Paradise Trinity Day

[Amigoo]

Re: "Cartesian Buffet" (new design)

"Buffet" is a delicious word for this title because its meanings (different pronunciations) could apply to this squared circle "cross reference".

I was curious if geometry could be created that correlates a squared circle having a diameter of 2 with another squared circle whose side of square has a length of 2. Of course, such geometry would not be simple (I searched anyway), but geometers will understand the resulting esoteric composition when the design and some of the wiggly numbers are posted in a day or two.

Cartesian Buffet might even have revelatory value, geometrically speaking. At least, the tasty objects in this Buffet include three circles all squared, with a 62.402.. degree radius confirming every square.

Rod

[Amigoo]

Re: "Cartesian Buffet" design
http://aitnaru.org/threepoints.html

Diameter of primary circle (yellow) = 2.
Side length of centermost square = 1.

All 3 concentric circles are squared
(confirmed by hypotenuse of red right triangle,
radius of primary circle at 62.402.. degrees).

The Buffet is open.
Try the new vesica piscis platter!
(share a hearty meal or just stuffit)

Rod ... ...

[Amigoo]

Re: "Cartesian Buffet" design

Changed a few colors for better coordination of related objects (primary circle is green; largest circle is yellow).
Also, largest square is now dark blue (men prefer less lavender on their visca piscis platter; who knew?)

Rod

[Amigoo]

Re: "Points of Balance" (new design)

Exploration of the geometry of Cartesian Buffet,
emphasizing the 27.597.. degree vertex of circle-squaring right triangles.

Rod

[Amigoo]

Re: "Vesica Squares" (new design)
"An august geometry soirée."

In addition to CSC concentric circles,
there's another option for studying the geometry of squared circles: vesica piscis.
But this Vesica graciously invited CSC twins to the soirée
... who immediately paired with the scalene twins!

Rod ... ...

[Amigoo]

Re: "Vesica Squares" design
"An august geometry soirée."

Theme for the first annual soirée:
"Patterns ~ the music of geometry"

Rod

[Amigoo]

Re: "Vesica Squares" design
"An august geometry soirée."

Vesica Squares soirée, indeed!

The latest design revision helps explain why vesica piscis gained such prominence in sacred geometry. The integration of the squared circles' scalene triangles, isosceles trapezoids, pentagons (4 blue sides), and isosceles right triangles (including the square root of Pi*), all hosted by a vesica piscis having CSC circles, is intriguing - even esoteric!

* large circles' diameters = 2

Rod

[Amigoo]

Re: "In The Area" (new design)

An interesting technical observation that may be useful in the study of squared circle geometry:

Given a circle with diameter equal to 2, the area of the isosceles trapezoid inscribed in the circle is 1/2 of the area of Pi (this circle's square). The square inscribed in the trapezoid is 1/4 the area of Pi (and 1/2 the area of the trapezoid).

rod

[Amigoo]

Re: "In The Area" design

Exploration of the circle/trapezoid/square relationships shows that these relationships are consistent (but not "revelatory") for other possible squares of a circle*. Long story short: months of research suggests that geometry must exist that precisely defines a squared circle; the unique scalene triangles and related trapezoids will be present. How to get there from here? Wise answer: "Next question!"

Moving on ...

Consider the ever increasing decimal digits of Pi, increasing without predictable numeric pattern. Because a circle's circumference and diameter are fixed (we're still trying to determine the exact ratio), the only reason the changing ratio does not reflect a predictable numeric pattern is the derivation of Pi. Otherwise, the square of a circle (calculated by the increasing number of Pi digits) should be increasing by equal and predictable values.

But "increasing" is a good clue in squared circle exploration!
This tells us what to look for in geometry that keeps advancing toward precise definition of a squared circle.

What's next? Wise answer: "Next question!"
(heading back to the sandbox to kick sand and look for new patterns)

* with the square root of Pi as the side length of a reference square, the exploration range is from a circle inscribed in this square to a circle enclosing this square (square is inscribed in the circle).

Rod ... ...

[Amigoo]

Re: "Equal But Opposite" (new design in development)

Too much curiosity quickly answered the "What's next?" question.
But this geometric juxtaposition of two squared circles is just a guide for later research
... and slow progress is anticipated since kicking sand is more than satisfying now.

Beach instead of sandbox would be a vacation upgrade.

Rod

[Amigoo]

Re: "Equal But Opposite" (new design in development)

For the record ...

When I reviewed that brief pontification about "predictable numeric pattern", I thought "Say what? Somebody's full of Pi!" But "out of the box" thinking permits mixed metaphors (if that's what that was all about).

In the creativity sandbox, the EBO (Equal But Opposite) design is progressing quickly but I'm not sure if it expresses the intended concept. Maybe another slice of Pi without metaphoric spice would help.

Then again, this geometry expresses other interesting aspects of squared circles, such as the 62.402.. degree angle in the large, red right triangle with a vertical side equal to the square root of Pi times 2. So, the design may express all of the Pi that's necessary for this recipe.

Rod

[Amigoo]

Re: "Equal But Opposite" design (completed)

To clarify, 62.402.. degrees refers to a diagonal line, often the hypotenuse of a right triangle;
27.597.. degrees refers to the smallest vertex of the right triangle that has this hypotenuse.

The 62.402.. degree radius has long remained a primary focus in this research
since it clearly identifies a squared circle!

Rod ... ...

[11light11]

Love it all, Rod!

[Amigoo]

Re: "Vertical Transition" (new design)

Visually, this geometry shows the squared circle transition from square inscribed in circle (top of design) to circle inscribed in square (bottom of design); effectively, the range of all possible squares of one circle.

The middle circle visually proves that one unique trapezoid (two overlapping scalene triangles) must exist when the circle is truly squared; all other squares, from the beginning to the end of this transition, are not the true squares of this circle.

After thousands of hours of research with CAD software, I have yet to find a way to draw this perfect trapezoid (re: the Greek rules for this challenge) or how to prove geometrically that the circle is squared!

"You can't get there from here!" seems to be the necessary conclusion, but complemented with:
"This is what 'there' looks like ... if you ever find a way to this geometric estate."

Maybe refilling the creativity sandbox with fresh sand will help ...
sand with fewer decimal digits in the dimensions of its grains!

Rod

[Amigoo]

Re: "VH Transition" (renamed from Vertical Tansition)

My image editor had other ideas about presentation of this CAD drawing, refusing to create crisp lines in the enlarged geometry.

So, VH (Vertical Horizontal) Transition is the better title. And I see that this presentation permits easier 1:1 comparison of the squared circle with its two siblings. "Who can tell?" Maybe this 1:1 perspective has potential for revelatory guidance!

The two red triangles and vertical line relate to trigonometry that calculates the area of the circle.
For the squared circle (copied from the May, 2012 document):

Area = (( Cos 27.597112635690604451732204752339.. ) x Diameter ) Squared
Circum. = (( Cos 27.597112635690604451732204752339.. ) x length of square's side ) x 4

The 55.194225271381208903464409504677.. degree Angle of Squaring Radii (ASR) refers to the lower vertex of a downward pointing isosceles triangle. The top horizontal side of this triangle is a portion of the top horizontal line of the circle's square.

The right half of the ASR is the angle of focus for this trigonometry:

55.194225271381208903464409504677.. x .5
= 27.597112635690604451732204752339.. ( cosine angle )
= 0.88622692545275801364908374167057.. ( cosine; Radial Pi or rPi )
= half the square root of Pi (value derived from cosine angle)


Here's how these wiggly numbers compare to Pi:
(selection of the 2 billion units diameter is arbitrary;
precision increases with more cosine angle digits)

No-Pi formula for area:
A = 0.88622692545275801364908374167057
x 2000000000 = 1772453850.9055160272981674833411 squared
= 3141592653589793238.4626433832795

Pi formula for area:
= 3141592653589793238.4626433832795 ( A = Pi x Radius squared )

No-Pi formula for circumference:
C = 0.88622692545275801364908374167057
x 1772453850.9055160272981674833411 x 4
= 6283185307.1795864769252867665585

Pi formula for circumference:
= 6283185307.179586476925286766559 ( C = Pi x Diameter )


Since 90 - 27.597112635690604451732204752339..
= 62.40288736430939554826779524767.. degrees,
a radius must exist that effectively squares the circle!

TMI? Tell me about it! In recent years, I've had all the Pi I want,
... but keep returning for more.

Rod

[Amigoo]

Re: "VH Transition" design

'Who can tell?", indeed! The required reformatting of the three circles left a seeming void in the composition
... which was filled by another perspective of the squared circle.

Now, the four circles appear in a "U" shape, with the 3-circle range beginning at the upper left of design, and ending with the fourth circle (upper right) which highlights the green isosceles right triangle from the CSC set.

With the red horizontal line of the fourth circle representing the square root of Pi, when the green triangle pivots counter-clockwise at the right end of the Pi line, it stops at the red perpendicular line. At this point (literally and figuratively), both the circle and its perfect square may be drawn.

Study of this transition hints to an out-of-the-box thinker that only two points are required to square the circle.
Undoubtedly, a research effort best transitioned to an upgraded creativity sandbox.

Rod

[Amigoo]

Re: "Lotsa Pi" (new design)
http://aitnaru.org/psymmetry.html

This geometry is a brief summary of the more interesting lines and objects in squared circles.

If the beginning object was the green square (similar to one inscribed in a circle having a diameter of 2) and the yellow isosceles right triangle (sides = 1/4 the square root of Pi) was added and allowed to pivot, the rest of the geometry can be created from these two objects.

It's obvious to geometers that these two components are sufficient to create a squared circle (it's not a "solution"), but the method highlights the intriguing geometry of squared circles ... especially the overlapping inscribed squares.

Rod

[Amigoo]

Re: "G-BIRT en paire" (new design concept)
Pair of Geometrically-Balanced Isoceles Right Triangles

Further exploration of the circle-squaring effect of a pair of CSC isosceles right triangles that pivot.
A lazy summer schedule is planned for this sandbox creativity, but the geometry is not complex.

Rod

[Amigoo]

Re: "Wings of Linneaus" (new design; renamed from G-BIRT en paire)

"Geometrically-balanced isosceles right triangles
appearing as wings of Linnaeus in a circle squared"

Who knew?! This geometry, once awakened, preferred to wax poetic,
rendering "lazy summer schedule" a midsummer's night's dream.

Rod

[Amigoo]

Re: "Wings of Linneaus" design

The isosceles right triangle (yellow) in the center of this Wings design has appeared in many recent studies of the geometry of squared circles. While "lazy summer schedule" is still on the agenda, a study of this right triangle now seems important for future research.

In the latest design ("The Ceneter of Pi"), this right triangle (red) is the focus of the study. In particular, I've been curious which geometry identifies the placement of this triangle in the squared circles design compositions, therefore identifying the "center of Pi".

"Ceneter" is creative allusion to a representative in a certain deliberative assembly. In The Ceneter design, the side length of the red triangle is equal to the square root of Pi; the largest circle has a diameter of 4.

The debate about Pi (its transcendental nature as well as the real ratio as observed in squared circle geometry) will likely continue for decades. So, consult the Ceneter whenever your mind needs expansion ... geometrically speaking.

Rod

[Amigoo]

Re: "Sibling Rights" (new design)

The Ceneter says ...
Consider a spiral of five circles, all effectively squared but not concentric; a CSCSCSCSC set.

This geometry emphasizes the close pair of right triangles appearing in every squared circle.
The spiral is the design, but selection of contrasting colors will take several days
... on a "lazy summer schedule".

Rod

[Amigoo]

Re: "iSibling Rights" (new design)

A revised title, alluding to Infinite Sibling Rights ("infinite" because of the apparent logarithmic spiral; "rights" referring to the two side-by-side right triangles in each of the seven squared circles (six effectively; the largest includes its complete square).

"Lazy summer schedule" was postponed again to capture the sense of project closure (or at least, major plateau). And closure (or plateau) indeed occurred when seven circles were found within 360 degrees of the spiral in this Cartesian landscape (see red spiral).

This geometry should make a good case study for investigation of the "proven" transcendence and irrationality of Pi. Certainly, with seven circles all squared and aligned in a logarithmic spiral, the infinity of Pi might be observed.

Rod ... ...

[Amigoo]

Re: "iSibling Rights" design

The geometry is similar but coloring was improved to highlight the replicating objects in the seven circles.

In particular, each circle hosts the unique scalene triangle with its 45 degree vertex, opposite the side with length equal to the side of a square inscribed in that circle. In fact, the foundational geometry can be viewed by completing and highlighting these squares that spiral around.

It's almost esoteric that seven squares (with their enclosing circles) precisely fill a 360 degree section of this spiral!
... and the seven squares have CSC relationship.

Somewhere in this 360 degree spiral of objects, the transcendence and irrationality of Pi must exist
... or they're arbitrarily superimposed upon the spiral's precise symmetry.

Rod

[Geoff]

Off subject here, but I noticed a screen flash when this thread opened, and I have seen it on others. I think someone has a faulty avatar or signature. Does anyone else see that flash?I am using Safari, I may try another browser and see if its the same.

hugs
Geoff

[Amigoo]

No avatar or signature in use on this page - just text and smilies.

But I sensed a flash or two off to the side of my vision several times this past week
... and one while not in front of the monitor. Perhaps, midwayers checking the circuits.

Rod