Paradise Trinity Day

[Amigoo]

Re: "Pi à la mode" design
(regarding "in style" once more)

Curious why four '7's should be geometrically significant,
I couldn't resist more exploration: the '7's are objects
within circles representing one-half of the spiral
(already observed by astute geometers).

Treat yourself to real pie à la mode if you're so gifted
... then explore the full potential of these patterns.

Rod

[Amigoo]

Re: "Pi à la mode" design
(regarding "in style" once more)

Speaking of "in style, once more" (this squared circle spiral was discussed last year) ...

"Pi à la mode"'s spiral (the four '7's occupy one-half of a 360-degree turn of the spiral) begs the question: What happens to transcendental Pi during the next 360-degree turn? Obviously (mathematically), a new transcendental Pi begins! So, how did the previous 360-degree turn of the spiral absorb all of Pi's transcendence? The "impossible" magic of quadrature

Or, perhaps, Pi is not really transcendental! ... or this centuries-popular constant
does not precisely describe the ratio of a circle's circumference to its diameter!

Methinks that mathematically-created Pi is truly transcendental, but that constant does not define the precise relationship of circumference to diameter. At least, these Cartesian studies (especially the spiral) geometrically shout that Pi is overdue for a reality check. Yes, proof exists that Pi is transcenderntal, but in these Cartesian Neighborhoods, that Pi is just a great recipe! ... aromatic for centuries!

Rod ... ... (cruisin' in style ... in laptop Neighborhoods)

[Amigoo]

Re: "Pi à la mode" design (regarding "in style" once more)
"Pi(s) à la mode (Psalm 7777), Suggested serving size: 1/16"

Notes:
Four '7's occur in one-half of a 360-degree turn of this squared circles spiral.
"Pi(s)" refers to geometric objects (squared circles) containing SoCS line lengths.
"1/16" refers to subdivision of Pi-related objects and their dimensions.
SoCS = Side of Circle's Square

What would it mean to claim that Pi is evenly divisible by 16

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

Rod

[Amigoo]

Re: "Pi à la mode" design

Suggested serving size: 1/16

Re: https://www.jpl.nasa.gov/edu/news/2016/ ... ally-need/

"For JPL's highest accuracy calculations, which are for interplanetary navigation, we use 3.141592653589793."

One more Pi digit (16 total) might be sufficient for intergalaxy navigation.

Rod

[Amigoo]

Re: "Pi à la mode" CE design
(CE = Classroom Edition)

This geometry speaks well for itself, but independent EGO
(Expert Geometer Groupthink) is required to decipher/interpret
what this squared circle geometry is actually communicating.

Two juxtaposed circle-squaring scalene triangles are identified,
unique triangles where the smaller has area 1/16 of the larger
and each of the three lines has length 1/4 of the larger
(geometers' secret: 4 x 4 = 16).

Rod

[Amigoo]

Re: "Pi à la mode" CE design (CE = Classroom Edition)
(revised since "EGO" was actually "EGG")

This geometry speaks well for itself, but independent EGO
(Expert Geometer Opinion) is required to decipher/interpret
what this squared circle geometry is actually communicating.

Two juxtaposed circle-squaring scalene triangles are identified,
unique triangles where the smaller has area 1/16 of the larger
and each of the three lines* has length 1/4 of the larger
(geometers' secret: 4 x 4 = 16).

* one line represents a side of that circle's inscribed square
and one line represents a side of that circle's area square
("impossible" marriage of sqrt(2) and sqrt(Pi) ).

Rod

[Amigoo]

Re: "Pi à la mode" CE design
(CE = Classroom Edition)

Every red '7' represents two lines of a circle-squaring right triangle whose three lines
all have sqrt(2) relationship to similar lines in the previous and following '7's.

Longest line of the '7' represents circle's diameter and long side of right triangle
represents a side of the circle's area square (SoCS, an increment of Pi).

Squared circle geometry that well-supports the ongoing mantra:
"Either sqrt(2) is transcendental or Pi is not!"

Rod ... ...

[Amigoo]

Re: Salient Points of Quadrature design,
Dr. SPoQ of "squared circle geometry that speaks for itself"?

Geometers' secret: there are 8 points in this central star
of squared circle geometry that highlights sqrt(2).

Is the circle squared Not without sqrt(2)

Rod

[Amigoo]

Re: Salient Points of Quadrature design

This squared circle geometry, obviously hosted by sqrt(2),
hints that the "three concentric circles" were meant to be
displayed with diameters in sqrt(2) relationship.

That would be a most salient point, complementing
wild conjecture that Pi may not be transcendental.

2-for-1 commentary on the popular "impossible"

Rod

[Amigoo]

Re: Salient Points of Quadrature design

squared circle geometry, obviously hosted by sqrt(2)

Geometers' secret: 8-pointed golden star "proves" that
a circle and its square have sqrt(2) relationship.

Rod ... ...

[Amigoo]

Re: Salient Points of Quadrature design

squared circle geometry, obviously hosted by sqrt(2)

Re: https://www.math.utah.edu/~pa/math/q1.html

"the Greeks discovered that the diagonal of a square whose sides are 1 unit long has a diagonal
whose length cannot be rational. By the Pythagorean Theorem, the length of the diagonal
equals the square root of 2. So the square root of 2 is irrational!"

However, when considering the Pythagorean Theorem,
half of a circle's inscribed square, and a diameter = 2:
2 = 2(sqrt(2)/sqrt(2)) = 2/1 (ratio of two integers),
proving that sqrt(2) is rational by equality of ratio

Rod

[Amigoo]

Re: 2:1 and 1:2 design and previous comment:

However, when considering the Pythagorean Theorem,
half of a circle's inscribed square, and a diameter = 2:
2 = 2(sqrt(2)/sqrt(2)) = 2/1 (ratio of two integers),
proving that sqrt(2) is rational by equality of ratio?

The non-quadraturial geometry of 2:1 and 1:2 shows that
sqrt(2) is "absorbed" by inscribed squares of the two circles.

With the red lines of the right triangle having sqrt(2) length,
the geometry proves that the inscribed square of the large circle
has twice the area of the inscribed square of the small circle
(precisely 2:1, the 2:1 and 1:2 reference)

Thus, when CSCSC concentric circles and their inscribed squares
are the foundation of quadraturial geometry, transcendental Pi
is well-contained within this "Pi Corral" ... IMO.

Rod

[Amigoo]

Re: 2:1 and 1:2 design

With these circle-squaring right triangles (CSRTs, two sides green),
the resulting Pythagorean isoceles right triangle (upper right)
confirms the precise sqrt(2) relationship of the green CSRTs
... and posits the "Pi Corral" of the sqrt(2) host.

Rod

[Amigoo]

Re: 2:1 and 1:2 design

If a circle's circumference = Pi when its diameter = 1
and the circumference = 2Pi when its diameter = 2,
then transcendental Pi can be doubled (finite value?).

Or is a transcendental number mathematical "shadow" to an integer
with the transcendental precisely tracking the integer operations

Squared circle CSCSC geometry posits existence of "shadow"
if Pi is truly transcendental. At least, we can still believe that
Pi / Pi = 1 (an integer ... unless this '1' is a shadow).

Rod

[Amigoo]

Re: 8 Moons of Q design

Quadrature so convincing that "speaks for itself" is understatement
Even the design name needs no explanation, but Q&A might follow
once "8 Moons" wax and wane o'er Cartesian Neighborhoods.

Rod

[Amigoo]

Re: 8 Moons of Q design

"Convincing quadrature, oft waxing and waning,
o'er Cartesian Neighborhoods X and Y plane-ing."

Geometers' secret:
"8 moons" refers to the '8' shape of some pairs of circles
and not to the number of "moons". Important relationships
are revealed in those geometric shapes.

Rod ... ...

[Amigoo]

Re: 8 Moons of Q design
(highlights circle-squaring scalene triangles)

"Convincing Quadrature, oft waxing and waning,
o'er Cartesian Neighborhoods X and Y plane-ing.
'What's the point (once inquired) of 1 in a million?'
Quadraturial scalenity, nigh golden in trillions!"

Rod

[Amigoo]

Re: 8 Moons of Q design

Who knew (for this Diameter only)
When D = 4.0, Circumference = Area

Given: Diameter = 4.0, radius = 2.0

Circumference = Pi x D
= 3.1415926535897932384626433832795.. x 4.0
= 12.566370614359172953850573533118..

Area = Pi(r^2)
= 3.1415926535897932384626433832795.. x 4.0
= 12.566370614359172953850573533118..

A squared circles "Pi Corral"?
aka "Pythagorean Perimeter"
aka "8 Points of Pi"

Rod

[Amigoo]

Re: 8 Points of Pi design

Highlights the 8 points of Pi's billions and trillions on a circumference
that define that circle's square (square rests upon 8 points only)
... and materialize convincing Pythagorean geometry!

A diameter of 4 proves that sqrt(2) is controller of the "Pi Corral"
by direct (and precise) association of circumference to square:

If Diameter = 4, Circumference = Area (C = A)
C = Pi x D = 12.566370614359172953850573533118..
A = Pi(r^2) = 12.566370614359172953850573533118..

Rod ... ...

[Amigoo]

Re: 8 Points of Pi design (w/ "ICU")
Updated in: http://aitnaru.org/images/Sqrt_Pi_Ratios.pdf

When you perceive that "New Pi"is looking at you,
that's not remarkable - even anticipated!

But when New Pi speaks to you, "That's the ticket!"
to comprehending squared circle geometry.

Rod

[Amigoo]

Re: 8 Points of Pi design (w/ "ICU")

But when New Pi speaks to you, "That's the ticket!"
to comprehending squared circle geometry.

Who knew?! In squared circle geometry where D = 4
(proves direct and precise relationship of circle to square),
sqrt(2) is geometric "anti-gravity" to transcendental Pi:

Either sqrt(2) is transcendental or Pi is not ...
in these Cartesian Neighborhoods of X and Y.

Until New Pi speaks to you, unobtanium!

Rod

[Amigoo]

Re: 8 Points of Pi design

But when New Pi speaks to you, "That's the ticket!"
to comprehending squared circle geometry.

Not to worry! New Pi has identity (check the Caller ID):

Ratio of diameter to side of circle's square
= 2(sqrt(1/Pi)) = 2/sqrt(Pi) = sqrt(Pi)/(Pi/2)
= 1.1283791670955125738961589031215..
= New Pi

Rod "Can you hear me now?"
Who's calling? Unobtanium?

[Amigoo]

Re: 8th Note of Pi design
(yet to be heard)

Who's playin' that song? Unobtanium

Perhaps, the 8th day of the circle represents "the power of the soul
to contact that light which totally transcends nature."

Rod ... ...

[Amigoo]

Re: On and On design

Climbing outside the box is easy, but ...
knowing the dimensions of your containment
is the required first step.

Just keep trying On and On, On and On
for a dog gets a bone oft with de-light;
you might read, you might dance,
you might sing, you might write

Rod

[Amigoo]

Re: 8th Note of Pi design (more flag)
In a Cartesian Neighborhood where numbers gossip
about circles squared, Pi is meat in a geometric sandwich:

1.4142135623730950488016887242097.. sqrt(2)
1.7724538509055160272981674833411.. sqrt(Pi)
2.0

Relationship of sqrt(Pi) to its enclosing values:

1.7724538509055160272981674833411..
/ 1.4142135623730950488016887242097..
= 1.2533141373155002512078826424055.. sqrt(Pi) x sqrt(2)

2.0 / 1.7724538509055160272981674833411..
= 1.1283791670955125738961589031215.. 2.0 / sqrt(Pi)

Pi's finite location between sqrt(2) and 2.0:

1.2533141373155002512078826424055.. sqrt(Pi) x sqrt(2)
x 1.1283791670955125738961589031215.. 2.0 / sqrt(Pi)
= 1.4142135623730950488016887242097.. sqrt(2)
x 1.4142135623730950488016887242097..
= 2.0

Apparent lesson: in a geometric sandwich,
it's OK for processed meat to stick out.

Rod ... ... (off to find veggie bologna)