sand
... then look for new patterns in the creativity sandbox.

Rod ... ...

[Amigoo]

Re: I-Lambda VIp (new design)
"Quintessential pattern of Pi"

In this design (also geometric model), Pi is represented by the long, vertical red line (equals square root of Pi for this circle's diameter). And both of the light blue circles are squared.

Imagine that, of the two inscribed squares in the large circle, the yellow is in fixed position; the green rotates counter-clockwise. Since the base of the red Pi line is attached to the lower corner of the green square, the Pi line's length changes as the green square rotates. This rotation models the increasing decimal digits of Pi.

Now, with the known trillion+ decimal digits of Pi, movement in this model (as more Pi digits are discovered) would be imperceptible - even with a microscope! But the model effectively shows the direct relationship of Pi to a circle and its square.

About the design name's "VIp": "VI" is simply the sixth version of the I-Lambda designs; "p" = pattern (specifically, "quintessential pattern of Pi"). That "VIp" could also mean "VIP" is coincidence! Perhaps, this pattern will some day qualify for such recognition.

Rod

[Amigoo]

Re: I-Lambda VIp design
"Quintessential pattern of Pi"

A Pi pattern would not be "quintessential" without this intrigue (another circle was discovered that is also squared). This circle's inscribed square is identified by half of the square root of Pi (the square of the small, light blue circle; small circle's diameter equals radius of large, light blue circle).

Not all lines are drawn, but the ones present describe this intrigue ("long story short"):
Left side of small circle's inscribed square defines a line that is part of the third (dark blue) circle's square.
The geometric facilitator of this intrigue is the angle of that defined line: 27.597.. degrees.

Thus, in this geometry, the circle's square defines the square's circle.
Translation: "To square the circle, one must circle the square"

Rod ... ...

[Amigoo]

Re: I-Lambda VIp design
"Quintessential pattern of Pi"

Squared Circles Humor

Point - Counterpoint
(instructions for final exam)

Teacher: What's the point?
Student: Which point?
Teacher: The ones on the circle.
Student: Which points?
Teacher: Precisley.
Student: "Precisley"?
Teacher: Good point!

Rod

[Amigoo]

Re: I-Lambda VIp (design simplified)
"Quintessential pattern of Pi"

Squared Circles Humor

Point - Counterpoint
(instructions for final exam)

Teacher: What's the point?
Student: Which point?
Teacher: The ones on the circle.
Student: Which points?
Teacher: Precisley!
Student: "Precisley"?
Teacher: Good point!
Student: What's the point?
Teacher: Which point?
Student: The ones on the circle.
Teacher: Precisely!
Student: Good point!

Rod (still wondering "What's the point?")

[Sandy]

Thanks for the chuckle Rod!
xxSandy

[Amigoo]

Re: I-Lambda VIp design
"Quintessential pattern of Pi"

A single 27.597.. degree angle was weak confirmation of squared circles, motivating further exploration of the geometry. With slight reorganization and color highlighting of the integrated geometric objects, the two inner circles now reveal identical right trapezoids ... with each hosting an identical pair of objects: right triangle and scalene triangle.

A squared circles version of "Where's Waldo?" (and his larger twin).

Rod

[Amigoo]

Re: I-Lambda VIp design
"Quintessential pattern of Pi"

The "magnifying glass" (light blue) seemed to be an important symbolic object during the development of this design, but the lines within the focus of this glass didn't appear that important (relative to the entire composition) ... until today.

Now, the V-shaped notch near the top identifies the right triangle whose trigonometry confirms the square of that circle:
diameter = 2 units, radius (hypotenuse) = 1, long side = 1/2 square root of Pi (0.8862269..), angle at center = 27.597.. degrees.

Rod

[Amigoo]

Re: I-Lambda VIp design
"Quintessential pattern of Pi"

The geometry was simplified slightly to highlight the three red "L" shapes (also right triangles). These objects having identical angles are consistent features of the three, tightly-integrated squared circles in this composition.

Rod ... ...

[Amigoo]

The Scalene of Pi (new design)
Visual perspective on the relationship of Pi to its circle-squaring scalene triangle.*

Large circle's diameter = 3.141592653589793238462643383.. units.
Scalene triangle's left side = 1.772453850905516027298167483.. units
(equals the square root of the diameter).

* This scalene triangle effectively squares all circles.

Rod

[Amigoo]

Scalene of Pi (more artistic presentation)
Visual perspective on the relationship of Pi to its circle-squaring scalene triangle *

Large circle's diameter = 3.141592653589793238462643383..
Scalene triangle's left side = 1.772453850905516027298167483..
(equals the square root of the circle's diameter).

The "exclamation point's" circle fits precisely inside the
red right triangle (where the green line would cross)
when the circle is expanded as CSC.

* This scalene triangle effectively squares all circles,
especially via the hypotenuse of the red triangle
(line is drawn at 62.4028873643.. degrees).

Rod ... ...

[Amigoo]

Scalene of Pi design (more lines added)

This revision highlights (golden lines) the direct
relationship of Pi to its circle-squaring scalene triangle (green):

largest circle's diameter = Pi
= 3.141592653589793238462643383..
scalene’s right side = square root of Pi
= 1.772453850905516027298167483..
lower golden diagonal = square root of Pi
= 1.772453850905516027298167483..
upper golden diagonal = half of Pi
= 1.57079632679489661923132169..

Rod ...

[Amigoo]

Re: Scalene of Pi design

Interesting design with good potential, but such wiggly numbers!
And where did they come from?

Translation: "Back to the drawing board!"
This geometry needs more analysis.

Later in the sandbox, the problem was isolated:
only two lines are described correctly.

Rod

[Amigoo]

Re: Scalene of Pi design

Oops again - someone forgot to factor in the 10,000
multiplied times Pi to expand the lines in the CAD drawing.

So, what's correct? Elefino
(but the numbers look better)

Rod

[Amigoo]

Re: Scalene of Pi design

As soon as the Elefino left the room, I rechecked the numbers.
All is well ... for a finite number of decimal places.

So, given a circle with a diameter of Pi, the circle-squaring scalene triangle
can be created geometrically. And that triangle's upper vertex defines the
the square's sides via the 62.4028873643.. degree radius.

At least, the numbers in this Cartesian plane seem to agree.

How many decimal digits of Pi are required? All of them!
But ask an impossible question and the Elefino returns.

Rod

[Amigoo]

Re: Pythagorean Pi (new design)

This design increases the number of pages in the PDF file to 88, a number having esoteric essence.
A memorable example: 8 + 8 = 16, 1 + 6 = 7

The geometry of the design hints of a "good & sufficient" plateau for this study of squared circles:

A circle whose diameter equals the (square root of Pi) x 2
(3.544907701811032054596334966..)
has a square whose side length equals Pi
(3.141592653589793238462643383..)

Preliminary analysis:
as the number of decimal digits of Pi increase (are discovered), the angles of the red right triangle will not change.
And this suggests that this Pythagorean Pi geometry correctly identifies Pi's circle-squaring scalene triangle.

Midwayer prompting fun:
9:26 was an intriguing morning prompt since it didn't appear to relate to anything ... except this geometry.
A few moments of reflection discovered this association:

I've been multiplying line length values times 10 million to force greater accuracy of angles and better CAD design.
Pi times 10 million = 31415926.535897932384626433832795 where "926" appears to the left of the decimal point!

Another symbolic association ( 119:8.8 ): "Nebadon, the chief of ten million inhabited worlds"
[more intrigue: "88" just made another appearance]

Rod ... ...

[Amigoo]

Nebadon, the chief of ten million inhabited worlds

Out of context, this quote suggests an incorrect name for our planet. Here's a better quote:
"Urantia is the sentimental shrine of all Nebadon, the chief of ten million inhabited worlds" ( 119:8.8 )

As I pondered the value of posting this explanation, I glanced at the clock: 1:55
Hmmm ... 1:55 seems to have some numeric association with the previous post.

Rod

[Amigoo]

Re: Scalene of Pi design (revised)
From whence the "blue moon" in the background?

Technically, I drew it there and assigned a color. So, the question should be:
"What in this geometry created the three points that defined this circle whose
diameter relates to the square root of 2"? (1.4142135623730950488..)
... especially since this circle is the last object drawn!

The answer might explain why Pi's scalene triangle squares a circle.

Rod

[Amigoo]

Re: Scalene of Pi design
From whence the "blue moon" in the background?

This had to answered ... or bedtime would be at sunrise.
And the answer was so obvious, with the right perspective:

The hypotenuse of the red right triangle
(length = 1.25331413731550025120788..)
has length equal to the square of the blue circle
(diameter = 1.414213562373095048801688..)

... and the perpendicular to the midpoint
defines the center of the circle.

Rod