Paradise Trinity Day

[Amigoo]

Re: MP 17.403 (new design)

"MP" could allude to the geometric magic of the "missing page" in the Zzzz story, but it is the first letters of "Master Pattern".

... I've been grasping for associations with the x:17 prompt when it (angle 17.403..) is all over my Cartesian work space!
Undoubtedly, "17" refer to this nearly ubiquitous angle in squared circle geometry.

The MP 17.403 design highlights this angular quotidian possibility.

Rod

[Amigoo]

Re: MP 17.403 design

For the record ...
The angle is 17.4028873643093955482677952.. degrees, plus a few more decimal digits.
How many? Close to the number of decimal digits in Pi.

But not to worry - when all of these angles are identical, the circle is squared.

Rod

[Amigoo]

Re: SG 17403 (new design)

MP 17403 was upgraded to the sacred geometry of squared circles
... and is great for meditation on: "What's the point?"

Rod ... ...

[Amigoo]

Re: MP3 (new design)

Such strange closure for a project! I keep thinking that I'm recording final commentary, diagrams, etc., when "just one more" design again emerges from the creativity sandbox. Today's MP3 design refers to the three equal line lengths which separate each of the three sets of points of two identical isosceles right triangles.

This is easy to visualize:

Inscribe a square in a circle, then divide the square in half diagonally. Allow the two halves to rotate independently at the center of their hypotenuses (circle's center) until each set of points is separated by the same distance. That same distance is equivalent to the square root of Pi (when the two triangles are perfectly repositioned, the circle is effectively squared!).

The MP3 design in development shows that perfect alignment.
The "MP" in "MP3"? "Mystical Pi" is tonight's leading contender for the title,
but the "Missing Page" myth energized this mini-adventure.

As for the recent x:37 prompts, consider this:
Three points of each triangle and the shared hypotenuse center = 7 points!

Rod

[Amigoo]

The not so mysterious explanation of this Number 37 riddle:
the three points of the triangles have the same separation
even when the triangles form the square.

Maybe there's a different geometric marker that confirms the circle's square.

Rod

[Amigoo]

Re: MP37 design (MP3 renamed)

I expected the MP3 design to highlight its geometry in a technical way, but it seized the opportunity to wax symmetrical.

The colorful waxing is impressive,
but the 17.403 degree angles are still all over this Cartesian plane.

Rod

[Amigoo]

Re: MP37 design
Well, some technical information is worth noting:

"MP37" (Magical Pi 3:7) refers to the seven points of two isosceles right triangles, rotating on the center of their hypotenuse. These triangles each form half of a square inscribed in the circle, with one square overlapping the other. The dark blue hypotenuse (diagonal of square) of each corner of the circle's square (not inscribed) hosts six 17.403.. degree angles.

Is the circle squared? I don't know!

My not-so-magical Pi calculator doesn't have enough digital fingers to count the decimal digits in the identical twenty four 17.403.. degree angles. How many decimal digits would I estimate? The number of decimal digits in Pi ... times 24.

Rod

[Amigoo]

Re: Ping? (final design)

NAK! (a closure clue I understand)

Rod ... ...

[Amigoo]

Re: MP37 design (further embellished)

Ping? ... ACK (but just to elaborate)

At a mountain retreat many centuries ago,
a geometry guru (gGu) and geometry grasshopper (gGr)
are discussing how to prove that a circle is squared
using the 24 17.403.. degree angles of the MP37 geometry:

gGu: In the future, man will learn about Pi and its trillions of decimal digits.
gGr: What is "Pi" and what is "trillions"?

gGu: A happy round belly and uphappy debt. ,
gGr: Is that the meaning of "squaring the circle"?

gGu: Yes and No.
Making a belly happy with Pi is easy.
Eliminating trillions of debt is impossible, IMO.
gGr: What is "IMO"?

gGu: In the future, man will substitute internet slang for real words.
gGr: Is "smexy" this internet slang?

gGu: Yes and No, but IANAL and DBA.
gGr: Will Pi help me to square a circle?

gGu: No, Pi will prove that this is impossible.
gGr: So, what do we do until Pi is invented?

gGu: Just prove that the angles are equal.

Rod

[Amigoo]

(currently uphappy with bifocals)

In my previous post ...
"uphappy debt" should be "unhappy debt".

Rod

[Sandy]

No worries Rod. I found my "bespectacled eyes" read it properly anyway, despite the miss spelling and I probably am not the only one.
XX Sandy

[Amigoo]

Re: Arc Arc Arc (new design)

All of the designs started as serious geometry study, then many of these CAD drawings were converted into an arbitrary design at the completion of the geometry study. Arc Arc Arc is more informative geometrically than artistic ... but that's its purpose.

Arc Arc Arc is a study of the three significant arcs that relate directly to the six 17.403.. degree angles associated with one corner of a squared circle in the MP37 design. Why these angles relate to a squared circle is visually apparent in this study.

The project seems ready for a nice long nap.

Rod

[Amigoo]

Re: Circle the Square (new design*)

Geometry Quiz

Given these two components of a squared circle:

1). A straight line with length equal to the square root of Pi.
2). A circle's diameter with length equal to 2.

1. Identify the steps required to construct the geometry shown in Circle the Square.
2. How do you know that this circle is squared?

* Technical information about squared circle geometry.

Rod ... ...

[11light11]

Rod, I just love your work! Please don't ever stop sharing it with us!

Lots of love to you,

Michele

[Amigoo]

Michele,

Thanks for the comment!

Regarding the trick question of the Geometry Quiz:
"How do you know that the circle is squared?"

Because it was stated at the beginning of the Quiz:
"Given these two components of a squared circle:"

But construct a geometric proof and you'll impress the world!

Rod

[Amigoo]

Re: Circle the Square design

The large square, one of a set of two that I've been studying, has an intriguing short history:

After working on this and related geometry for a week, I had opened this CAD file to continue research. When I did, this square was the only object in the file! What?! How did I erase all of the other geometric objects?

I was psychologically prepared to reconstruct the Circle the Square geometry when I discovered that the square was the final addition and had size sufficient to make rest of the geometry a pinpoint. Once the square was erased, the previously existing geometry became visible.

I still don't know how this happened (a Midwayer clue?), but this single square became the better representation of the geometry of a squared circle; two squares with one overlapping was not the foundational geometry.

"Who can tell?"

Rod

[Amigoo]

Re: Circle the Square design
The three halves of proof ...

In the center of the design,
"just one more line" was added to show:
- half of the side length of the circle's inscribed square (golden).
- half of the square root of Pi (green).
- half of the circle's diameter (red).

Arc Arc Arc!

Rod

[Amigoo]

Re: Circle the Square design ("just one more line" What? Again?!)

I've long suspected that this squared circle geometry might offer important symbolism for the future. Curiosity about the "three halves" (two circle-related components joined by half of the square root of Pi) led to discovery of a fourth half (dark blue line in the center of the design).

These four lines (red, green, blue, golden), connected in a butterfly shape, well promote the message of "transformation". And in cultural myth, the butterfly is a symbol of transformation because of its inspirational metamorphosis.

Why did I select the primary colors (red, green, blue) and the resulting mixture (golden; near white) for this butterfly of transformation?
Intuition or guidance but not because I remembered "primary colors" as I selected contrasting colors to highlight the geometry.

Rod

[Amigoo]

Re: Circle the Square design (more exploration of the geometry)

In this next version, imagine that the dark blue circle rotates on the vertex of the 45 degree angle of the stationary, green scalene triangle*. Only one point of the circle causes it to align as shown. When thus aligned, a chord is created that has the same length as a side of a square inscribed in the circle. And the horizontal green line (another chord of the circle) then has the same length as the square root of Pi!

Interestingly, circles in a range of diameters can also be squared this way. What? Pi squares other circles?! Why not? Pi is a ratio of a circle's circumference to its diameter - this ratio applies to other diameters as well!

* a 45 degree angle until the circle (diameter = 2) is correctly positioned, then the unique, circle-squaring scalene triangle is effectively created.

Rod

[Amigoo]

Actually, three points (not just one) of the circle are important. And the circle's chord, having length equal to the side of an inscribed square, exists for various sets of those three points.

Translation: This design is a good visual reference for further exploration of the geometry of a squared circle.

Rod

[Amigoo]

Re: Fiesta Pi (new design)
"Celebrate the ratio of endurance."

I keep searching for confirmation of the precise geometry that defines a squared circle, but have to be satisfied with entertaining designs. This Fiesta Pi highlights the continuing effort and hints that occasional breaks to "smell the roses" (or fragrant Pi) are always refreshing.

Olé! comes to mind as I search for my sandals and summer shades.

Rod ... ...

[Amigoo]

Olé! I found my summer sandals and shades.
(Not winter sandals ... in north Texas?!)

Regarding the circles squared by Pi (a line length equal to the square root of Pi) ...

Range of diameters relative to the square root of Pi
(a circle having a diameter of 2 is within this range):
- from square root equal to side length of a square inscribed in the circle.
- to square root equal to side length of a square enclosing the circle.

An easy visual (for the circles in this range):

Imagine the square root of Pi as a horizontal line, then lower this line (midpoint aligned on the circle's center) into the circle from the top. When the two points of the line touch the circle, draw diagonal lines at 135 degrees (from right point of Pi line) and at 45 degrees (from left point of Pi line). Where these diagonals meet the circumference of the circle are points that define diagonals whose length equals the side of a square inscribed in the circle.

The resulting isosceles trapezoid contains two similar scalene triangles that define the square of this circle.

What about all other circles? Draw them centered on the circle already squared. The magical 62.4028873643093955482677952.. degree radius is the easiest geometric marker for squaring other circles (Geometry 101 skills required).

Rod

[Amigoo]

What?! Where's the 62.403.. angle in circles not having a diameter of 2?

We could call this a documentation error (the method only works for circles having a diameter of 2), but the lesson is that the square root of Pi defines the square of one circle only: a circle having a diameter of 2.

So, you ask: "If I have a line length equal to the square root of Pi and circle with a diameter of 2, I already know how to square the circle without drawing the scalene geometry?!"

Exactly!
So, "What's the point?!" (of all this research).
To identify unique geometry existing in a squared circle,
for if such geometry exists, the circle can be squared!

Rod

[Amigoo]

Speaking of unique geometry ...

While the square root of Pi (as a line length) cannot be used to square a circle not having a diameter of 2, the left and right sides of the trapezoid (two overlapping scalene triangles) defining the square of the circle have consistent angles: 72.597.. degrees (left) and 107.403.. degrees (right).

All squared circles contain this angles-similar trapezoid with its diagonal side lengths equal to the length of a side of a square inscribed in the circle. And this consistent pattern clearly implies that geometry does exist to "square the circle".

Rod

[Amigoo]

Re: As Above, So Below (new design; visual summary of the geometry)

"As Above, So Below" refers to a unique feature of this geometry and its circle-squaring isosceles trapezium:
the red right triangle above circle's center has same dimensions as red right triangle below center.

Rod