Paradise Trinity Day

[Amigoo]

Re: Trapezoidal Concentricity design
"Squared Circle Geometry that speaks for itself."

"Y(es!)" (design completed)

Rod

[Amigoo]

Re: Trapezoidal Concentricity AI design
(AI = ad infinitum)

With squirrels runnin' all over the neighborhood this week
(not the Cartesian neighborhood), it was easy to rationalize
one more Trapezoidal version ... albeit circle-squaring
scalene and right triangles boldly appear.

About the influence of squirrels ...
"Aw nutz! - draw another squared circle!"

Rod ... ... (off to get vanilla and walnuts)

[Amigoo]

Re: Scalene Concentricity design

Another version of Trapezoidal Concentricity, but highlighting
the circle-squaring scalene triangles. The similar triangles of
D =2 (largest circle) and D = 1 (smallest circle) beg proof of
the oft-touted transcendental Pi.

Rod

[Amigoo]

Re: Scalene Concentrizity design
(Scalene Concentricity renamed ...
and with greater busyness)

The 'Z' in the center explains the rename.

Rod ... ... (off on other busyness ...
after completion of early morning Zzzzzzz)

[Amigoo]

Re: Concentrizity Squares design

Study of the isosceles right triangle whose hypotenuse has length
equal to a side of the square of the largest circle and whose siblings
(lines) become the sides of the squares of the two smaller circles
(triangle begets smaller triangle begets smaller triangle).

Say what?! "Impossible" geometry needs be a bit complex.

Rod

[Amigoo]

Re: Concentrizity Squares design

The math of the "triangle begets" series:
(SoCS = Side of Circle's Square)

Diameters: 2.0 / sqrt(2) / sqrt(2)
SoCS: sqrt(Pi) / sqrt(2) / sqrt(2)

Rod ... ...

[Amigoo]

Re: Concentrizity Squares design
“Impossible” geometry needs be a bit complex ...
when hosted by a certain isosceles right triangle.

This geometry tracks the squares of three nested circles (CSCSC),
and shows* that the square of the smallest circle (D = 1) has precisely
1/4 the area of the largest circle (D = 2), begging the question:
Whence transcendental Pi in this geometric precision?

* shows to serious squared circle geometers
.. who boast that "It's so eaZy!"

Note: The 'e' in 'eaZy' is not related to this 'e':
https://www.mathsisfun.com/numbers/e-eulers-number.html
... but the graph hints that it might be.

See also: https://en.wikipedia.org/wiki/E_(mathematical_constant)

Rod

[Amigoo]

Re: Concentrizity Squares design
“Impossible” geometry needs be a bit complex ...
when hosted by a certain isosceles right triangle.

Who knew?! A squared circles soirée! ...
with well-integrated trapezoids, parallelograms,
scalene and right isosceles triangles ... and more!

"Lines and triangles and squares, oh Pi!"

Rod

[Amigoo]

Re: Concentrizity Squares design
“Impossible” geometry needs be a bit complex ...
when hosted by a certain isosceles right triangle.

Current portfolio: http://aitnaru.org/images/Trapezoidal_Transcendence.pdf
Long story short: http://aitnaru.org/images/Concentrizity_Squares.pdf

Rod ... ...

[Amigoo]

Re: Concentrizity Squares design
“Impossible” geometry needs be a bit complex ...
when hosted by a certain isosceles right triangle.

Re: http://aitnaru.org/images/Concentrizity_Squares.pdf

About the horizontal white lines ...
The longest has length = sqrt(Pi)
Shorter two have length = sqrt(Pi)/2

Significance: Help correlate squares of circles
via the "certain isosceles right triangle"
(and parallelograms and trapezoids).

CSCSC geometry is required to prove squared circles
(via the correlating sqrt(2) geometric linkage)
... apparently.

Rod

[Amigoo]

Re: Concentrizity ICU design
“Impossible” balance of sqrt(Pi) / sqrt(2)
Squared circle geometry, quintessentially esoteric.

Rod

[Amigoo]

Re: Concentrizity ICU design
“Impossible” balance of sqrt(Pi) / sqrt(2)

Color adjusted to include a geometers' clue:
Length of two red vertical lines = sqrt(Pi).
Imagine that the leftmost line slides diagonally
until that certain geometric balance occurs.

Rod ... ... (off to slide on a ride)

[Amigoo]

Re: Concentrizity Too design

A study of the juxtaposition of two irregular pentagons,
similar (and consistent) circle-squaring objects ...
and inclusive of similar trapezoids, et al.

Rod

[Amigoo]

Re: Precisely Scalene design
"A geometric 'P.S.'" (about the 3 points required to square a circle)

Golden line has length equal to the circle's radius.
Does "P.S." suggest end of story.

Rod ... ...

[Amigoo]

Re: Precisely Scalene design
"A geometric 'P.S.'" (about the 3 points required to square a circle)

This geometry also makes a statement about transcendental Pi:

In this juxtaposition of two circles (D = 2, D = 1),
the right triangle that squares the small circle has 1/4 the area of the right triangle that squares the large circle;
the isosceles triangle that squares the small circle has 1/4 the area of the isosceles triangle that squares the large circle.

Thus, we may inquire ... whence transcendental Pi within these geometric "bookends"?

Rod

[Amigoo]

Re: Precisely Scalene design
"A geometric 'P.S.'"

Speaking of points (about proving squared circles) ...

Two diagonal parallel lines were added to show trapezoidal correlation:
the trapezoid that squares the small circle is similar to the trapezoid
that squares the large circle.

Trapezoids plus right and scalene triangles correlate two circles squared,
where their diameters equal 1.0 and 2.0, respectively.

A paradisiacal trinity of geometric circle-squaring objects.

Rod

[Amigoo]

Interestingly, the following UB quote does not comment on "squaring the circle" - just states a fact
(interpretation): By definition, a geometric object cannot be both circle and square at the same time
(but could contain the same area as either object according to this geometry research).

"Within the time-space frame and from the intellectual reference point of mortal comprehension,
even the infinite God cannot create square circles ..." (118:5.1)
See also: http://www.theub.org/part-iii.html#P118_5_1

Wisdom: "A ride can become a walk when it's not a ride, but a walk cannot become a ride
without roadside assistance ... and no one can walk to Paradise."

Rod ... ...

[Amigoo]

Re: Precisely Scalene design
"Geometric 'P.S.' with Euler enlightenment.
Paradisiacal trinity of circle-squaring objects."

See also: http://aitnaru.org/images/Concentrizity_Squares.pdf
"Impossible" geometry needs be ESOTERIC ... without apology.

Geometers' clue:
Vertical and long diagonal red lines have sqrt(2) relationship.
Vertical line (within D = 2) has length = sqrt(Pi).

Rod

[Amigoo]

Re: Precisely Scalene design
"Geometric 'P.S.' with Euler enlightenment;
paradisiacal trinity of circle-squaring objects."

"Final"^2 “... not even a mouse;”

Rod

[Amigoo]

Re: Lines Of Linkage design (LOL)

Who knew?!
Scalene balance does exist in a CSC relationship!
(golden lines have the same length)

Rod

[Amigoo]

Re: Lines Of Linkage design (LOL)
"Final, final" update.

The geometry of LOL confirms that, when the circle is squared,
sqrt(2) will be there, correlating the circle-squaring objects.

Example: The golden lines of equal length connect two scalene triangles
(one inscribed in each of the two CSC circles) and the respective sides
of these similar triangles have sqrt(2) relationship.

Rod ... ... (cruisin', LOL)

[Amigoo]

Re: Spiraling Scalenity design
Perhaps, revealing another "golden ratio" ... of circle-squaring essence.

Re: http://www.liquisearch.com/golden-ratio ... _triangles

“If triangles had a god, they would give him three sides.”
—Charles Louis de Secondat Montesquieu (1689–1755)

If circle-squaring scalene triangles had a god,
it would be the three gods of the Paradise Trinity.

Rod

[Amigoo]

Re: Spiraling Scalenity design

Who knew?! There's a greater "eye of the pyramid"
(design enlightened with more circles, decreasing by sqrt(2)

Rod

[Amigoo]

Re: Ro of Fourths design
"Squared-circle patterns extraordinaire!"

"Ro" spelling is intentional (is both "row" and one-time email typo for "Rod")
and fortuitously, "Fourths" sounds like "force". So, "Row with the force"
... or "Ro, Ro, Ro your boat ..."

"Fourths" refers to the 1/4 decreasing area size of each green square,
relative to the next larger square.

Rod

[Amigoo]

Re: "Ro of Fourths" design in http://aitnaru.org/images/Concentrizity_Squares.pdf
"Squared-circle patterns extraordinaire"

Intriguing geometry that highlights 1/4 math permutation
since this infinity suggests evidence of a "Pi corral".

Consider this precise association:

... .25(.25(.25(circle's diameter
... .25(.25(.25(circle's square

Rod