Paradise Trinity Day

[Amigoo]

Re: Transcendental Transition design (replaced)

These squared circles (large circle's diameter is twice small circle's)
display the defining geometry of a squared circle via the inscribed
circle-squaring scalene triangle and the 'N' shape lines:

From left to right (green, red, green)
- diameter of circle
- square root of circle's square
- side of inscribed square

Note the parallel and half-length lines of the small object set.

Rod

[Amigoo]

Re: Transcendental Transition design ("new & improved")

An intriguing short dream this morning, just before awakening on my last day of formal vacation (my friend from Finland returned today to those 187,888 lakes (re: Wikipedia)).

In the dream, I had received a very brief email:
From: (unknown), Subject: Squalene Triangle, Text: Enough said.

Why "squalene" instead of "scalene", the triangle in focus for several months? All day, I tried to interpret this dream (especially, "Enough said.") as commentary on the geometry, such as "Salient geometry of squared circles has been discovered - there's nothing more to say about them (as design)."

Yet, "squalene" was such a distinct word in this dream! I plan to slowly wander back to the geometry exploration, but am somewhat unsettled by another possible interpretation of the dream's message (re: future response to ebola, et al.):

Re: http://articles.mercola.com/sites/artic ... posed.aspx

"Injecting organisms into your body to provoke immunity is contrary to nature, and vaccination carries enormous potential to do serious damage to your health."

"The purpose of an immune adjuvant added to a vaccine is to enhance (turbo charge) your immune response to the vaccination. Adjuvants cause your immune system to overreact to the introduction of the organism you're being vaccinated against. Adjuvants are supposed to get the job done faster (but certainly not more safely), which reduces the amount of vaccine required per dose, and the number of doses given per individual."

Who knew?! Squared circle geometry proffers social commentary! Or does it?
BTW: Another short dream segment, two mornings previous: a dark tornado
with a mushroom-shaped top was approaching quietly.

Rod

[Amigoo]

By the way ...
Dr. Helmut Sander has been aware of the 8 points of a squared circle
(the points upon which rest the square of the circle):
http://www.emis.de/journals/NNJ/Sander.html

Rod

[Amigoo]

Re: TT Squares en Paire (new design)
" 'What's the point?' Indeed! "

Reflection on the Transcendental Transition geometry, et al.

Rod ... ...

[Amigoo]

This triad of squared circle compositions (Transcendental Transition, TT Squares en Paire, TT Triangles en Paire) may include the necessary geometric objects for unlocking the secret of "squaring the circle".

In particular (re: TT Triangles), the placement of the 45 degree vertex of the scalene triangle in the 90 degree vertex of the isosceles right triangle hints that a certain juxtaposition of these two objects is sufficient to define a circle and its square. The hypotenuse of the right triangle has length equal to the circle's diameter and the side of the scalene opposite its 45 degree vertex has length equal to one side of a square inscribed in the circle.

But "What's the point?"

Rod

[Amigoo]

Re: TT Triangles en Paire design
Better color coding, aka Carpe Diem!

Rod ... ...

[Amigoo]

Re: http://aitnaru.org/psymmetry.html

Of course, the enticing aroma of Pi always wafts in these geometric compositions.

Rod

[Amigoo]

Re: Carpe Diem design ( http://aitnaru.org/psymmetry.html )

Continuing curiosity (What geometry exists in a squared circle?) inspired this additional complexity in Carpe Diem. At least, it confirms obvious circle-squaring patterns as replicated in a smaller, integrated circle having half the diameter of the larger circle.

Visually, TMI indeed! but useful for later research, including interpretation by advanced math.
"Who can tell?" Now, it's intriguing geometric composition - geometers' eye candy (or Pi).

Rod

[Amigoo]

Re: Carpe Diem design (new Pi color)

Apparently, some geometers prefer lemon Pi.
"La la how the life goes on"
... on planet 606.

Rod

[Amigoo]

Re: Carpe Lucem (new design)

A lite version of Carpe Diem.
(whence this unintended paronomasia?)

Rod

[Amigoo]

Re: Carpe Lucem design
Such a braggart, this squared circle geometry!

As soon as I doubted "Is this geometry really unique?" another circle appeared,
creating a CSCSC set and aligning the side of its inscribed square with the diameter
of the next largest circle (side and diameter have 1:2 length relationship).

Also, the diameter of this smallest circle has 1:2 length relationship
with the diameter of the largest circle ... and has length equal to
the side of a square inscribed in the middle circle of the CSCSC!

Now, that's an enticing lemon Pi!
Bake this one for November 10th
(next Pi Day; 314th day of year)

Rod ... ...

[Amigoo]

Hmmm ... two Pi days per year? Why not!

Bake round Pi for March 14th to celebrate the apparent infinity of the ratio;
square Pi for November 10th (or 9th) to celebrate squared circles.

Rod

[Amigoo]

Re: two Pi Days

These two design titles suggest celebration motif:
March 14th (Carpe Diem) - a daytime celebration with lighthearted merriment.
November 10th (Carpe Lucem) - an evening celebration, perhaps more formal.

... and tasty Pi competitions both days (or weeks)

Why "Lucem" instead of "Noctem"?
We illuminate evening celebrations.

Rod

[Amigoo]

More Pi all year ...
Another option, incorporating two Pi Days each year:

March 14th, daytime celebration with lighthearted merriment.
November 10th, evening celebration, perhaps more erudite.

An 8-month season of Pi, promoting the concept of geometric duality
and alluding to the 8 points upon which rests a circle's square.

... with tasty Pi competitions both days (or weeks).

Rod

[Amigoo]

Re: Carpe Lucem design (updated)
Who knew? There's a geometric center in this CSC composition!

Every squared circle has an identifying "N" shape inscribed within the circle; one line represents the side of a square inscribed in the circle and the other line represents the diameter of the circle. A point on the diagonal connecting the two lines identifies the circle's geometric center (relative to its placement in the CSC continuum).

Perhaps, the sublime essence of "transcendental" in squared circles.

Rod

[Amigoo]

Re: Carpe Lucem design

BTW: the diagonal of the "N" has length equal to a side of the circle's square.

Rod ... ...

[Amigoo]

Re: Carpe Lucem design

So many wiggly numbers!
The other diagonal (connects the other ends of the lines of the "N") ?

It represents the side of an isosceles right triangle
whose hypotenuse has length equal to an increment of the square root of Pi.
But with two diagonals, the "N" also has shape like a butterfly.

Hmmm ...
"transcendental" and "butterfly" (metamorphosis) seem related!
Perhaps, the essence of squared circles, at least in this universe.

Rod

[Amigoo]

Re: Carpe Lucem design
"whose hypotenuse has length equal to an increment of the square root of Pi"

Hmmm ...
Something about these wiggly numbers is implied by this statement.

Could it be that the smallest circle has a diameter of 2?
(making such a statement true for this CSC set)

Rod

[Amigoo]

Re: Pi à la mode(l) (new design)

Well, I'm still unable to find transcendence (at, least numeric) in a CSC geometric model of Pi.
And this hints that the geometric model of Pi (its derivation) is the source of its transcendence
- not the natural, geometric relationship of a circle's circumference to its diameter.

No problem! I've always been fond of Pi(e) à la mode.
And in this summer heat, à la mode is transcendent.

Rod

[Amigoo]

Re: Pi á la Mode(l) design (completed)

A refreshing model of Pi's transcendence.
(geometric exploration "down the rabbit hole")

The lengths of the horizontal lines of the
dark blue isosceles right triangle, red "arrowhead",
and squares of the inner and outer circles
are related in increments of 1,2,4.

Where's (what's) the 8th increment?
8 points of a circle upon which its square resides.

Rod

[Amigoo]

Re: Prominent Points of Pi (new design)

Looking down the rabbit hole (some might say "donut hole", "concentric scalene triangles", or even "symbolic portal to infinity"), geometrically-integrated circle-squaring patterns can be easily observed.

This CSC composition is both fascinating and intriguing - even esoteric - because the identified "points of Pi" (smallest vertices of the three inscribed, concentric scalene triangles) identify evenly spaced points along one side of the inscribed square of the largest CSC circle.

A good example of squared circle geometry that cautions:
Transcendence is only relative to your preferred model of Pi!

Rod ... ...

[Amigoo]

Re: Prominent Points of Pi design

Curiosity (again) causes more design complexity ... and completion delay.

Wondering why, only two vertices of consecutive (in their concentricty) scalene triangles were points on the line (side of square) but the third vertex only pointed to a point, led to another discovery: "arcs of Pi" (1/4 of a circle).

The underlying pattern is now obvious: a spiral mapped by the smaller vertices of the concentric scalene triangles. Fascinating geometry but difficult to represent with clarity in a multi-color design. Apparently, well worth a serious attempt during the days ahead.

Just yesterday, I was satisfied with Pi á la mode(l) and now have to carve precise arcs of Pi to complement the mode(l).
What is the sensory savor of this "á la mode(l)"? Premium Madagascar Bourbon Vanilla Beans?

Rod

[Amigoo]

Re: Prominent Points of Pi design (nearing completion)

Who knew? Surf's up! This wave of Pi is an infinite tube.
Hang 11 if you dare but the geometers' green room is quite public.

As for me, I'm still busy eating it (Pi á la mode(l))
but ordered the saucy seniors' shubee catalog just in case.

Rod

[Amigoo]

Re: Prominent Points of Pi design (completed)

In this local universe of Pi ...
Surf's up! Catch your wave.

Hmmm ... enticing numbers!
Three concentric scalene triangles start the momentum
and seven spiraling arcs outline the rolling surf.

Rod ... ...

[Amigoo]

Re: Pi2b (new design)

After a week's sabbatical, I asked myself if there was anything else in this CSC geometry.
Pi2b answers that question.
And considering this week's anniversary, a timely distraction.

So, I left the computer to head to the kitchen for another cup of coffee.
Ooops! Not so fast, geometry genius - a pit stop had signaled priority.
Backtracking, I noticed 12:34 PM on the bedroom clock.

Rod