Paradise Trinity Day

[Amigoo]

Re: Symmetry of Circles Squared design

More insight (aka, "wild conjecture") ...
Pi is transcendental unto itself

This "wild conjecture" also alludes to suspicion several years ago:
the geometric model for Pi (how Pi was first calculated) is itself
mathematically transcendental.

"Say what!?" Self-fulfilling prophecy.

Rod ... ...

[Amigoo]

Re: Ley Lines design

Now developing in the Cartesian neighborhood.
Simplicty for the magic of squared circle geometry
... and more sqrt(2) overcontrol.

Rod

[Amigoo]

Re: Ley Lines design
"Triangular Pi - Es la ley!"

Squared circle geometry with simplicity ... finally!
And a good model for creating a drafting cSquare.
(stretch diameter along long yellow line)

Note relationships of lines in circle-squaring isosceles right triangle:

Diagonal yellow line to upper red side of triangle
= 2(sqrt(1/Pi)) = 1.1283791670955125738961589031215..

Hypotenuse of triangle to diagonal yellow line
= sqrt(Pi)/sqrt(2) = 1.2533141373155002512078826424055..

(obviously) Hypotenuse of triangle to upper red side =
sqrt(2) = 1.4142135623730950488016887242097..

Rod

[Amigoo]

Re: Triangular Pi LL design
(LL = ley lines, but PiLL is "medicinal")

The geometry of Ley Lines finally revealed line relationships in the geometric "flutterby"
(looks like a butterfly). This flutterby geometry confirms the circle-squaring magic of
a unique scalene triangle, inscribed (or inscribable) in each integrated circle.

Need flutterby enlightenment? Consider the Triangular PiLL,
a scalene morsel only consumable "outside the box".

Rod

[Amigoo]

Re: Triangular Pi LL design
(LL = Ley Lines, but PiLL is "medicinal")

The great Mystery Of Pi (also "MOP")

An isosceles right triangle whose sides have length = sqrt(Pi),
has an hypotenuse whose length = sqrt(2) x sqrt(Pi).
sqrt(2)(sqrt(Pi)) / (sqrt(2)(sqrt(Pi)))/2 = 2.0

2.0 = length of dark blue diagonal that is the diameter
of the circle squared, creating three siblings (two are twins)
that are squared by the same non-isosceles right triangle.

Geometry sleight of hand? or local universe MOP?

Rod ... ...

[Amigoo]

Re: Triangular Pi LL design
"The great Mystery Of Pi (also 'MOP')"

About the isosceles right triangle whose sides
have length = sqrt(Pi), hypotenuse = sqrt(Pi)(sqrt(2)):

1.7724538509055160272981674833411.. sqrt(Pi)
x 1.4142135623730950488016887242097.. sqrt(2)
= 2.506628274631000502415765284811.. sqrt(Pi)(sqrt(2))

Relationship of triangle's sides/hypotenuse
to triangle's diagonal (squared circle's diameter):

1.7724538509055160272981674833411.. sqrt(Pi)
x 1.1283791670955125738961589031215.. sqrt(Pi)(Pi/2)
= 2.0
x 1.2533141373155002512078826424055.. (sqrt(Pi)(sqrt(2)))/2
= 2.506628274631000502415765284811.. sqrt(Pi)(sqrt(2))

Pi, as an isosceles right triangle,
defines the circle squared by this Pi ...
with the support of sqrt(2).

Rod

[Amigoo]

Re: Triangular Pi LL design
"The great Mystery Of Pi (also 'MOP')"

Hmmm ...
1.1283791670955125738961589031215.. = sqrt(Pi)/(Pi/2)
and not sqrt(Pi)(Pi/2)

Rod

[Amigoo]

Re: rTPi design (pronounced "Arty Pi")
aka, "Transcendentally Arty Pi Aligned Symbiotically (TAPAS)"

Geometry that contrasts the right triangle portion of the unique,
circle-squaring scalene triangle in each of two integrated circles,
right triangles in geometrically-similar sqrt(2) relationship.

Visual, squared-circle appetizers extraordinaire!

Rod ... ... (off for Happy Hour TAPAS)

[Amigoo]

Re: rTPi design (pronounced "Arty Pi")
aka, "Transcendentally Arty Pi Aligned Symbiotically (TAPAS)"

Geometers' tip: ("know therefore this Day ...", at this time)
"right triangle portion of the unique, circle-squaring scalene triangle"
does not refer to isosceles triangles ... and now another appears
amongst the red flutterby.

Rod ... ...

[Amigoo]

Re: Thin Red Line design
"Still holding firm"

... with the support of sqrt(2)

Rod

[Amigoo]

Re: Thin Red Line design
"Still holding firm"

Sqrt(2) secret of the Thin Red Line

Red lines of equal length in lower half of this design are part of an isosceles
right triangle whose hypotenuse is one side of the square of the largest circle
(three concentric circles whose diameters decrement by sqrt(2)).

With hypotenuse of isosceles triangle having length equal to sqrt(Pi)
[large circle's diameter = 2.0], side of square of the next smaller circle
has length equal to sqrt(Pi)/sqrt(2), equal to the lengths of the sides
of the isosceles right triangle ... and begging the question:

Whence transcendental Pi in this sqrt(2) quadrature

Rod ... ...

[Amigoo]

Re: Thin Red Line design
"Still holding firm"

A Yin-Yang design in the lower center was intended to suggest "geometric tension"
between beliefs of Pi as transcendental or not, but the symbolic '8' keeps shouting
that the 8 points on a circle, upon which its square rests, is the true message.

Indeed, all of the geometry displayed in these recent designs derives from
the "bottomless geometry toybox" that arises from these 8 points.

The right triangle in the upper right quadrant highlights relationships
of the red and green lines in this circle-squaring Pythagorean sibling.

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

Rod

[Amigoo]

Re: TRL Redux design (TRL = Thin Red Line)
"Another sqrt(2) squared-circle signature"

... and one more example of "bottomless geometry toybox"

Rod ... ... (signature wheely)

[Amigoo]

Re: Quadrature Quixotica design

A geometric path to Quadrature Quietus?

Rod

[Amigoo]

Re: Quadrature Quixotica design
"Onward to Quadrature Quietus"

A few more lines to emphasize trapezoidal integration: In these CSC circles,
the midpoint-to-midpoint line length is equal to the corner-to-corner diagonal
line length of the next smaller circle in the CSC integration.

Rod

[Amigoo]

Re: Quadrature Quixotica design
"Onward to Quadrature Quietus"

This time, a few lines were removed to better display the true trademark
of circles squared: the large scalene triangle is subdivided into 4 similar triangles,
with each enclosed by its own circle and the 3 outer circles each sharing a point
on the perimeter of the circle enclosing the large scalene.

Noticeably esoteric is the geometric highlighting (by the 3 circles) of each
of the three unique-length sides of the inner scalene triangle (1 of 4).

Rod ... ... (tunin' the chariot for the ride to Quietus,
a "nearby" destination in the local universe)

[Amigoo]

Re: Symmetry of Circles Squared design

The rest of the story (about the center's object) ...

Marketing advised me to refer to the geometric object in the center
as a "plus sign with longer vertical length for geometric direction"
(thus making this object a religious Rorschach Test).

The real story ...

The CAD software draws circles in straight line segments,
so merge points are not so precise under magnification.
The plus sign identifies the exact center of the design
with longer vertical length for Cartesian direction.

Rod

[Amigoo]

Re: QQQ design (Quadrature Quixotica Quietus)

With sqrt(2) (and its association with isosceles right triangles)
all over this Cartesian neighborhood, I was enlighted* to find
three similar right triangles (of circle-squaring persuasion);
geometric linkage of the three CSC circles (aka, QQQ).

* "delighted", my first choice, was grammatically correct,
but contextually incorrect (re: the word "de-lighted"),
... so I improvised.

Rod

[Amigoo]

Re: Quantophrenia kNot design
("kNot" alludes to both "knot" and "not")

Exploration of QQQ geometry (intense study of patterns)
revealed that the geometric patterns within QQQ extend
outside the current Cartesian boundaries of QQQ. ...

So the Neighborhood was expanded, not quantophrenially,
but by pattern replication throughout the Cartesian space.
And a certain scalene triangle is Quantophrenially kNot
featured throughout the kNew Neighborhood.

Rod (after reviewing unusual Q words, then
learning to spell/pronounce "quantophrenia")

[Amigoo]

Re: Quantophrenia kNot design
("kNot" alludes to both "knot" and "not")

Hmmm ... a '77' pattern is obvious (one '7' is reversed).
“Not only seven times but even to seventy times and seven"
comes to mind.

Rod ... ...

[Amigoo]

Re: Quantophrenia kNot design

a '77' pattern is obvious

Such esoteric geometry ... with a mysterious '77'!

Now, patterns reveal the next geometers' secret:
The long red diagonal on the left is the side
of a square inscribed in a circle.

May the convincing force of Quantophrenia kNot
be with your exploration of "impossible" circles squared.

Rod

[Amigoo]

Re: Cartesian Spring design
"Dawn of the new 'possible'!"

Discover the geometric fragrance!

Rod ... ...

[Amigoo]

Re: Cartesian Spring design
"Dawn of the new 'possible'!"

Such esoteric geometry, with so many possibilities!
(presence of GOD is in the mind of the beholder)

Rod

[Amigoo]

Re: CSiQ design
aka: Circle Squaring intelligence Quotient
aka: Circles Squared, indeed Quiescent
aka: Cartesian Spring in Quadrature

Geometers' tip: Design is subtle reference to midpoint-to-midpoint
length of the sides of the golden trapezoid, a sqrt(2) value in relation to
the isosceles right triangle within the circle-squaring scalene triangle.

"Say what?!" Quadrature 101 ... in this Cartesian neighborhood.

Rod ... ... (sans training wheels)

[Amigoo]

Re: CSiQ design

aka: Circle Squaring intelligence Quotient
aka: Circles Squared, indeed Quiescent
aka: Cartesian Spring in Quadrature
aka: Circles' Symmetry Eye Q

Yesterday ... Easter candy
Today ... Geometers' eye candy

"There's more!" Geometers' mind candy:
Each circle-squaring right triangle has two sides
(hypotenuse and long side) in 2(sqrt(1/Pi)) ratio
= 1.1283791670955125738961589031215..

Rod