Paradise Trinity Day

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Re: Paradise Trinity Day

Post by Amigoo » Fri Mar 11, 2011 6:44 am

Interesting possibilities (7.070707, 7.77, etc.), but I'm looking for naturally occurring line lengths within symmetrical geometry. I haven't seen 7.77 or 7.0707, but could draw such a line length. Where to place that line is a mystery - it could also be a puzzle piece that doesn't belong to this puzzle.

Just after discovering a 7.07107 line length in October, my x:07 prompts began (7:07 was first). But these prompts have not been that directive - x:07 could mean something very specific to the geometry or could be a simple project-related "Hello, we're observing what you're trying to do." And that does not necessarily imply that a solution is possible.

Reinforcing the idea that a solution is possible may be another prompt which began appearing several months ago: x:53. My only mental association for x:53 is this geometry which was created last summer ("No. 53" is part of the arbitrary name): http://www.createstainedglass.com/geome ... rcles.html The recurrence of x:53 as a prompt seems to suggest that something in this stained glass pattern is an important clue. And if that interpretation is correct, then a solution to the geometry may be possible.

Fortunately, I realized last October that these prompts would become confusing if I kept changing the diameter of the primary circle (it has remained 10 units). The "square" of such a circle is calculated as 8.8622692545275801364908374167057 units per side. Discovering geometry which naturally creates such a square is the goal, but manually creating such a square is not permissible. However, it is a useful technique for "reverse engineering" exploration.

And then there's the nagging problem of Pi, an infinite value. If Pi has to be part of the solution, a solution is impossible. But that is the key to my theory that only 8 points on a circle must be known in order to specify the square of the circle (each 2 points define one side of a square). Even if the circle has a trillion points, only 8 of those points must be known (and not all of these are required).

Rod

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Re: Paradise Trinity Day

Post by Amigoo » Fri Mar 11, 2011 9:37 am

Maybe the purpose of the x:53 prompt ...

After reviewing the "No. 53" geometry dozens of times, I finally noticed that 10 was not the diameter of that circle - it was 11.28379. And this allows for a square of 10 units per side, obviously part of last summer's "reverse engineering". In this geometry, 7.07107 is the length of a diagonal line drawn from the midpoint of one side of the square to the midpoint of an adjacent side.

If I'm interpreting x:53 correctly, this original diameter may provide geometry with more obvious clues. I could consider the past 5 months as wasted effort, but the accumulated observations may have been a necessary path.

Rod

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Re: Paradise Trinity Day

Post by LurkerAbyss » Fri Mar 11, 2011 10:24 am

Rod,

Although the past 5 months may appear to be wasted effort, I think there is always something to gain from the creative thought processes and activity that went into the effort anyhow! :D

It's like that joke about the chicken crossing the road.. no one ever talks about what's on the other side, the important part is just that he crossed the road anyway!

Or it's like that Frogger game where... okay it's not really like that because I sucked at that game and always died.

Love
Lucky
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Re: Paradise Trinity Day

Post by Amigoo » Fri Mar 11, 2011 6:04 pm

I once read a meditative version of the Chicken question:
"Which arrived first, the chicken or the road?

Regarding my thesis that not all of the 8 points on a circle are required to determine the circle's square ...
I restored the decorative and intriguing image on the site page which visually shows why this is possible:

If points on a circle can specify whereupon to place the square of the circle, then two inscribed squares may be placed precisely within the circle ... and thereby specify all 8 of the required points. And with two inscribed squares, only two points must be known: the points for one corner of each square (since the other 6 points are automatically specified).

My x:53 prompt (10:53) just appeared as if to comment on the "two points" perspective.

Rod

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Re: Paradise Trinity Day

Post by Amigoo » Fri Mar 11, 2011 7:33 pm

This morning's point (an observation) ...

Only one point must be known in order to square a circle:

A point away from the center of the circle (its distance equal to the length of one side of the square of the circle) allows an equilateral triangle to be placed on the circle. One side of that triangle aligns with one side of the square of the circle.

And this highlights an intriguing geometric progression ( 1,2,4,8 )
when considered in reverse order of a solution to the geometry:

One point (the point away from the center) ...
defines 2 points (one side of the square) ...
which define 4 points (sides of the square considered as 4 points away from the center of the circle) ...
which define 8 points (the two inscribed squares).

Rod

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Re: Paradise Trinity Day

Post by Amigoo » Mon Mar 21, 2011 6:07 pm

Progress continues to be slow, but the x:07 prompts occur even after I try to convince myself that the goal may not be achievable. And this continued prompting suggests otherwise - the goal is achievable!

Yesterday's observation seems to provide an important clue ...

For any square, the length of a diagonal line from the midpoint of one side to the midpoint of an adjacent side is in constant relationship to the length of a side. And that relationship (ratio) is .707... As I recall, the x:07 prompts began last October when I selected a square having a side length of 10 units. While I've been researching the significance of the 7.07107 diagonal length for months, the real significance may be the ratio!

Interestingly, this ratio seems to hint that a formula may be created to calculate the "square" of a circle without using Pi. Some day, such a formula might be discovered ... as well as the geometry which shows how to "square a circle".

Rod

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Re: Paradise Trinity Day

Post by Amigoo » Fri Apr 01, 2011 7:39 pm

One-day geometry entertainment for April 1st:
http://www.aitnaru.org/homepage/freewill.html

Is this serious geometry?
How can I tell? The cat's got my tongue!
Did Einstein stick out his?

A whiskered riddle about this design:
No toes but eyes nose
and 3 circles squared
- see one without toes.

Seriously ...
With limited research, it appears that the cat's eyes are adjacent and do not overlap only when the circle is squared. And this hints that a solution for "squaring the circle" might be hidden in this whimsically whiskered geometry. This cat's eyes clue relates directly to the position of the center of one side of both inscribed squares.

It's not an immediate quest - I'll be taking a symbolic cat nap from this long study of squared circle geometry.

Meow!

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Re: Paradise Trinity Day

Post by LurkerAbyss » Fri Apr 01, 2011 8:29 pm

Ha!

You have succeeded in summoning a smile from me oh dear Rod :) They say curiosity killed the cat, but apparently it has circled and squared it too!

Great to hear from you by the way :D

:sunflower:

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Re: Paradise Trinity Day

Post by Amigoo » Mon Apr 18, 2011 8:17 pm

Today's insight: http://www.aitnaru.org/homepage/freewill.html

... and a riddle for this design:
Give yourself a golden star if you recognize the proportions in this geometry.

There are many "7"s in this design, but they have a horizontal top and vertical side (notice the dark blue lines with 45-degree angles). This pattern of "7"s (the blue lines) can be extended in either direction ... infinitely, perhaps.

Rod

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Re: Paradise Trinity Day

Post by Amigoo » Mon Apr 18, 2011 8:33 pm

My comment should have read "90-degree angles" not "45-degree angles".

My 1040 tax form (prepared on the last day for filing) was stressful;
not as easy as "squaring the circle".

Rod

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Re: Paradise Trinity Day

Post by theunim » Mon Apr 18, 2011 10:07 pm

It's okay, Rod, at least it's done for the time being. Kudos!
Oh, my friend,
all that you see of me
is just a shell,
and the rest belongs to love.

— Rumi

The pure love of one soul can offset the hatred of millions. ~ Gandhi

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Re: Paradise Trinity Day

Post by Amigoo » Fri Apr 22, 2011 9:22 pm

Re: http://www.aitnaru.org/homepage/freewill.html

This design shows why the triangle is the better focus for a Golden Ratio spiral than the square or the rectangle. Imagine the dark blue triangles rotating on their 90-degree angles with their side lengths changing as they rotate:

To rotate 90 degrees counter-clockwise, multiply the lengths by 1.6180339887
To rotate 90 degrees clockwise, multiply the lengths by -1.6180339887

The dark blue lines in this online image show how these triangles change every 90 degrees. The triangles were used to construct the golden spiral, but the construction geometry was removed for better presentation of the image. The Cross of Lorraine was added to emphasize the geometry as well as to honor the current holiday.

See also: http://en.wikipedia.org/wiki/Golden_ratio

Rod

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Re: Paradise Trinity Day

Post by Shield » Sun Apr 24, 2011 2:25 pm

LurkerAbyss wrote:Rod,

Although the past 5 months may appear to be wasted effort, I think there is always something to gain from the creative thought processes and activity that went into the effort anyhow! :D

It's like that joke about the chicken crossing the road.. no one ever talks about what's on the other side, the important part is just that he crossed the road anyway!

Or it's like that Frogger game where... okay it's not really like that because I sucked at that game and always died.

Love
Lucky
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:D :D :D :D Frogger was not my game either. :lol:

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Re: Paradise Trinity Day

Post by Amigoo » Sun Apr 24, 2011 4:44 pm

Lucky,

Yes, the effort keeps proving its value and often includes the necessary lessons of patience ...

I'm convinced now that the "7", conveyed as symbolism by the prompts, is the triangle formed by a Golden Mean rectangle (picture a right triangle with its 90-degree angle in the upper right, the horizontal short side pointing to the left and the vertical long side pointing downward).

The ratio of the long side to the short side is the Golden Ratio ("an irrational mathematical constant, approximately 1.6180339887"). This triangle can be created with simple geometry for any square. Since a square can easily be drawn to enclose a circle or be inscribed within the circle, the Golden Ratio appears to have significance in the quest to "square the circle".

This web page is a good reference: http://en.wikipedia.org/wiki/Golden_ratio . Its diagram, "Construction of a golden rectangle:", shows this "7" triangle when you imagine a line drawn from the lower right corner of the rectangle to the upper left corner.

The design on this page ( http://www.aitnaru.org/homepage/freewill.html ) shows how those triangles (dark blue lines) are used to construct the Golden Spiral in the design. Once the triangle is placed on an X and Y axis, creating the adjoining triangles is simply a matter of drawing 90-degree lines until they reach an axis. "Squaring the circle" might prove to be this easy!

Rod

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Re: Paradise Trinity Day

Post by LurkerAbyss » Sun Apr 24, 2011 8:22 pm

Forgive me for losing sight of this topic, but where and how does the Golden Spiral come into play?

I mean, it's obvious that it has its calculated and geometrically purposeful place in there, I just don't remember really hearing anything about its relation to the rest of the geometry or even seeing it in previous designs for that matter.

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Lucky
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Re: Paradise Trinity Day

Post by Amigoo » Sun Apr 24, 2011 9:36 pm

Lucky,

In my search for geometric objects which would provide clues for "squaring the circle" (specifically those which relate to the "7" shape in the numerous x:07 prompts that I've been receiving since last October when a 7.07107 line length appeared in the exploratory geometry), the "7" shape of the triangle in the Golden Ratio (re: my previous posting) seems to be an important clue. And therefore the Golden Ratio itself seems promising for this geometry exploration!

I've forgotten the mental associations which recently guided me to an investigation of the Golden Ratio, but the Ratio's association with similar ancient Greek-inspired geometry is well-documented:

Re: http://en.wikipedia.org/wiki/Golden_ratio

"The golden ratio has fascinated Western intellectuals of diverse interests for at least 2,400 years. According to Mario Livio: Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece ... to present-day scientific figures ... have spent endless hours over this simple ratio and its properties. But the fascination with the Golden Ratio is not confined just to mathematicians. Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics.

Ancient Greek mathematicians first studied what we now call the golden ratio because of its frequent appearance in geometry."


The consistency of the x:07 prompts (and the occasional x:53 prompts which relate to an earlier geometry design for a stained glass window) convince me that the Midwayers have some interest in this project. I'm still not certain about the final goal, but the path has been educational and growth producing, especially regarding the "extra-curricular adventures" such as this Golden Ratio.

I do not qualify as even a minor mathematician, but I well-relate to the "spent endless hours" (for several years) on this and related geometry exploration.

Rod

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Re: Paradise Trinity Day

Post by LurkerAbyss » Sun Apr 24, 2011 10:59 pm

Thanks for the response Rod. It is clear to me that I have some extra reading and research to do when I get the chance!

As well, thank you as always for sharing your findings and information with us. Somewhere down the line, you will be one of many whose work has and will continue to assist my own. :)

Love
Lucky
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Re: Paradise Trinity Day

Post by Amigoo » Mon Apr 25, 2011 1:07 am

Lucky,

I forgot to mention how the Cross of Lorraine was added to the geometry. While searching for information about the Golden Spiral, I found this page: http://www.thevesselofgod.com/crosslorraine2.html . The two-bar Cross is important symbolism for that design; that spiral is a Fibonacci spiral which is closely related to the Golden Spiral (that's about the limit of my math expertise).

However, curiosity (or guidance) caused me to investigate if the Squared Circle, Golden Spiral and Cross of Lorraine could all be integrated in one geometric drawing. Consider the resulting design a visual draft of "proof of concept" since numerous repetitions of "reverse engineering" were required to finalize the design.

The design's greater value may be to inspire reconsideration of currently unanimous expert opinions that "squaring the circle" is an impossible mathematical task. I believe that belief and faith are "things" which persist even after the graduation of their once-living mortal repositories to the mansion worlds. And those "things" provide a good foundation for further elaboration and embellishment.

In the universe, we continually contribute to the previous work of others. If this were not true how would an inhabited world advance toward its golden era of light and life?

Rod

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Re: Paradise Trinity Day

Post by LurkerAbyss » Mon Apr 25, 2011 1:59 am

I couldn't agree with you more about the persistence of the "things" of faith and belief.

It is these, I believe, which help to establish and propagate the ideas and paradigms of generations, the new radical paradigms that a generation such as the one of today desperately needs and is surely on the edge of experiencing.. the advancement towards light and life :)

Love
Lucky
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Re: Paradise Trinity Day

Post by theunim » Mon Apr 25, 2011 7:22 am

Rod, I'd like to interject with my own thoughts here on your work. I've yet to really comprehend all of what you've spoken about in your past posts myself, but I do enjoy reading and seeing what you contribute on the thread.

I have had an interest in the Golden Ratio and the like of that part of mathematics, along with a minor love of Geometry going right there, too, since my eighth grade math teacher decided to dedicate an entire unit to teaching us about it after the subjects for standard testing were done and over with, so I did like reading what you posted about it. Might I also add that it has been researched by others in the past that Leonardo da Vinci's work has had the Golden Ratio applied to it quite perfectly? (Though, if it has been mentioned before in this thread, I apologize, my memory is not very nice at times.)

I would think, after all you've done so far, that your path is a very good example of how life should be lived in general: one of regarding what is also along the way to the goal, besides regarding the goal itself? :D
:hithere Theunim
Oh, my friend,
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and the rest belongs to love.

— Rumi

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Re: Paradise Trinity Day

Post by Amigoo » Fri Apr 29, 2011 6:52 pm

Re: http://www.aitnaru.org/homepage/freewill.html

This design gradually wandered away from a focus solely on the geometry. As the geometry developed, the inherent symbolism of the design appeared to change as well. Although "Patriarchal Cross" and "Cross of Lorraine" are common names for the two-bar cross, the design eventually "mandated" another name for this cross: the Cross of Belief and Faith.

By considering other symbolism and by renaming the cross, its geometry was no longer controlled by the historical proportion formula (length of bars to each other, to the post and to the golden spiral). Instead, this design now emphasizes geometric balance: the upper bar of the cross is centered on the long side of the dark blue triangle. Interestingly, that upward sloping diagonal line offers additional symbolism: our final ascent to Paradise.

Also, if the outer squared circle is considered as subsequent attempts to "perform the impossible", the alignment of that magenta circle with its green square, the light blue vertical post of the cross and the large dark blue triangle indicates improving geometric precision. And this easily symbolizes our increasing perfection as we ascend in the direction of Paradise.

In this final design, the new "cross of belief and faith", in addition to its subtle religious symbolism, becomes symbolic of the cross we bear when attempting to perform impossible feats ... including the discouraging influence of others who try to dissuade us from those attempts.

Rod

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Re: Paradise Trinity Day

Post by Amigoo » Mon May 02, 2011 10:33 pm

The final update to this design incorporates improved geometric balance.

Although this caused the upper bar of the cross to no longer be centered on the diagonal line of the largest triangle, the geometric symbolism increased: the upper left quadrant of this portion of the cross now reflects the golden ratio (the length of the vertical line to the horizonal line is 1.6180339887). And this hints that the two-bar cross itself has returned to the historical proportion formula. However, in this final design the cross is freed from its historical alignment relative to the golden spiral (unintended but intriguing symbolism, this "freedom").

There seems to be no need to change this design again. It is what it is ... and is already very symbolic of its journey. If Midwayer prompting was meant to be a guide to different geometry, then that design will develop some other day. The x:07 prompts have occurred recently but not as frequently, so I'll need to reflect later on their continued purpose.

Re: http://www.youtube.com/watch?v=e9MwNm0gXd8&
Yesterday, a friend sent me this link "Fibonacci numbers - The Fingerprint of God".
This video seems to explain why a cross complements this geometry design so well.

In the design, the dark blue lines aligned within the spiral are the diagonals of the Fibonacci rectangles which are shown in the video. Two adjacent sides of each rectangle are outlined by the light blue lines.

Today's prompting includes x:52, 53, 54, 55 (I mentioned in a previous post that x:53 referred to last year's stained glass design). I'm considering these prompts today to be motivation to get on with life on earth - another geometry design can wait for life's balance.

Rod

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Re: Paradise Trinity Day

Post by Amigoo » Fri Jun 03, 2011 5:27 pm

I didn't expect that "get on with life" might refer to "get on with the project" (and quit drooling over the golden spiral).

Re: http://www.aitnaru.org/homepage/freewill.html

The new image on this site page was flipped vertically to better illustrate the following math. Although the math should be considered theoretical, the numbers look impressive. What does this theory suggest? That the square of a circle can be calculated without using the esoteric Pi.

And the supporting trigonometry helps prove the project's beginning thesis: the square of a circle is defined by only 8 points on a circle (actually only 2 points relative to the center of the circle must be known). The 2 points in focus for this math are represented by the horizontal side of a right triangle (within the golden triangle in the design; a golden radius forms the hypotenuse side of the right triangle). The "angle of squaring radii" mentioned in the math is the lower vertex of the golden isosceles triangle.

How does this relate to the continual Midwayer prompting (x:07, x:27, x:53)? I don't know - this unexpected math adventure doesn't seem to be the goal of the project (and that may be why the prompts continue). Perhaps, this is a necessary step toward that goal. At least, once a circle and its square are united in trigonometric harmony, new possibilities for reaching the goal may appear.

The many numbers ...

Angle of squaring radii = 55.19422527138266
x .5 = 27.59711263569133 (angle below 90 degrees in right triangle)

New formula to calculate the square of a circle without using Pi:
Area = ((Cos 27.59711263569133) x Diameter) Squared

Example: Diameter = 1128379.16709552 units, square = 1 trillion units

Cos 27.59711263569133 = 0.8862269254527521812007657454843
x 1128379.16709552 = 999999.99999999999999999999999967
squared = 999999999999.99999999999999999933

using Pi = 1000000000000.0131624263503764498 (A = Pi x Radius squared)


So, which calculation of the circle's square is more accurate?
Since I like pattern repetition and symmetry, I favor the many 9s.

What would such a new math constant (the cosine) be called?
How about Radial Pi (rPi) to suggest its derivation and use.


PostScript:

This math adventure is already suggesting an intriguing title for a book:
The Poor Little Bulldog Who Ate Pi

Rod

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Re: Paradise Trinity Day

Post by Amigoo » Sat Jun 04, 2011 6:14 am

Just curious if greater precision could be obtained (still using Windows XP 5.1 calculator).
Many zeros seem to have better repetition and symmetry than many 9s:

Angle of squaring radii = 55.19422527138265163926944545242
x .5 = 27.59711263569132581963472272621 (angle below 90 degrees in right triangle)

New formula to calculate the square of a circle without using Pi:
Area = ((Cos 27.59711263569132581963472272621) x Diameter) Squared

Example: Diameter = 1128379.16709552 units (square = 1 trillion units)

Cos 27.59711263569132581963472272621 = 0.8862269254527521812007657454843
x 1128379.16709552 = 1000000.0000000000000000000
squared = 1000000000000.0000000000000000000

using Pi = 1000000000000.0131624263503764498 (A = Pi x Radius squared)

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Re: Paradise Trinity Day

Post by Amigoo » Sun Jun 05, 2011 3:08 pm

Here's the example of calculating the circle's circumference using the same Diameter (1128379.16709552)

New formula: C = ((Cos 27.59711263569132581963472272621) x length of square's side) x 4

0.8862269254527521812007657454843 x 1000000 x 4
= 3544907.7018110087248030629819372

= 3544907.7018110553843896069515809 (C = Pi x Diameter)

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