Paradise Trinity Day

[Amigoo]

Re: Source and Center design

The irregular, symmetric octagon (light blue) was drawn last
while attempting to find geometric justification for the large
squared circle that surrounds the central busyness.

Points and lines in this Cartesian universe do not appear to define
the outer squared circle, but the integrated octagon proves that
quadrature exists upon this inhabited planet.

Rod ... ...

[Amigoo]

Re: Trivial Pursuit design

After the geometric grandeur of Source and Center, this latest geometry
seems such trivial pursuit, but it does show relationship of the octagon
to the large squared circle:

Hypotenuse of golden isosceles right triangle x 2 = diameter of large circle;
side of isosceles triangle x sqrt(Pi) x sqrt(2) = side of large circle's square.

Rod

[Amigoo]

Re: SoQ design (pronounced "So Q!")

More "quadrature simplified", developed from study of Trivial Pursuit.

This new geometry could be called Trivial Pursuit II, but displays
intriguing circle-squaring similarities of the two circles,
one a sqrt(2) sibling of the other.

Rod ... ...

[Amigoo]

Re: SoQ design ("So Quadraturial!")

intriguing circle-squaring similarities of the two circles

... but now there are three similar circles!
(typical geometric replication in circles all squared)

Rod

[Amigoo]

Re: SoQ design ("So Quadraturial!")
Now with three circles all squared.

Perhaps, geometric proof that ...
either sqrt(2) is transcendental
... or Pi is not.

Current portfolio:
http://aitnaru.org/images/The_Right_Triangle.pdf

I knew that "squaring the circle is impossible",
but did not know that it's so difficult!

Rod ... ...

[Amigoo]

Re: SoQ design ("So Quadraturial!")
An opportunity moment for a new English word
(from "squaring the circle", aka "quadrature")

Quadraturial - Something so difficult to achieve
that it's believed to be impossible.

Sample usage (husband to wife, both recently retired):
Let's go on our second honeymoon and start a new family

Wife: Part I sounds tempting and Part II could be quadraturial!
Husband: Let's try for one child - not four!

Rod

[Amigoo]

Re: SoQ design ("So Quadraturial!")

BTW: "quadraturial" was used earlier in relation to the Three Pi Vise design:
http://aitnaru.org/images/Golden_rPi.pdf

Incidentally, "too quadraturial" seems redundant.

Rod

[Amigoo]

Re: P's & Q's design

Re: https://en.wikipedia.org/wiki/Mind_your_Ps_and_Qs

"Another proposed origin is from the English pubs and taverns of the 17th century. Bartenders would keep a watch on the alcohol consumption of the patrons; keeping an eye on the pints and quarts that were consumed. As a reminder to the patrons, the bartender would recommend they 'mind their Ps and Qs'."

So ... what's the reason for this topical association

"P's & Q's" is the name of the latest geometry design, featuring the Cartesian conjunction of Pythagoras and Quadraturial.
Obviously, in Cartesian Neighborhoods of circles all squared, geometers must mind their P's and Q's.

So ... where's Pythagoras in this Neighborhood a^2 + b^2 = c^2
(evidence that sqrt(2) is transcendental if Pi is transcendental)

Rod

[Amigoo]

Re: PP&Q design
(Pi, Pythagoras, & Quadraturial)

2(sqrt(1/Pi)), 2/sqrt(Pi), sqrt(Pi)/(Pi/2), etc.
is the "new & improved" Pi constant

2(sqrt(1/Pi)) = 1.1283791670955125738961589031215..

Rod

[Amigoo]

Re: PP&Q design
(Pi, Pythagoras, & Quadrature)

Such "wild & crazy" squared circle geometry
... that highlights 2(sqrt(1/Pi)), et al.

Best geometric focus: golden zigzag where each straight line
increments or decrements by 2(sqrt(1/Pi)) ... et al.

Have a taste of this cool Cartesian conical concoction
... geometrically speaking.

"Is your circle squared?"
Is your ice cream cool?

Rod ... ...

[Amigoo]

Re: PP&Q design
(a presentation of Pi, Pythagoras, & Quadrature)

"When thinking outside the box ...
seize upon the circuit of quadrature!"

And with more squared circles simplification,
the presentation has geometric je ne sais quoi.

Entertainment for the holidays (C'est si bon!)

Pop Quiz:
Given that the largest circle is squared,
prove that the next smaller circle is squared.

Extra Credit:
Prove that points exist to square the smallest circle.

Rod

[Amigoo]

Re: As Above, So Below design

Who knew
With so much attention on THE circle-squaring scalene triangle
(= 45, 27.597.., 62.403.. degrees), a different scalene exists
that incorporates the other in "As Above, So Below" geometry
... and highlights the long side to short side ratio of another
circle-squaring object, the right triangle:
1.9130583802711007947403078280203..

Rod

[Amigoo]

Re: As Above, So Below design

a different scalene exists

Who knows where that "different scalene" went?
Apparently, a stepping stone to the current geometry
now presented as AASB (long story short):

Salient ratios of the two circle-squaring triangles:
1.1283791670955125738961589031215..
1.9130583802711007947403078280203..

Rod ... ...

[Amigoo]

Re: As Above, So Below design

I kept trying to simplify this complex geometry,
but it wants to be esoteric; maybe prophetic.

For the Cartesian record, it's just ...
The ratios of two circle-squaring triangles:
1.1283791670955125738961589031215..
1.9130583802711007947403078280203..

Rod

[Amigoo]

Re: PPoP design (in development)
aka "Persistent Power of Parallelograms"
nicknamed "Double-P oP" (aka "Double Pop")

Who knew that so much squared circle redundancy
exists in these Cartesian Neighborhoods.

Double PoP may be declared the "bottom of the creativity box",
such is the geometric weariness promoted by this redundancy.

Rod

[Amigoo]

Re: PPoP design
aka "Persistent Power of Parallelograms"
nicknamed "Double-P oP" (aka "Double PoP")

With 3 circles, squared and well-integrated, "Double PoP" is an intriguing nickname
But this geometry seems to shout that the number '3' has local universe significance
... perhaps with similar significance throughout the Grand Universe.

Rod ... ...

[Amigoo]

Re: PPoP design (aka "Double PoP")

Geometers' secret about the "persistent power":
The 3 parallelograms, derived from the circle-squaring scalene triangle
in each circle, identify that circle's radius (therefore, its center).

How the radii are derived is not so obvious ...
as well as which set of parallelograms

And "Double PoP" + "like Father, like Son"
might require eternal contemplation.

Rod

[Amigoo]

Re: Here! There! design
(as in "You can't get There from Here)"

... but There exists! and looks like this:
Lotsa geometric objects with 3 circles squared
... and all squared circles linked by sqrt(2).

There's that number '3' again, perhaps with this message:
If Here is planet earth and There is Paradise (which is achievable),
then a "metamorphosis" needs to occur during our journey.

Rod ... ...

This design will require three alpha mornings to complete!
(best progress at alpha level upon morning awakening)

[Amigoo]

Re: Here! There! design
(as in "You can't get There from Here)"

The foundational geometry is certain: 1,2,3
(1 begats 2 begat 3 ... 5 circles squared)

"... there are just seven associative possibilities,
and only seven, inherent in three ..."

Rod

[Amigoo]

Re: Here! There! design
(as in "You can't get There from Here)"

Plan A was cancelled (aka "Start over!") because of a calculation error,
rendering the Cartesian collection inaccurate for conversion to design.

How this Plan B relates to Plan A is a mystery! Who can tell
This design might be intervention by some squared circles Gaia.
At least, the 1,2,3 concept was preserved!

Geometers' secret:
The set of 3 circles are adjoined on their inscribed squares,
sans display of the two exterior inscribed squares. See?

Rod

[Amigoo]

Re: Here! There! (and everywhere!) design

They say ... Once you've heard enough about squared circles
you'll start seeing faces in the "impossible" geometry.

Rod

[Amigoo]

Re: Here! There! (and everywhere!) design

Final hair (dark blue) confirms that six circles squared truly exist
and that the diameter of the largest circle is calculable! (derived
from two of the diameters of the smaller circles)

Design updated in:
http://aitnaru.org/images/The_Right_Triangle.pdf

Rod ... ...

[Amigoo]

Re: Here! There! (and everywhere!) design

Required disclaimer:

While "squaring the circle" (by the Greek rules) is not yet proven,
the geometry in these Cartesian Neighborhoods is proving that
sqrt(2) is the perfect unifier of integrated circles squared
... Here! There! and everywhere!

Required mantra:

Either sqrt(2) is transcendental or Pi is not.

Rod

[Amigoo]

Re: Here! There! (and everywhere!) design

The Art Department proffered this embellishment for geometric balance,
but I don't see how they knew with so much smoke in their studio!

At least, they kept the golden flutterby shape (leftmost "eye")
whose left and right sides feature sqrt(2)+1 line length difference.

Rod

[Amigoo]

Re: Here! There! (and everywhere!) design

This morning's alpha moment (aka "mental flutterby"):

If Here! There! displays geometry incorporating sqrt(2)+1 increment/decrement,
then the "non-repeating" sequence of Pi's decimal digits might indeed repeat
at the sqrt(2)+1 boundaries "Who can tell?"

Or maybe this sqrt(2)+1 value identifies the effective "limit" of Pi.

Rod