Paradise Trinity Day

[Amigoo]

Now, 8 days later, I can report that the smoke alarm event was limited to one day - the alarm has remained silent since that day.

I surmised that the event was a demonstration of Midwayers' ability to arouse prompting attention in various ways. After a few alarm "tests", I had been quickly conditioned to look at a time display when I heard the alarm.

This "various ways" concept seemed to be confirmed yesterday when I had started a computer backup: I walked away from the computer, then returned twice while the backup was active and noticed the estimated backup time remaining: 1:33, then 1:22. Apparently, sequential prompts can occur in decreasing time displays!

Besides enjoying the prompting entertainment, I've learned that I'm a good ol' dog (re: Pavlov).

Rod

[Amigoo]

Recalling my post of November 17th ...

"So, I start wondering about this ':13' prompt. Is it related to the geometry research? Could be ... ':13' in the patterns previously identified associates with 'the square root of 2'. But I sensed that this was not the purpose of this morning's 11:13 am alarm."

... and considering this early morning discovery at 4:13 am ...

I've named this design "The God Particle of Geometry":
http://aitnaru.org/homepage/freewill.html

Rod

[Sandy]

That's a very good and fitting name, Rod! You truly make me wish I had looked a little closer for the "joys of discovery" while sitting in Geometry class eons ago. Your passion for the numbers and truly, what is behind the numbers shines through and enlightens all of us... regardless of our ability to completely undertand the complexities of what you are accomplishing... Well done!

I found "The God Particle" stunning.
Love,
Sandy

[Amigoo]

Sandy,

The design is still not perfect geometry, but appears to be excellent for the study of geometric relationships between the square root of Pi and the square root of 2, relationships which might some day reveal a solution to this geometry challenge. Like the search for the atomic "God Particle", this research is progressing slowly but may need to continue for a long time.

Perhaps, the search - the quest - has greater value than the discovery since it seems to cause increasing spiritual awareness and growth. If our spiritual ascent to Paradise requires commitment and progress for a very long time, such quests now (including the short-term failures and setbacks) give us a sense of the effort that will be required to reach the shores of Paradise.

Rod

[Amigoo]

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

This revision of the design features more revealing geometry.

The green "7" is highlighted for its symbolism of half of the square root of 2 (0.70710678...), an important line length (x 10) in this geometry, for its relationship to the 7:07 pm prompt which occured at the beginning of this geometry research and for its association with the whimsical name of this design: "The God Particle of Geometry".

Other symbolism related to the number 7:

http://www.angelfire.com/az/rainbowbridge/seven.html

"'It is DONE' is another expression found in connection with the number Seven."

http://www.granbychurchofchrist.org/Bib ... lation.htm

"The number 7 came to symbolize the meaning of totality or completeness associated with God's authority on the earth. It is believed by many that the number 7 is a product of adding the number 3 which represented the complete divine, to the number 4 which symbolized the whole earth." [the equation, 3 + 4 = 7, also relates to the Pythagorean Theorem for a right triangle (disguised in this design as the dark blue peak behind the green "7")].

Subtle symbolism in the design: a dark blue door appears to have opened to the area behind the magenta circle.

Rod

[Amigoo]

Regarding other symbolism related to the number 7 ...

I just noticed that there are 7 circles in this design: the center magenta circle acts as a unifying member of two different sets of circles (the three smaller ones in diagonal sequence and the three concentric ones, two of which are light blue).

"God Particle" was intended to allude to the geometry quest - not to the design itself. But this unifying aspect of the magenta circle could easily symbolize the uniting of the material and spiritual worlds in this new era of spiritual awakening.

Rod

[Amigoo]

For some, this math may be more interesting than the design ...

The green "7" is inscribed within a circle having a diameter of 10 units
(this is the hypotenuse of a right triangle).

A vertical line connecting the two points of the "7" would be the Adjacent Side,
by measurement = 8.862269254... (= half of the square root of Pi x 10).

The horizontal top of the "7" is the Opposite Side,
by measurement = 4.632513751... (also, C squared - A squared = B squared).

The circle's Area formula (A = Pi x R squared) calculates the same Adjacent Side length:
Side length = Square root of (Pi x R squared) = 8.862269254...

Therefore (conjecture):
A right triangle of these dimensions, inscribed within a circle having a diameter of 10 units,
effectively squares the circle.

This is not geometric proof, but does show the research value of using line increments of 10. Incidentally, a perpendicular line drawn from the bottom point of the "7" to the side of the largest circle has a length of half of the square root of 2 x 10 (7.071067811...). This is a good example of the design showing a relationship between half of the square root of 2 and half of the square root of Pi.

[Amigoo]

Re: http://aitnaru.org/homepage/freewill.html
The PDF file attached to this image was updated.

Diagram 10 shows the supporting geometry of this design, especially the "T" shape which seems to relate to the "Texas T" turnpike logo that was mentioned earlier in the year(and which I sensed to be a prompt when I drove past it several times).

Rod

[Amigoo]

Re: http://aitnaru.org/homepage/freewill.html
The beginning portion of this description of the geometry is included in Diagram 10 of the PDF file attached to the freewill web page:

A "Quadrature of the Circle", reflecting the cosmic inspiration of the Pythagorean Theorem on this "insoluble" challenge from antiquity. The ratio of the line lengths of the large blue circle's inscribed "T" equals the square root of 2. For this circle, having a diameter of 15 units, the length of the vertical portion of the "T" is 10 units; the length of the top portion is 14.14213562... units (the square root of 2 x 10).

The two smaller circles have diameters of 5 units each. This "resting T" geometry provides the vertical green line which helps identify the correct angle (longer magenta lines in "V" shape) for the intersection of the largest blue circle, effectively "squaring the circle" with the two intersection points.

The portion of the design on the right side that looks like a logo is an artistic representation of another technique for trisecting a line (a triangle within a triangle; the top horizontal line is the trisected line). This diagram sits on the adjacent side of a right triangle whose adjacent side length equals half of the square root of Pi x 10 (= 8.862269254...); the length of the hypotenuse is 10.

In this geometry, the upper half of the resting "T" has a length equal to half of the square root of 2 x 10 (= 7.071067811...). This line, the right triangle previously mentioned, the large circle and its square illustrate a unique mathematical relationship: the square root of 2 and the square root of Pi, a relationship that seems to be necessary in geometry in order to "square a circle".

Rod

[Amigoo]

Re: http://aitnaru.org/homepage/lifeischoice.html

The "line trisection" of my previous post reminded me that angle trisection was the previous geometry study (which ended in July); I had not reviewed that geometry again until today. It took me an hour to determine if this was a proposed solution to the Greek challenge or just geometry design (I had forgotten the beginning steps).

Diagram 11 was added to the freewill page's PDF file to keep the solution's proposal alive in these research notes ... and to illustrate another use for an equilateral triangle.

So, what's the first step? If you said, "Draw an angle", you're qualified to go to step #2 (Draw an equilateral triangle within the angle to be trisected). Diagram 11 shows a 50-degree angle - an angle greater than 75 degrees was not tested, but may be unnecessary with compensating geometry.

Rod

[Amigoo]

Testing this afternoon on other angles indicates that this angle trisection geometry is design - not a solution. I suspect that the triangle size was based on the angle size (suggesting "reverse engineering"). If I don't find a correction today, I'll remove (or replace) Diagram 11 of the PDF file.

Rod

[Amigoo]

Well, I did both: removed the design from the PDF file, then replaced it the next day with a geometry study (which provides a good plateau for next year's resumption of this angle trisection research).

"Three pennies on an arch" was the original concept that these coins could be aligned on a sheet of paper to cause the trisection of an angle drawn on the paper. I saved the pennies from that Summer 2008 impromptu test during my lunch hour at the office, but the concept never gave birth to a solution to this geometry challenge.

However, today this drawing clearly reveals that significant lines are present in the geometry and appear to validate the concept. As for a solution, align your pennies now ... before they disappear in this new era of electronic money.

The angle? 60 degrees, exactly 20 degrees per segment.

Rod

[Amigoo]

The premise of the pennies and the arch was not explained ...

If a penny (a circle of arbitrary diameter) were to be placed in the well of an angle (in the vertex), could the penny specify by various geometric lines how to trisect the angle? In the drawing, that specification is "engineered" (the end result is assumed - the pennies were placed on a specific arch exactly as required for the trisection).

Another geometric object was added today (large blue circle on the arch) that may facilitate this specification by the penny in the well. Other exploratory lines which support this perspective were removed, but consider this clue about the missing lines: imagine that the blue circle and its geometric objects originated above their current position and were lowered into position. Note: the blue circle has a diameter equal to the diameters of two pennies.

Drop a penny into the well and see if your blue solution bubble rises above the arch.

Midwayer prompting for several years strongly suggests that these geometry challenges are indeed solvable ... and that the symbolism of disproving "the impossible" will benefit humanity for many generations. The names of geometers (and mathematicians) who provide the proofs will be remembered long after my contributions have been forgotten.

Rod

[Amigoo]

Toss a penny into the well and discover how to trisect an angle with three concentric circles ...

Diagram 11 of the PDF file was replaced by this new concept. It's not a solution to the geometry challenge ... unless the size of the green triangle in the center can be determined by the penny in the vertex of this 50-degree angle.

Rod

[LurkerAbyss]

Hey there Rod

S'been awhile since I last posted in here, but I check up on this thread every time that I log into the board, just so you know.

Although I do not excel in mathematics and may not be able to completely understand the complexities of your accomplishments, as Sandy worded it, I still have a great appreciation for your work. Just as with everything around me, I find it very easy to make intellectual, philosophical, and spiritual connections with this geometry (and geometry in general). There have been quite a few times, and I really do mean this, where I have pulled up the different designs on your page and just sat and either meditated, pondered, or just simply admired. I have also had times where I've pulled up a simple paint program and started doing my own designing, although this has been more to entertain or explore general connections and ideas rather than any serious geometric/mathematical work. I am pleased that you have continued to maintain this thread as I am always inspired by your attitude and progress. I very much feel the way you do about my own personal "work" and what I am trying to contribute to this species, and couldn't agree more that "the symbolism of disproving "the impossible" will benefit humanity for many generations". And although it may not be geometers, I too only ask and hope that there will be names who provide the proofs for the ideas and theories I have long after my contributions are forgotten. I can fully relate to and admire your humble attitude; you and I and many others like us are living in a time where humanity has been in the infant stages of evolution for so long and will one day begin to utter its first word and take its first real baby step, and we do not seek to be famed and glorified forever in human epics or change the entire species in the course of one lifetime, but rather we are happy to be the caretakers who loved, nurtured, and fostered that infant humanity so that it could become a healthy toddler long after our time.

Love
Lucky

[Amigoo]

Lucky,

I am certain that I do not fully understand the "complexities" of the geometry since I stay focused on the designs (patterns) ... and have learned just enough geometry to create authentic designs.

Now that the winter season is upon us in the DFW area (I'm also certain of this since I've delayed using my furnace - 49 F this morning in the house), a "long winter's nap" from the geometry is overdue. This latest design (Diagram 11) offers a good creativity plateau for rest.

I emailed this Holiday Respite to friends this morning (the geometry design that was attached to that email is now appended to the PDF file online):

~~~~~~~

If trisected angle geometry were a "holiday tree", it might look like this Three Pennies image.

The angle is trisected if the length of the bottom line of the small green triangle
is equal to the diameter of the penny in the vertex of the large angle.
But the geometry must be drawn - not copied.

Given: the angle is 50 degrees

That's all you need to know (50-degree holiday trees are very popular)

Ho! Ho! Ho!

~~~~~~~

Rod

[Amigoo]

Another clue ...

The angle is trisected if two lines, each drawn from the center of the three concentric circles to the intersection point of two pennies at the top, form a 50-degree angle.

The description below the design in the PDF file was changed to "As above, so below" to reflect the above/below consistencies of portions of the geometry. However, if others choose to associate this expression with the creator of all and those things and beings created ... the geometry would not protest.

Rod

[Amigoo]

This personal symbolism is unique and worth sharing in the context of this geometry topic. My wife passed away in 2005 from a lifetime of diabetes complications. Having had two kidney transplants, her list of daily medications was extensive.

This morning, this mental picture occurred just before I awakened. I saw an equilateral-triangle-shaped arrangement of her many medication
pills. The pill on the lower right had this word imprinted: LOVE

(the triangle shape relates to my geometry study; I knew in the dream that those were her pills).

Rod

[Amigoo]

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

This design is my favorite for this web page (which is why I replaced the previous design yesterday). But when I focused on the heptagon embellishment in the center this morning, I began to understand that the embellishment may be the real inspiration for the replacement.

I had tried several years ago to find a solution to the geometry challenge of the heptagon, but decided that the final design ( http://aitnaru.org/homepage/lightandlife.html ) would have to be the creativity plateau. And then (this week), I was inspired to try again ... although I had already convinced myself that any further progress might take years. It didn't - I'll add the "work in progress" to the PDF file this week or next.

Interestingly, two circles having equal diameters and wrapped around a heptagon may be part of a solution. This could be the reason that I've been seeing x:26 and x:52 as prompts for months. I had not associated these prompts with each other and had wondered if x:26 related to a departed friend's birthday (January 26th). Also, x:52 occasionally appears in sequence: x:52, x:53, x:54.

[a prompt just now caused me to notice 10:53 am on the clock]

So, the two identical circles may explain the 26/52 mystery: x:26 relates to "half of the square root of Pi" (0.8862 26 9254..., from the established patterns of these geometry prompts) and Pi is easily associated with circles. Since 52 = 26 x 2, this might be a clue that two circles would be part of this geometry solution and that these two prompts were forecasting this resumed study of the heptagon.

Good entertainment for the holidays! Maybe "drooling" (over this geometry possibility) is the correct description for this emoticon which appeared amongst my typing debris as I corrected words and punctuation: ).-

More fun: the colon ending the previous sentence added eyes to the emoticon : ).-

[they like the stories: 11:11 am now on the clock,
and after spending time editing the emoticon explanation, 10:26 am appeared]

Rod

Even more fun ...

After posting the story, I headed to the kitchen to get a very late breakfast. I passed a clock (in a different room) which needs to be reset to the correct time: 12:52 am. I should have expected this prompt - the story was about both x:26 and x:52.

[Amigoo]

After an afternoon of x:26 and x:52 prompts, taking advantage of another alignment seemed permissible: Diagram 13 was added to the PDF file on this 13th day of December. The description below the image contains a "reflective" trinity:

“If 10 is reflective of the material universe
and 7 is reflective of the spiritual,
3 is reflective of their trinity union.”

Rod

[Amigoo]

A few comments about x:43 and x:53 ...

If these prompts are evidence, my current study of the regular heptagon (a geometric object with 7 equal sides) is progressing. I've been using inexpensive CAD software for drawing lines and for making relative measurements. More expensive CAD software would be required to "prove" geometric relationships, but this software has been more than satisfactory for sketching ideas.

Including 7 circles with equal diameters upon the sides of the heptagon led to an intriguing software "malfunction" yesterday (or operator error ... or ?). The software allows the user to specify a default color for the lines drawn and a color that is used while drawing the lines. For days, the default color has been black. But when I drew a set of 7 circles yesterday, 3 of them remained red after being drawn instead of converting to the default black.

I was puzzled by this "malfunction", but could not prove what had happened. While contemplating the mystery, I began to suspect that the 4/3 ratio of colored circles might relate to the x:43 prompt that I've been seeing. Indeed, after spending several hours today on the geometry and incorporating this 4/3 ratio in the exploration, I discovered that an isosceles trapezoid is the perfect geometric object for uniting the 7 circles in this heptagon while maintaining the 4/3 relationship.

For the past few years, these geometry-related prompts have never been direct clues for analysis of the geometry, but have remained very effective "nudges" ... for both exploratory motivation and direction. Interestingly, the discovery of the trapezoid occurred a day after "53" reappeared as a significant number from the past (a year ago):

This design ( http://aitnaru.org/theub.html ) was first posted online here ( http://www.createstainedglass.com/geome ... rcles.html ) and was titled "Squared Circle No. 53". This was actually design #3 of set #5, a way of organizing this series of drawings. At that time, "53" was not a recurring prompt for me. Using "53" in the title was subtle humor: the many hours spent on this exploration seemed sufficient for me to have created 52 other squared circle designs.

These potential comments seemed unimportant until both x:43 and x:53 appeared as prompts (on various clocks, some not set to display the correct time) as I debated whether to post them today. [just now, 3:39 pm; I often associate the x:39 prompt with Midwayers in general] If anything, the comments will help document a few more days of the unfolding adventure.

Rod

[Amigoo]

Another comment about x:43 and x:53 ...

Summing the numbers of these prompts gives x:07 and x:08, both being frequently observed prompts en mi casa. Whereas x:07 is directly related to the geometry adventure, x:08 has remained a personal prompt for several years (related to birth time and to house number).

[5:55 pm just now as I type]

This complex interassociation of prompts, having occured for months, leads me to believe that a greater event is unfolding "behind the scenes". I'm a willing particpant, but sense that I don't have exceptional comprehension of the event(s). And I'm well-aware that I don't have expertise in geometry, but have been learning just enough when appropriate to keep moving forward.

This continues to be great entertainment for someone who occasionally needs to justify hours and hours in front of a computer screen ... day after day, week after week, etc. If this is unintended material for a future book or movie, I'll just wait for the DVD (and probably order it online). Until that release, trying to interpret/explain what I observe is at least good writing experience. The risk for just posting imagination is minimal (Plan B: I'll claim that it was fiction ... if x:39 does not object).

Rod

[Amigoo]

On my way to the kitchen, I contemplated whether or not the "x:39" ending comment would be considered humor by the Midwayers. As I entered the kitchen, I passed a clock: 6:39 pm. Humor or not, the comment seems to have been noticed.

So, I rushed back to the computer to add this comment. And now as I type: 6:43 pm. That's either humor or a "Get back to work!" Fortunately, I don't have to respond - I'm on break ... and going back to the kitchen by a different route.

Rod

[Sandy]

Looks like you got your answer!
What are you working on in the kitchen, Rod? I am getting hungry just thinking about it!
Love,
Sandy

[Amigoo]

Sandy,

The last serving of Baked Pumpkin Pudding with a large dollop of whipped cream.
(I sent the recipe to your email address).

Rod