Paradise Trinity Day

[Amigoo]

Re: Points of Order design
“What’s the point?” indeed!”

The red "X" is more complicated than first perceived
and was expanded to display its real potential!

The dark blue slanted "e" is symbolic representation of the "X"
and possible "heavenly Pi" symbol for a conjectured "new & improved"
mathematical constant. This "e" is not drawn like Euler's.

Rod

[Amigoo]

Re: Points of Order design
“What’s the point?” indeed!”

The dark blue slanted "e" is symbolic representation of the "X" and possible
"heavenly Pi" symbol for a conjectured new & improved mathematical constant

The new symbol was redrawn to incorporate a defining geometric signature:
This is the circle-squaring scalene triangle inscribed in a circle. The midpoint of
the vertical side (not displayed) determines the tip of the left arc.

The symbol's current nickname: "ePidigm" (epic Pi paradigm).

Answer to trick question about drawing a line to square the small blue circle:
The vertical red line already identifies the square of the circle (identifying
a point on the circle for another line gives equal credit).

Rod

[Amigoo]

Re: Points of Order design
“What’s the point?” indeed!”

Regarding ePi (probable short form of "ePidigm"), one more line was added
to highlight the salient geometric objects: a smaller "+" pivots at the center of
a larger "+" (without its left arm) with circle-squaring balance ... apparently.

Vertical line of smaller "+" = side of inscribed square of largest circle
Vertical line of larger "+" = diameter of largest circle.
ePi ++

Rod ... ...

[Amigoo]

Re: Points of Order design
“What’s the point?” an ePidigm?"

"++" became a mandate to upgrade the ePi symbol.
"Been there! Done that!" ... with humor in mind:

Student: "What kind of Pi does an ePidigm eat?"
Old Lao: "That's irrational and transcendental."

Rod

[Amigoo]

Re: Points of Order design
“What’s the point?” an ePidigm?"

a smaller "+" pivots at the center of a larger "+" (without its left arm)
with circle-squaring balance ... apparently

While the equal lengths of two lines appear to be the balancing
mechanism, those lines stay equal as the smaller "+" pivots" (convert
the "++" to squares to see how this works). Instead of equal length,
the perpendicularity of those two lines (only when ++ balanced)
is the tastier clue, geometrically speaking.

"I knew that squaring the circle was impossible,
but I did not know that it would be so difficult!"

Rod

[Amigoo]

Re: Points of Order design

the perpendicularity of those two lines
(only when ++ balanced) is the tastier clue

Crafty wording, that geometer, but those lines stay perpendicular
through the 45 degree range of motion being studied.

So, what else is tastier in this recipe? Maybe "singularity" would have been
a better word since "the singularity of those two lines" is esoteric enough
to have mysterious (or no) geometric meaning.

Although "ePidigm" may have to wait until another epic,
some Pi shoes keep shuffling along in squared circles.

Rod

[Amigoo]

Re: Points of Order design

So, what else is tastier in this recipe?

"That's the ticket!" This is still a "starter recipe",
but still missing the secret ingredient ... perhaps.

The green diagonal was added (points to center of red "X")
because it's a line of special interest.

Regarding "ePidigm", the old billboard pre-advertising
comes to mind: "Watch this space" (but not today)

Rod ... ...

[Amigoo]

Re: Points of Order design
Interestingly, "singularity" can be applied to this geometry!

Re: https://en.wiktionary.org/wiki/singularity
"1. the state of being singular, distinct, peculiar, uncommon or unusual"

Recalling that two overlapping inscribed squares exist in a squared circle
caused me to review the red "X" and how it associates the side of a square
inscribed in the circle and a diameter of that circle:

The side is a portion of one square and the diameter is the diagonal
of the overlapping square. When the circle is squared, this geometry
(overlapping squares, circle, etc.) seems to present a singularity!

At least, it is unique Cartesian display of an integration of lines and
objects that is "singular, distinct, peculiar, uncommon or unusual".

Rod

[Amigoo]

Re: Points of Order design

"singularity" can be applied to this geometry!

Consequently, I was persuaded to study the mysterious green line
that points to the center of the red "X", the geometry that associates
the side of an inscribed square with the diameter of the same circle,
including a vertical line with length equal to the circle's square!

Arc! Arc!

In the geometry, this line "floats" parallel to the top and bottom lines
of the green square and its left end follows the arc that is a portion
of a circle that would enclose this line as a square to the left ...
and thus caressing the center of a vesica pisces!

Rod

[Amigoo]

Re: Points of Order design

"singularity" can be applied to this geometry!

Such pressure, that "singularity"!
However ... "Been there! Done that!"

Now the design is an embellished and impressive
geometric singularity of three circles squared.

Rod

[Amigoo]

Re: Points of Order design
Such complexity, that "singularity"!

And such impressive geometry that I'm convinced that
it reflects "three circles squared". But I recall when falling down
the rabbit hole, I asked a local: "How do I get there from here?"
He explained mysteriously: "Well, you can't get there from here,
but you can go back there from here."

And that's when I was persuaded to reverse course and fall up
up the rabbit hole; such is the benefit of sanitas cyclometricus!
If one's research were morbus cyclometricus, one would now
be neither here nor there!

Say what?! Geometric singularities are impressive visuals,
but difficult (maybe impossible) to catch.

Say what? "Back to the drawing board!" ...
after sharpening pencils and getting a round toit.

Rod ... ...

[Amigoo]

Re: 2^3 design

"Back to the drawing board!"

A few hours of relaxing doodling at the drawing board was interrupted by
recollection of the 8 points on a circle upon which rests the circle's square.

2^3 is a very quick study of those 8 points, the original Points of Order.

Rod

[Amigoo]

Re: 2^3 design
"Eight points of order.
Three points of perfection."

While "back to the drawing board" (and not very inspired), I wondered:
"2^3" (=8) seems appropriate as a title, but where is "3" in the design?
Points for THE scalene triangles exist all around this geometry!

So, the large, symbolic scalene triangle was drawn - "symbolic" because
this is the triangle (its Cartesian placement) that I first noticed and
used often for these squared circle geometry studies.

And what closure that 3 and 8 were shown to relate
... geometrically and perfectly in circles all squared!

Geometer's tip: With the length of this base equal to Pi (3.14159..),
the sides of an isosceles right triangle (base equals hypotenuse),
have the same length as the diagonal blue lines.

Rod

[Amigoo]

Re: End of Journey (new design concept)

Rather than renewed excitement and energy
(geometry is very promising and unexpected),
this design seems to educe a bit of sadness.
What's this emotion all about?
An ending? Beginning? Both?

And why the recent inspiration to view this?:
https://www.youtube.com/watch?v=n1hQdgJOMpA

About the apparent geometry: Pi contains within itself
the ability to both define and square the circle
... and is demonstrable with geometry!

Rod

[Amigoo]

Re: End of Journey (new design concept)

Pi contains within itself the ability to both define and
square the circle ... and is demonstrable with geometry!

Well ... that was a mirage (it's no longer "obvious" several hous later,
but should be interesting exploration this week).

Yet, "End of Journey" might still be appropriate since
rummaging in a bottomless toy box has its risks.

Rod

[Sandy]

Never watched this before, Rod. ( https://www.youtube.com/watch?v=n1hQdgJOMpA ) but I find it strangely comforting, like a visit home. Strange, considering my roots lie on far distant shores... or do they? We are all one...and well, "Winnie the Pooh" was and still is my favourite children's book.
It will be interesting to see how the geometry progresses.
love,
Sandy

[Amigoo]

Re: End of Migration (design concept renamed from "End of Journey")

"End of Journey" might still be appropriate

Associating "journey" with "migration" better describes the evidence
that squared circles have definite, precise, and observable geometry.
However, "How to get there from here?" (via the Greek rules
for this challenge) is yet to be determined (then TBA).

It's intriguing - even fascinating - that a geometric "butterfly"
would correlate nested circles* all squared. This design includes
the foundational geometry of Points of Order but highlights the
migrating geometric Lepidoptera ... and scalene triangles.

* these four circles have geometric CSC.. relationship:
circle inscribed in square inscribed in circle ..

And speaking of metaphor, I'm amused that some migrating butterflies
may not return to their first home since this associates metaphorically
with falling in the rabbit hole: some journeys down the rabbit hole may
convert a roundtrip ticket to one-way (allusion to morbus cyclometricus).

Upon entering the rabbit hole, how does a squared-circle chaser confirm
the essence of the ticket purchased? The direction of travel doesn't change.
Not to worry: a CSC stamp proves roundtrip authorization and guidance.


TMI (unless you're fond of Lepidoptera), but alludes
to the magical complexity of squared circle geometry:

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

"Butterflies are part of the class of insects in the order Lepidoptera."
"Butterflies exhibit polymorphism, mimicry and aposematism"

polymorphism: http://en.wikipedia.org/wiki/Polymorphism_%28biology%29
mimicry: http://en.wikipedia.org/wiki/Mimicry
aposematism: http://en.wikipedia.org/wiki/Aposematism

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

"In some cases the individuals that migrate in one direction
may not return and the next generation may instead migrate
in the opposite direction."


Rod ... ...

[Amigoo]

Re: Open Box of Pi (new design concept)

With the geometry of squared circles obvious (IMO), exploration has become
redundant - even tedious (but not yet boring). How much Pi can one consume?!

I'm entertained that this design recalls the "open box" observation
from months ago and finally decided that redundancy is acceptable
when one is thinking outside the box ... that is always open!

Rod

[Amigoo]

Re: Open Box of Pi design
"Who let the Pi out?"

Thinking outside the box with scalene perspective.

Rod ... ...

[Amigoo]

Re: Open Box of Pi design (supersized)
"Who let the Pi out?"

Next year's bus trip to the Sanitas/Morbus* Cyclometricus border (on Pi Day 3.14)
will likely inaugurate a new tradition for Pi aficionados: Just prior to a feast of
supersized Pi with all of the toppings, guests will take turns (or not) yelling
across the chasm: "Who let the Pi out?"

This year's exploration established that the echo returns in 3.14.. seconds.
Unfortunately, first wave participants were unable to measure the final echo
and surmise that the sounds simply fade but will not end until the final
decimal digit of Pi has been calculated.

* On the difficult trip to the border, guests are lectured long and hard about
not abbreviating Sanitas/Morbus as "S&M", despite the social repercussions of
trying to square the circle when this has already been proven "impossible".

Rod

[Amigoo]

Re: Open Box of Pi design (supersized)
"Who let the Pi out?"

Landmarks of the inauguration:

Side of largest square (magenta)
= 3.1415926535897932384626433832795.. = Pi
Base of smallest scalene triangle (yellow)
= 1.5707963267948966192313216916398.. = Pi/2
Diameter of smallest squared circle (light blue)
= 1.4142135623730950488016887242097.. = sqrt(2)

Rod ... ...

[Amigoo]

Re: Open Box of Pi design (supersized)
"Who let the Pi out?"

Next year's Pi Day 3.14 seems far away,
but consider this tasty perspective ...

Who knew? sqrt(2)
was invited to join
inaugural too!

Landmarks of the inauguration:

Side of largest square (magenta)
= 3.1415926535897932384626433832795.. = Pi
Base of smallest scalene triangle (yellow)
= 1.5707963267948966192313216916398.. = Pi/2
Diameter of smallest squared circle (light blue)
= 1.4142135623730950488016887242097.. = sqrt(2)

While March 14 is traditional Pi Day,
Julian day 314 (November 10, 2015) is Leftover Pi Day*:
http://www.fs.fed.us/fire/partners/fepp ... lendar.pdf
(* reheated frozen Pi from 3.14)

Rod

[Amigoo]

Re: Open Box of Pi design (supersized)
"Who let the Pi out?"

Speaking of "S&M", there are some things more difficult than trying to square a circle: positioning a digital antenna in an attic surrounded by a radiant barrier (reflects the incoming signals). The good news is that there is an ideal position; the other good news is that twenty trips up the ladder to the attic (each with a 10' crawl on hands and padded knees and face mask on to filter out particles of disturbed fiberglass insulation) is better exercise than concentrated computer mouse manipulation.

But I digress ...

The inaugural, supersized Pi had room for one more topping (another geometric object: a similar light blue circle). With so many toppings already, I had resisted "just one more line" until its geometric value was obvious: the diameter of these two circles represent the radii of a circle that is the squared circle complement* to the square that would be created if the left diagonal (yellow) of the large scalene triangle were the side of such square.

Say what?! The quantity of toppings whose humor geometers would appreciate when they order supersized PI: "Would you like some Pi with those toppings?"

* diameter = 2(sqrt(2)); side of square = 2(sqrt(Pi/2))
TWIMC: http://aitnaru.org/images/Three_Point_One_Four.pdf

Rod

[Sandy]

Hi Rod,

Speaking of "S&M", there are some things more difficult than trying to square a circle: positioning a digital antenna in an attic surrounded by a radiant barrier (reflects the incoming signals). The good news is that there is an ideal position; the other good news is that twenty trips up the ladder to the attic (each with a 10' crawl on hands and padded knees and face mask on to filter out particles of disturbed fiberglass insulation) is better exercise than concentrated computer mouse manipulation.

I can laugh because I am not the one doing it...but if it makes you feel any better I empathize because I experienced something similar every year for nearly 20 when I crawled up into the attic and brought down the gazillion boxes of Christmas decorations positioned towards the back of the attic and then back up they'd go... a smart person probably should have repositioned them in a better, more accessible position but my "pack ratedness" didn't allow for the room at that time.

That is a fascinating design. I find I loose myself in its "movement" ..."veddy interesting" (Sargent Shultz voice)

love,
Sandy

[Amigoo]

"veddy interesting"

I was certain about the positioning of my digital antenna in the attic
when the channel with Sergeant Schultz appeared.

All this talk about Pi and toppings has me convinced:
make the box edible and just pile on the toppings!

It's 2:00 am and I'm heading to the kitchen for serious nibbles
... while the spaghetti sauce is warming!

Rod ... ...