Paradise Trinity Day

[Amigoo]

Shaelan,

The simple explanation is that it doesn't relate directly, but the geometry research had been ongoing for several years when I started the PTD topic (just before 10/10/10). At that time I had posted (Oct 13, 2010): "Speaking of 'trinity', a cube is a geometric trinity."

Long story short: I began posting occasional, diary-like comments about this research and soon reached a point where it seemed best to keep accumulating to this topic. Besides, I had experienced (IMO) a lot of geometry-related midwayer prompts (explained in comments posted as the months went by) and I wanted history to note this influence in case the research ever had significant future value.

Rod

[shaelan]

Rod

That is amazing brother and I say don't stop because your threads got me thinking as well. I thought about all the connections to all of the things that need correcting and started making those corrections in me first.

Thank you
Shaelan

[Amigoo]

Re: Symmetry a la Scalene design
"Equidistant points of Pi"

Easy geometry! and good reference for those scalene triangles
that, when overlapped with juxtapositional precision, speak well
for this "out of the box" concept of squared circle reality.

Rod ... ...

[Amigoo]

About the design title, "Deus Ex Testimonium" ...

"Deus ex machina" is an old expression referring to stage plays, where "God" (or similar symbolism in physical form) suddenly appears on stage near the end of the play to resolve the complicated, loose ends of the plot.

So, "Deus ex testimonium" might mean that God is revealed through testimony (evidence) in the geometry. And this relates to the concept of a squared circle being the symbolic meeting of heaven and earth. The Deus Ex Testimonium geometry might be Part I of a "Cartesian play" with Symmetry a la Scalene Part II ... where "heaven and earth meet" (at least, are revealed as coexisting in local universe time and space).


About the midwayer geometry-related time prompts (began about 2010) ...

This table of 2-digit number associations was constructed after I had gained awareness of the patterns.
x:53 refers to a digital time display for any hour having 53 minutes: 5:53 pm, 7:53 am, 2:53 pm, etc.:

x:53 - 1.7724 53 8509.. square root of Pi
x:26 - 0.8862 26 9254.. half of square root of Pi
x:13 - 0.4431 13 4627.. (half of square root of Pi) / 2

x:13 - 1.4142 13 5623.. square root of 2
x:06 - 0.7071 06 7811.. half of square root of 2
x:53 - 0.3535 53 3905.. (half of square root of 2) / 2

These patterns (time prompts) confirmed (IMO) ongoing midwayer observation of the research and eventually convinced me that a marriage of the square root of Pi and the square root of 2 would need to be evidenced in any claimed geometry of a squared circle. The geometry of these two simpler designs, created for Pi Day 2016, provide such evidence ... perhaps.

Rod

[Amigoo]

Re: Symmetry a la Scalene design
"Equidistant points of Pi"

"Letters to Eduard, Unanswered" (a philosophical perspective) would make an intriguing choice of title for a real stage play, perhaps destined for a grand opening at the ACU Theater in Abilene, Texas (first choice of venue): http://www.acu.edu/legacy/academics/cas ... index.html

After a real Eduard inquired in 2009 "What's the point?", I started searching for a few good points and soon discovered that a circle's area square rests upon only 8 symmetric points* along that circle's circumference. Got a symmetric squared circle? Then you got many geometric objects of special influence, including those of Pythagorean persuasion! * The 8 symmetric points are very obvious in "Symmetry a la Scalene".

The geometry time prompts have been less numerous in recent months,
but the x:07 continuum is gaining frequency (x:07, :17, :27. :37. :47, :57)
... and "7" is a number of special influence in the Grand Universe,
especially as "associative possibilities of three Deities" (16:0.1)

Rod

[Amigoo]

Re: Symmetry a la Scalene design
"Equidistant points of Pi"

"ePidigm" - derived from late-night contemplation of "epic, Pi, paradigm";
alluding to any Cartesian squared circles integration of sqrt(Pi) and sqrt(2).

Symmetry a la Scalene drifted to symmetric complexity (symmetry from
the perspective of squared circles). At least, the geometry is a good
sampling of the salient objects in such Cartesian neighborhoods.

Rod

[Amigoo]

Re: Neighborhood sTrolley design

Squared circle geometry is so stable now that it's time to plan a large interactive, walkabout museum. Of course, an indoor/outdoor, all-season, walkabout museum should also provide transportation having memorable charm. After creating a draft pathways map for the walkabout and mini trolley line, I tested its appeal with the very young crowd at the local Read-A-Book club.

"Ho Hum", Zzzzz, and vacant stares were the expected majority response, but a future mass transportation engineer asked about the Bullet Train line. "That's the Cartesian Neighborhood's Trolley line." I tried to explain. Immediately, the young crowd awakened and clamored "We want to go on the Neighborhood sTrolley!"

Hmmm ... "Neighborhood sTrolley" does have good marketing appeal! And can be cross-advertised as the Neighborhod scalene Trolley for savvy geometers who want to cruise the neighborhood for other savvy geometers. Of course, I want to be the first to sit in front and toot the horn - might even be a good retirement job, depending on benefits. But in today's economy, tooting your horn is often a major benefit, usually enjoyed at happy hours.

About the geometry of sTrolley ... "Been there! Done That!"
Looks like a promising scalene spiral, trapezoidally (PSST)

Rod ... ... (off to check my tires after hearing PSST)

[Amigoo]

Re: Whence Line Three design

Squared circle geometry is so stable now

Contemplation of this quote, post-design completion, inspired curiosity about "stable geometry". After all, the spiraling scalene triangles (in CSC geometry*) had been incorporated into a few designs during the past year. So, these were not the geometric objects that caused a recent sense of "stable geometry". What is so special about this recent geometry research

I had "just settled down for a long winter's nap" (i.e., the oft-typical 5:00 AM bedtime), was meditating on the three lines of the scalene triangle (as effective as counting sheep), puzzled by the third line's lack of describable identity, and was determined not to get out of bed to explore any new ideas!

Well ... 5:00 AM somehow rationalized a "few more minutes" on the laptop, especially since the mouse was still awake (who wouldn't be, running in squared circles day after day for hours and hours?!). Besides, Daylight Savings Time had just awarded us an extra hour of morning darkness (finally getting to bed before morning light helps one fall asleep faster at those hours (re: pineal gland and melatonin)).

Long story short (and since morning darkness would soon be fading [Hmmm ... poetic expression!] ): I did climb out of bed to explore a "new idea". Whereas the first line of the scalene triangle (side of circle's inscribed square) and the second line (side of circle's area square) had describable identity, the third line (side) was just some irrational line length ... until this final morning darkness [Hmmm ... more poetic expression!].

Back to LSS: the third line is the diagonal of a circle squaring trapezoid!
Of course, this awareness mandated visual explanation, Whence Line Three,
an easy, late late night clone from Neighborhood sTrolley.

* CSC = Circle inscribed in Square inscribed in Circle ...

Rod

[Amigoo]

Re: Whence Line Three design

the third line is the diagonal of a circle squaring trapezoid!

I knew there was a reason why I should have that first cup of coffee before writing:
This line is the longest line (on perimeter) of a circle-squaring, isosceles trapezoid!
Well ... it's daylight now and I can find my way to the coffee pot.

Rod ... ... (PSST - the tires were OK, but ...
the battery keeps losing its charge - an electrical PSST)

[Amigoo]

The portfolio: http://aitnaru.org/images/Tripartite_Soul.pdf

Typical workout (p. 61, COSmic Entanglement design)
Current "last one" (p. 32, Whence Line Three design)
Up and up! (p. 52, Jacob's Ladder design)
Amen! (p. 40, iMetamorphosis design)

Rod ... ... (battery's charged - time to ride)

[Amigoo]

Re: Whence Line Three design

“‘Line three’? Whence lines 1 & 2?
OIC: It’s a scalene triangle! ...
whence increments Pi by sqrt(2)”

Sqrt(2) insisted that its obvious control of this Pi corral
be displayed geometrically since Pi, a transcendental number,
appears to increment by sqrt(2) in this CSC neighborhood.

Rod

[Amigoo]

Re: Transcendental Spiral design

Just mention a CSC neighborhood with scalene triangles
and an infinite spiral appears! A design now in competition
for "last one" since sqrt(Pi)/2 = four equal servings!

Must be advanced math since the geometry shows
division of the primary scalene into four similar triangles.
Maybe the CSC pie pan is the recipe's secret ...
when sqrt(2) Pi shortening is used.

Rod

[Amigoo]

Re: Transcendental Spiral design
"Scalene triangulation with CSC concentricity"

This geometry tempted exploration of an infinite scalene spiral,
but having "Been there! Done that!" several times, I quickly framed
the Cartesian canvas before the morning's darkness began to fade.
Another satisfying scoup of sqrt(2) Pi shortening.

Rod ... ...

[Amigoo]

Re: Transcendental Spiral design
"Scalene triangulation with CSC concentricity"
(blue "plus sign" identifies center of geometry)

In the CSC three circles set, largest diameter = 2, smallest = 1
and D = sqrt(2) for center circle.

Whatever the shape of the triangle that "squares the circle",
the triangle for D=2 is 4 times the size of the triangle for D=1.
(also note enclosing squares in the background)

The "Pi shortening" controversy ...

How does Pi remain transcendental with the precise difference
in the sizes of these two diameters? Or does sqrt(2) become
transcendental because of Pi's seductive aroma?

Methinks that, in a CSC recipe, sqrt(2) is Pi shortening.

Rod

[Amigoo]

Re: Transcendental Spiral design (updated)
"Scalene triangulation with CSC concentricity"

Looks like a promising scalene spiral, trapezoidally (psst)

Psst ... the finity of Pi might be revealed
in this 360-degree scalene points spiral.

Rod ... ...

[Amigoo]

Re: Transcendental Spiral design
"Scalene triangulation with CSC concentricity"
Psst ... (restated as a "press release")

Who knew?! The finity of Pi (where the ratio stops being transcendental)
might be delineated by an eight-circle, 360-degree, scalene points spiral.
(the identifying geometric pattern repeats every 360 degrees)

Rod

[Amigoo]

Re: Transcendental Spiral design (about the numbers)
"Scalene triangulation with CSC concentricity"
"Lines and triangles and squares! Oh Pi!"

"Visually enspiraling!" would say Archimedes (250 BC),
the first mathematician to calculate a value for Pi.

Concentric circles squared by decreasing increments of sqrt(Pi):
(some geometric objects not displayed for better design clarity)
Diameters = 2, 1.414.., 1, 0.707.., 0.5, 0.353.., 0.25, 0.176..
(circles 9, 10 not represented in design; D = 0.125, 0.088..)

01_ 1.7724538509055160272981674833411.. sqrt(Pi)
02_ 1.2533141373155002512078826424061.. sqrt(Pi)/sqrt(2)
03_ 0.88622692545275801364908374167057.. sqrt(Pi)/2
04_ 0.62665706865775012560394132120307.. sqrt(Pi)/sqrt(2) / 2
05_ 0.44311346272637900682454187083529.. sqrt(Pi)/4
06_ 0.31332853432887506280197066060154.. sqrt(Pi)/sqrt(2) / 4
07_ 0.22155673136318950341227093541764.. sqrt(Pi)/8
08_ 0.15666426716443753140098533030075.. sqrt(Pi)/sqrt(2) / 8

09_ 0.11077836568159475170613546770882.. sqrt(Pi)/16
10_ 0.07833213358221876570049266515037.. sqrt(Pi)/sqrt(2) / 16

This design online: http://aitnaru.org/lighttruth.html

Rod ... ... (off to crunch on non-numbers)

[Amigoo]

Re: Transcendental Spiral design (about the numbers)
"Lines and triangles and squares! Oh Pi!"

Re: http://www.aitnaru.org/images/Tripartite_Soul.pdf
For clarification of the numbers of Transcendental Spiral,
another page was added; 16 is the mysterious delimiter*.

Dimensions for Transcendental Spiral design
Diameters = 2, 1.414.., 1, 0.707.., 0.5, 0.353.., 0.25, 0.176..
(circle 9 begins next 8-circle pattern; D = 0.125)
(circle 10 not represented in design; D = 0.088..)

* 16 may be a clue for delimiting transcendental Pi.
Perhaps, the model for Pi is indeed transcendental,
but the actual in vivo Cartesian ratio is not.
Say what? "What?"

Rod ... ... ("Been there! and Psst)

[Amigoo]

Re: Transcendental Spiral design (about the numbers)
"Lines and triangles and squares! Oh Pi!"

Case closed! (see "Dimensions for Transcendental Spiral Design")
Obviously, 256 is the value related to a transcendental Pi delimiter.

... and this explains the tiny little square in the bottom right
of the Transcendental Spiral design (1 of 256 squares).

Rod

[Amigoo]

Re: Transcendental Spiral design (about the numbers)
"Lines and triangles and squares! Oh Pi!"

"Coffee Table" version of the concept:
http://aitnaru.org/images/Transcendental_Spiral.pdf
"sqrt(2)^16 = 256"

How the 256 squares relate to a Pi value is still unknown
as well as their relationship to 256 Pi decimal points.
So, crunch on this during your coffee breaks.

Rod

[Amigoo]

Re: Transcendental Spiral design
"Lines and triangles and squares! Oh Pi!"

While updating this design to allude to the inner/outer scale of the squares (re: 1:16; one set for inscribed squares and one for circles' area squares), my mind wandered outside the box (or further outside and around the corner), daring to whisper occasionally "I think I'm thinking!" (It It).

Hmmm ... "It It" (prefix to "Is Is"? It It Is Is! or suffix? Is Is It It?)
... but I digress ("I think I'm thinking!")

It It finally returned to conscious space and proffered a new word game (Entrepreneur Alert!)
This game concept derived from a scenario easily visualized:

On the wharf, a fishmonger and fishwife (potential buyer/competitor) were admiring a large fish with glistening scales that had been hoisted up
with two not-so-glistening scales (a very heavy fish). Said the fishmonger to the suspicious fishwife: "Fond of the scales?"

The game: The fishwife was permitted a one-word answer that was not "Yes" or "No", but still conveyed her Yes/No intent. In this word game, every word of intent can be used but once. The questions can be impromptu or supplied by the game. As each question is asked, one person asks the question and one person responds; all participants record the response, deciding privately if the response meant "Yes" or "No" (responder also records their intent).

When the round (one or multiple questions) is over, the scores are tallied according to the correct interpretations. A Fishmongering Period of Discussion (Fish POD) about the round is encouraged, with responsible-adult beverages quaffed at leisure (a good Biergarten game!).

Entrepreneurs! Start inventing! The anxious quaffers are quailing!
(not that outdoors hunting sport) "quailing" = quizzical wailing

Sample question: "Fond of the scales?"
(re: weighing equipment - not fish covering)

Possible answer: "Precise"
(one would not say this about fish covering)

Rod ... ... (off to practice quaffing & quailing for questioning)

[Amigoo]

Regarding the Fishmonger Qs game ...
Careful crafting of the questions makes the game more difficult!

Since focus is on responder's interpretation of question, ambiguity
is important [e.g., fish scales (equipment) and fish scales (fish covering)].
Responder's one-word response should help resolve the ambiguity.

Some elaboration (setup) on the question is permitted.
This concept does not require a purchased board game,
depending on creativity of the host and participants.

Definitely, a mind-exploration game!
"What do you have on ... your mind?"

Rod

[Amigoo]

Re: Transcendental Spiral design
About the "i/o scale 1:16" ...

This geometry effectively displays* two integrated, descending spirals upon a concentric 9-circle CSC foundation:
- one of similar scalene triangles representing decrements of sqrt(Pi) and, therefore, area squares of the CSC circles.
- another spiral of the inscribed squares of the 9 CSC circles.

* Inner circles and related objects are not displayed for better design clarity.

The scalene triangles (one side represents sqrt(Pi) and one side represents sqrt(2)) create the "merged spiral" for the two individual descending spirals ["descending" because this is how the geometry was first drawn, but could have been drawn as an ascending spiral]. This geometry shows the precise associations of sqrt(Pi) to sqrt(2) as the objects decrease in size by sqrt(2).

Whatever the "perfect" shape of this scalene triangle, it maintains that same shape through one 360-degree turn of the spiral ... at which point the sqrt(2) squares' pattern begins another 360-degree spiral. Because sqrt(Pi) is integrated (associated) with this squares' spiral, it begins another 360-degree spiral.

But ... what does it mean to say that a transcendental number "begins another spiral"? Can a transcendental number have identifiable segments? This squared circle geometry suggests that such Pi-related segments must exist. However, since no continuous repeating patterns of Pi decimals have yet been discovered, what does this CSC squared circle geometry insinuate (or prove)? "Lines and triangles and squares (and spirals)! Oh Pi!"

The "1:16" scale? The geometric objects resulting after one 360-degree turn of the descending spiral have 1/16 the size of the similar objects that were present when this turn of the spiral began. "i/o"? "inner/outer" (reference to objects in this Cartesian neighborhood).

Rod (need a spiral, cold & creamy, descending into waffle cone)

[Amigoo]

Re: Transcendental Spiral design
About the "i/o scale 1:16" ...

Regarding line lengths of inner/outer scalene triangles* ...
(D = 2 for outer circle and D = 2/16 = .125 for inner circle)

1.4142135623730950488016887242097.. sqrt(2) [shortest side]
0.088388347648318440550105545263106.. sqrt(2) / 16

1.7724538509055160272981674833411.. sqrt(Pi) [side of circle's square]
0.11077836568159475170613546770882.. sqrt(Pi) / 16

1.9084505148775338043018185280297.. [longest side]
0.11927815717984586276886365800185.. [longest side] / 16

* since parallel lines exist for lines of inner/outer triangles,
simple geometry can show 1/16 of the longer lines.

Rod

[Amigoo]

Re: Transcendental Spiral design
About the geometric "i/o scale 1:16" ...
(indirectly related to Fishmonger Qs)

While applying the final design update , contemplation of the
concentric CSC 8-circle set generated another out-of-the-box idea:
The spiral 8-circle set is a geometric "delimiter" for the irrational
sqrt(2) and sqrt(Pi) ... and perhaps the transcendental Pi.

Say what? In a CSC 8-circle Cartesian neighborhood, "irrational"
and "transcendental" are tolerated but not promoted.

Rod ... ...