Paradise Trinity Day

[Amigoo]

Re: Pi2b design

Pi2b-1:2 is a more precise title since the three concentric scalene triangles (with other lines, points and angles within this CSCSC object set) identify many 1:2 line length relationships (especially between objects) which confirm three squared circles in geometric replication.

Say what?! How about "Geometers' eye candy for squared circles aficionados"

Rod ... ...

[Amigoo]

Re: Pi2b design
Wherefore art thou Pi?

Life is like a slice of tasty Pi ...
grab a piece whenever you can.
It's not impossible.

Pi2b or not 2b ... that is the answer.

Rod

[Amigoo]

Nuff Said (new design)

With this being Day 1 of the Captain's Table challenge (see "TUB Study Group" topic), playtime hours in the geometry sandbox are minimal. Nuff Said, an extract of objects from the iTrapezoid design, is a simplified replication of the scalene triangle (half of a trapezoid) that squares the circle.

The base (longest line) of the inscribed isosceles trapezoid has length equal to one side of the square of that circle. In this geometry, the identifying 1:2 line length relationships are clearly displayed by the trapezoids in the inner and outer circles (diameters = 40,000,000 and 80,000,000 units).

Obviously, the irrational, transcendental Pi is trapped within this CSC trapezoidal corral. Enough said!

Rod

[Amigoo]

Nuff Said design

A few more lines were added to highlight inherent symmetry (diamond shapes)
and to showcase the Greek lambda (11th letter of that alphabet)

Considering the Greek challenge of "squaring the circle",
the lambda symbol has intriguing historical signifcance:

"the reconciliation of opposites"
"a state of balance, requiring adjustment"
"the light of knowledge shining into darkness"

Rod

[11light11]

Hi Rod! I've been enjoying catching up on your work this morning! It's always such a lift for me and a great way to start the day! Thank you for sharing and keep up the excellent work!

With love to you, Michele

[Amigoo]

Michele,

Your timely comment about work reminded me that I had one more task remaining: the morning after review (next morning after the day that I'm "convinced" that the design is finished).

I found the symmetry of the many lines having length equal to the radius of the small circle so compelling that I had to increase the design's complexity yet again. However, such symmetry keeps complex geometry from appearing so cluttered with lines and objects!

Interestingly, the abstract lambda symbol (red) may be the final unifier in the debate about squaring the circle ...

Whereas advanced math "proves" that the circle cannot be squared, this geometry visually proves that squared circles do (must) exist in the universe. The identical angles of the similar objects in the large and small circles and the 1:2 line length relationships (plus diameters equal 40,000,000 and 80,000,000 units, precisely) seem to confirm this theory!

Thus, lambda's "state of balance, requiring adjustment" seems to agree that Pi is irrational and transcendent, but only within the finite boundaries of this CSC trapezoidal corral. And, regarding the debate, this proffers that we can have our cake (Pi) and eat it too!

Rod ... ...

[Amigoo]

Nuff Said design (one more line added)

the Greek lambda (11th letter of that alphabet)

I couldn't resist the symbolism: a double lambda (1111)
Besides, it's inherent in this CSC geometry!

Rod

[Amigoo]

Lambda Lines design (more complex)

Lambda lines (red) provide evidence of a squared-circles Cartesian plane.
They are formed when right triangles preempt circle-squaring trapezoids.

Say what? The border of morbus cyclometricus.
Say what? "Impossible" squared circles.
Say what? The finite essence of Pi.
Says who? This geometry.

Rod

[Amigoo]

Re: Lambda Lines design

Contemplation of past x:52/x:53 prompts
(x:52 immediately followed by x:53) ...

From the table of observed associations:
x:53 - 1.7724 53 8509.. square root of Pi,
x:26 - 0.8862 26 9254.. half square root of Pi
(x:52 = two lengths of x:26)

Analysis:
These prompts refer to the isosceles right triangle (largest red lambda) whose hypotenuse squares the largest circle (D = 80,000,000 units).
Half of this hypotenuse (plus a vertical line of equal length) form a smaller right triangle whose hypotenuse squares the circle whose
diameter has length (L = 56568542.4949.. units) equal to the side of a square inscribed in the large circle.

Translation:
These prompts identify geometrically related isosceles right triangles, directly associated with the square root of Pi
and which confirm the "impossible" circle-squaring in this unique Cartesian plane.

Say what? Squared circles exist!

Rod

[Amigoo]

Re: Lambda Lines design (new & improved)
Now reflecting the "geometric DNA" of squared circles.

In this CSC set, the isosceles right triangle of the large red lambda
and the lambda's enclosing isosceles trapezoid have replicated features:

1. The isosceles trapezoids in every circle have identical angles.

2. The side of the isosceles right triangle has length equal to the
circle-squaring hypotenuse of the next smaller isosceles right triangle
(which is part of that circle's inscribed isosceles trapezoid).

3. The midpoint-to-midpoint length of the sides of the trapezoid
is equal to the diagonal length of the next smaller trapezoid.

Some lines are not displayed for better design presentation.

Rod ... ...

[Amigoo]

Re: Lambda Lines design (another triangle)

More symbolism of universe balance represented by squared circles:
The lower right triangle with its circle-squaring 62.402887.. degree vertex
provides the fulcrum for a symbolic gyroscope (smaller circles).

Rod

[Amigoo]

Re: I-Lambda (new design)
"Architectural essence of the universe."

Allusion to the Greek concept of universe equilibrium,
a reconciliation of opposites requiring continual adjustment
... by unseen beings and forces who excel in this role.

Rod

[Amigoo]

Re: I-Lambda design
"Architectural essence of the universe."

Enhanced with essence of sacred geometry (7 circles, all squared)

Rod

[Amigoo]

Re: I-Lambda III (new design)

"CSC = Circle inscribed in Square inscribed in Circle ..
(aka Cyclometricus via Symmetry and Concentricity)"

Rod ... ...

[Amigoo]

Re: I-Lambda III design
How does it know?! A Pythagorean anomaly?

The light blue isosceles right triangle has equal identity
with the three red lambdas' left-slanting diagonal lines:

1. Hypotenuse has length equal to the shortest diagonal line
2. Hypotenuse has half the length of the longest diagonal line.
3. Combined length of the sides of this isosceles triangle
equals the length of the middle diagonal line.

This triangle rocks 'n' locks the square ... IMO.

Rod

[Amigoo]

Re: I-Lambda III design

How does it know?! A Pythagorean anomaly?

Today, I vote for "Pythagorean anomaly":

Each of the three lambda diagonals* form one side of a square. When the three squares are aligned on a common center (when they're concentric), they create the foundation of a CSC set** (draw a circle inside each square to see the complete set: CSCSC).

Therefore, an isosceles right triangle has direct geometric relationship to a unique CSC set.

* which equal the three lengths of the triangle: hypotenuse, two sides combined, hypotenuse x 2.
** CSC = Circle inscribed in Square inscribed in Circle ...

Rod ... ...

[Amigoo]

Re: I-Lambda III design (cross discovered)

In this local universe, a cross might authenticate
a concentric trinity of squared circles.

Rod

[Amigoo]

Re: I-Lambda III design
(recolored to highlight the "cross")

The diagonals of this cross have length equal to the radius of the small circle
and three diagonals connect the midpoints of the sides of the two identical, small
scalene triangles having vertically reversed placement.

Both of these scalene triangles effectively square the small circle,
and the one with the inscribed cross has 1/4 the area of the largest scalene triangle
(an easy visual: there are 4 identical scalene triangles in this large scalene).

"We shall see" ...
I had a dream where a note was written that October, 2014
is the month that this scalene will achieve recognition.

Rod

[Amigoo]

Re: I-Lambda III design
(blue isosceles right triangle in center enlarged)

Imagine that the midpoint of one side each of two inscribed squares in the largest circle pivots at one of the 45 degree vertices of the right triangle. The distance between the opposite ends of these sides effectively equals the square root of Pi when the sides (only one is shown) are positioned as drawn.

Say what?! There's obvious circle-squaring symmetry in this complex design.

Rod

[Amigoo]

Re: I-Lambda IV (new design)
"Linear correlation in CSC cyclometricus."

A solid research plateau and anniversary tribute to the first post for this topic:

"If not already declared On High,
10-10-10 presents favorable symbolism for a Paradise Trinity Day.

Rod ... ...

[Amigoo]

Re: I-Lambda IV design
"Linear correlation in CSC cyclometricus."

I felt inspired to add one more vertical yellow line to the geometry*, then noticed that these three lines appeared as crosses, but only two were equal in length. Reflecting on what symbolism might explain this anomaly, I remembered a certain paragraph (187:4.1).

And the two diagonal red "crosses" seem related, being symbolic of the travel along that difficult road.

* the straight yellow and red lines have 1:1 and/or 1:2 length relationships.

Rod

[Amigoo]

Re: I-Lambda V (new design)
Is the Pi Corral finally locked?

I knew that a "geometric lock" would be required to convince disbelievers that Pi, represented as a circle's inscribed scalene triangle, could not be transcendental. So, what comprises such a lock for a Pi Corral on a remote Cartesian plain? Tonight's research says 5 circles in CSC concentricity.

For when the identical scalene triangle is replicated 4 times (5 circles) with complementary concentricity, the outermost and innermost inscribed scalenes have parallel lines with perfect linear correlation (outer lines have 4:1 length relationships with inner lines; diameters = 40,000,000 units (largest circle) and 10,000,000 units (smallest circle)).

This replication pattern suggests that, by geometric representation (not mathematics), "transcendental" is an essence of squared circle mathematics but not an essence of geometric reality.

Say what?! I-Lambda V may explain this better with colored patterns
... and 5 circles, all effectively squared.

Rod

[Amigoo]

Re: I-Lambda V design (in development)
Is the Pi Corral finally locked?

Despite the recent and past nonsense posted about Pi not being transcendental, I-Lambda V and its 5 circles with replicated scalene triangles beg a disclaimer ... possibly infused with more nonsense:

Pi is both transcendental and irrational ... because this is inherent in its derivation from a geometric model(s). However, the Pi constant (ratio of a circle's circumference to its diameter) seems to be imprecise relative to apparent geometric reality in certain CSC concentricity. Nevertheless, such "imprecision" is most impressive - trillions of decimal digits impressive!

Today's analysis can only be: "Long live Pi!" ... until the lock becomes hardware.
(the lock on this Pi Corral is not closed and this lock is more concept - theory - than hardware)

Note to self: Pi will still live into eternity because that's its design.

Rod

[Amigoo]

Re: I-Lambda V (design completed)
Is the Pi Corral finally locked?

When introduced to scholars and aficionados of esoteric geometry,
this design is described as a "Pi Corral on a remote Cartesian plain
having CSC concentricity with five inscribed isosceles trapezoids
at 45 degree increments and having quintessential similarity."
Invariably, the first question asked is "What's in the bucket?"

Rod ... ...

[Amigoo]

Re: I-Lambda V design (more esoteric)

Who knew? Squared circles wax quizzical!
If squaring the circle is a drop in the bucket,
"What's in the bucket?"

Rod