Paradise Trinity Day

[Amigoo]

Re: Long Story Short design
(aka, Arc de Triomphe ...
in this Cartesian neighborhood)

For D = 2, sqrt(Pi) / Pi/2 = 2(sqrt(1/Pi))
= 1.1283791670955125738961589031215..

Rod ... ...

[Amigoo]

Re: Long Story Short design

Oops! A short story now longer ...
but subdivision of the golden, circle-squaring scalene triangle is informative!
What does it say? "Triangles rule the local universe!"

Hmmm ... points of three.

Rod

[Amigoo]

Re: Pi rSquare design
(pronounced "Pi are square")
(aka "Pi right-angle Square")

Who knew?! Geometry exists to square a number!
Discovered while exploring properties of 2(sqrt(1/Pi))

"Geometry that speaks for itself"
(a convenient mantra for now)

Rod

[Amigoo]

Re: Pi~ra~Square design
(pronounced "Pi are a square" or "Pi rah Square")
(aka "Pi right-angle Square")

Note: Title is now more descriptive.

Geometers' tip: This design is a geometric tool for finding
both square of a number and square root of a number.

Say what ?! How does this prototype work ?! Dunno
It may play (in the mind) more than "work".

Rod ... ... (rah! rah! rah!)

[Amigoo]

Re: Pi~ra~Square design
(pronounced "Pi are a square" or "Pi rah Square")
(aka "Pi right-angle Square")

About that light blue tubular object in the center ...

A handle that lifts up for placing this wired geometry tool
on the working blueprints for the pyramid.

Rod

[Amigoo]

Re: Pi~ra~Square design
"Final, final" version ...

Current collection: http://aitnaru.org/images/Scalene_Salience.pdf

Marketing wanted to know if this device is male or female.
Say what?! Why would they ask such a question
I said "Yes" and tossed the ball back to their court.

Lengths of the golden diagonals in lower right quadrant
are the square roots of a number; lengths of vertical red line
(circles' diameters) are the number referenced ...
according to the hieroglyphic instructions.

... ... (rah! rah! rah!)

[Amigoo]

Re: Pi~ra~Square design
A geometers' tip from the hieroglyphics:

For the large D = 4.0 circle, two similar right triangles are formed
by the red lines and outer golden lines (30, 60, 90 degrees).

Elementary geometry expertise is required to construct
the Pi~ra~Square square-to-square-root "calculator";
using it only requires construction of a right angle ...
but intervening decimal values can be esoteric.

Rod

[Amigoo]

Re: Pi~ra~Square design

“'Calculator' to square a number or find square root.
A geometric juxtaposition of two right triangles:
2(sqrt(1/Pi)) and 30/60/90 degrees (D = 4.0)”

As soon as updating description to include "geometric juxtaposition",
I remembered George Jetson from the animated TV series (1960s).
"Hooba-dooba-dooba!"

How intriguing! I was focused on the ancient past (re: hieroglyphics)
when the distant future appears (re: Skypad Apartments in Orbit City).
Obviously, the geometry in Pi~ra~Square is "other-worldly".

Rod

[Amigoo]

Re: Pi~ra~Square design

"Juxtaposition" of the two right triangles is especially
intriguing because the smallest light blue circle seems
to communicate advanced geometric significance.

Further analysis hints that this significance relates
to transcendental Pi as constrained by this "Pi corral".

Geometry of the smallest light blue circle confirms
that 2(sqrt(1/Pi)) has precise value in this "calculator".

Regarding line lengths of right triangle inscribed in smallest circle ...
hypotenuse (diameter) = 1.2732395447351626861510701069801.. (2(sqrt(1/Pi)))^2
right triangle's long side = 1.1283791670955125738961589031215.. 2(sqrt(1/Pi))

Rod

[Amigoo]

Re: Pi~ra~Square design

Myth records that Pi~ra~Square was offered to embellish
the interior of the Roswell exploration disc. "No thanks!!"
was the unanimous response from their Design Dept.
"It looks like Greek to us!"

Rod

[Amigoo]

Re: Pi~ra~Square design
"Final, final"^3 (re: brain anatomy)

Design enhanced after reflection on the Pi~neal gland:
https://www.youtube.com/watch?v=4gKkY1qLFJo

Got Pi~vision yet?

Rod

[Amigoo]

Re: Simpli_city design

Of course, study of Pi~ra~Square was not going to lead
to geometric simplicity (the goal). At least, the replicating
quadrilateral identifies the CSC simplicity ... within this
familiar city neighborhood of circles all squared.

"Know therefore this day, and consider it in thine heart"

Rod ... ...

[Amigoo]

Re: Simpli_city design

replicating quadrilateral identifies the CSC simplicity

OMG Squared circle quadrilaterals contain a provable puissant property!
Straight lines connecting the midpoints of the Q' lines form a parallelogram.
Such power of the 'P' (words)! , especially considering ...

A straight line connecting opposite points of the parallelogram
divide this geometric object in half ... precisely. Since these Qs
are composed of two triangles, one right and one very right (isosceles)

... and since the one right contains a long line (not hypotenuse)
representing a side of the circle's square, transformation of
the Q to P then /2 effectively divides the Pi portion in half
... precisely Now this is transcendental 'P'.

Rod

[Amigoo]

Re: Pi~ra~Square design

I was determined to inscribe a parallelogram into the Pi~ra~Square circles, but ...
"This geometry wants to speak for itself":

With the two sets of adjoined right triangles, this neighborhood was groomed
for highlighting another circle-squaring object - the scalene triangle.

Long story short: Once the two scalenes were identified, it was obvious that
their geometric similarity is special: Each line length pair of the similar scalenes
has a positively puissant ratio: Pi (= 3.1415926535897932384626433832795..).

Rod ... ... (tunin' fer ridin' to Simpli_city)

[Amigoo]

Re: Simpli_city design

The first draft clearly indicates that Simpli_city is not "simplicity"!
When the parallelogram was inscribed in the Pi~ra~Square circle,
matching quadrilaterals and circles-all-squared arrived.

Obviously, this city's boundary is defined only by the quantity
of creative energy in my ride! So, I'll plan on an early exit with a
comment that I'm saving myself for old age , drawing just
this Cartesian neighborhood's "inner loop".

Rod ... ... (scoutin' the inner loop,
deep within the local universe)

[Amigoo]

Re: Simpli_city design
"Redefining 'geometric symmetry' for the new millennium"

The two adjoined right triangles are convincing evidence that squared circle geometry
(this quadrilateral in particular) is now redefining "geometric symmetry", at least
for Cartesian neighborhoods. Computers seem necessary to expand this symmetry
into 3-dimensional space where "other worldly" creations are possible!

Consider this quadrilateral (which includes the circle-squaring scalene triangle)
as a building block and the geometry toy box becomes bottomless!

Rod

[Amigoo]

Re: Simpli_city design
"Redefining 'geometric symmetry' for the new millennium"

Rod ... ... ("Been there! Done that!")

[Amigoo]

Re: Simpli_city design
"Redefining 'geometric symmetry' for the new millennium"

A new Uber driver (unemployed geometer) asked me to highlight scenic routes
within the inner loop of Simpli_city. "Always start your meter!" I cautioned.
"On these routes, you might be drivin' in squared circles for hours!
And then you'll have to think outside the box to get home."

Rod ... ... ("Been there! Done that!"
... and returned home!)

[Amigoo]

Re: Simpli_city design (further exploration )

The circle-squaring right triangle's 2(sqrt(1/Pi)) hypotenuse/long-side ratio
defines a set of two integrated spirals, both having a growth factor of 2
per quarter turn (and sqrt(2) per 1/8 turn) ... apparently.

Since hypotenuse is circle's diameter and long side a side of circle's square
(a Pi value), the identical growth factor reflects on transcendental Pi
... apparently.

This observation might become the next design,
perhaps named Squarely Entwined.

Rod

[Amigoo]

Re: Simpli_city design (further exploration )

both having a growth factor of 2 per quarter turn (and sqrt(2) per 1/8 turn)

A sqrt(2) increment would not be 1/8 turn, relative to the 1/4 turn.

Rod

[Amigoo]

Re: Squarely Entwined design

"Double spiral of 2(sqrt(1/Pi)), with both
having growth factor of 2 per quarter turn"

This Pi is transcendental

Rod

[Amigoo]

Re: Squarely Entwined design
"Double spiral of 2(sqrt(1/Pi)), with both
having growth factor of 2 per quarter turn"

Cartesian mapping of the two longest sides of a certain circle-squaring right triangle,
where longest line represents circle's diameter and other line side of circle's square
(a precisely integrated Pi value). "Say what?!"

Either Pi is not transcendental or Sqrt(2) is transcendental
... since they share the same spiral growth factor of 2.

Rod ... ... (Hooby-dooby!)

[Amigoo]

Re: Squarely Entwined design

Caution: To prove that Pi is not transcendental (a design is not a geometric proof)
might be an "emperor has no clothes" moment in the long history of math.

See also: https://answers.yahoo.com/question/inde ... 6050209041

Rod ... ... (too naïve to understand ... advanced math)

[Amigoo]

Re: Squarely Entwined design
(Cartesian mapping of the longer sides of the circle-squaring
right triangle's 2(sqrt(1/Pi)) hypotenuse/long-side ratio)

Today's update: http://aitnaru.org/images/Squarely_Entwined.pdf
"squared circle geometry 'outside the box'"

Rod

[Amigoo]

Re: Squarely Entwined design

Today's squared circles meditation (short & sweet) ...

It should be no surprise that sqrt(2) is associated with Pi
since sqrt(2) is the essence of a circle (diameter is hypotenuse
of an isosceles right triangle).

And if this right triangle is symbolically associated with this
1111angels topic (Paradise Trinity DaY), then the question is
answered: Which came first? The circle or the triangle?


Rod