Paradise Trinity Day

[Amigoo]

Re: Scalene of Pi design
aka “Pi are square(d)”

New question: From whence the square of Pi?

The large blue square was drawn last and this is the square of the large golden circle whose diameter has length equal to Pi. The blue circle was drawn next to last according to the geometry of the red right triangle.

Apparently, this geometric composition did not include any object directly related to the square root of 2 until the red right triangle appeared. As of the moment, the only thing I know for certain is that “Pi are square(d)”.

Rod ... ...

[Amigoo]

Re: Scalene of Pi design
aka “Pi are square(d)”

Geometer's secret (aka "Note to Self"):
The first line drawn in this geometric composition is the golden diagonal at 45 degrees (length equals half of Pi).
This diagonal line and the square root of Pi are the only values needed to begin constructing the composition.

But what happens next?

Vesica Piscis dances on this Cartesian plane, followed by a circular presentation of the long-honored Pi.
Square and inscribed is the essence of the next performer who then introduces the diagonal twins
of common chord, and these two then introduce a most impossible guest of scalene reputation!

Say what? A piece of cake (or Pi)!

Rod

[Amigoo]

Re: Scalene of Pi design
aka “Pi are square(d)”

Geometer's secret (aka "Note to Self" and "the rest of the story"):

Vesica Piscis dances on this Cartesian plane, followed by a circular presentation of the long-honored Pi.
Square and inscribed is the essence of the next performer who then introduces the diagonal twins
of common chord, and these two then introduce a most impossible guest of scalene reputation!

Soon, an isosceles right triangle and two squares of different stature claim inherent filiation
with the impossible scalene, but the plane fails to ascend - the geometric association of the
square roots of Pi and 2 is still a tenuous metaphor of the uniting of heaven and earth.

Suddenly, a Pythagorean moment of enlightenment descends upon the plane - all objects gasp
when a right triangle appears above and proclaims its own filiation ... and more! Now, this
Cartesian neighborhood is treated to a more believable vision of the impossible.

An enclosing circle captures this Pythagorean moment, confirming the vision with a square of Pi
(sometimes called "serving size"). Naturally, the objects continued to debate the merits of cake
and Pi vis-a-vis the popular expression "have your cake and eat it too".

Unexpectedly, "have your Pi and eat it too" became the new rage when "too" was associated
with a certain square root. However, math celebrations in Gregorian March or Julian November
will always be blessed with both cake and Pi since aficionados prefer just desserts.

The red circle in the center has CSC filiation with the right object above.

Rod

[Amigoo]

Re: Scalene of Pi design
aka “Pi are square(d)”

When reviewing the geometry once more, a cold wind blew across this Cartesian plane:
It's a charming fairy tale, but the geometry proffered is not completely accurate;
the end of the tale still wiggles, numerically speaking.

Specifically, "confirming the vision with a square of Pi"
is overstatement; a sample - not a serving.

The design needs revision and will be replaced
... when (if) the wiggles are ever tamed.

Rod

[Amigoo]

Re: Scalene of Pi design
aka “Pi are square(d)”

the end of the tale still wiggles, numerically speaking

But wiggles no more! The dimensions of Pi square had to be corrected, effectively returning the "blue moon" to waxing intrigue. However, this is not the desired end of the tale, for if it were penned today the words must disclaim: "You can't get there from here, but this is what 'there' looks like."

Say what? The creativity sandbox needs fresh sand.

NTS: the 62.4028873643093955.. degree angle still identifies a squared circle.
(hypotenuse of red triangle is the geometric complement: 117.5971126356906045..)

Rod

[Amigoo]

Re: Scalene of Pi design
aka “Pi are square(d)”

No wiggly tale today - the geometry has been corrected and verified.
But the mystery remains: why is the square root of 2 inherent in Pi?

On this Cartesian plane, are Pi square(d)?
Is the proposed rPi constant equally definitive?
And "What's the point?"

Rod

[Amigoo]

Re: Scalene of Pi design
aka “Pi are square(d)”

Such delayed closure on the Texas 'T' prompt!

The design by this name will show that a right triangle whose long side equals Pi,
hypotenuse equals the (square root of Pi) x 2, has a vertex of 27.597112635690604.. degrees.

This right triangle identifies a circle having a diameter of 4 and is squared by the hypotenuse.
(hypotenuse equals length of side)

Therefore, Pi, 2 and their square roots are intimately related ...
according to this sandbox visualization.

Rod

[Amigoo]

Re: Texas 'T' Prompt design
"Life everlasting journey of belief and faith."

Allusion to midwayer prompting via a local road symbol
and display of Pythagorean circle-squaring geometry.

This design completed so quickly that I suspect "guidance"
to keep the complex symbolism once it appeared.

Rod

[Amigoo]

Re: Texas 'T' Prompt design
"Life everlasting journey of belief and faith."

Allusion to midwayer prompting via a local road symbol
and display of Pythagorean circle-squaring geometry:

Right triangle whose long side equals Pi, hypotenuse equals the (square root of Pi) x 2,
and a vertex of 27.597112635690604.. degrees. This right triangle identifies a circle
having a diameter of 4 and is squared by the hypotenuse (hypotenuse = length of side).
Replication of this geometry creates the smaller circle (side length of its square = 2).

And now with symmetry to promote the journey inward.

Rod ... ...

[Amigoo]

Re: Texas 'T' Prompt design
"Life everlasting journey of belief and faith."

Relationship of circle-squaring right triangle in large circle to
circle-squaring right triangle in small circle (long side : hypotenuse):

large circle: ((sq rt Pi) : 2)
small circle: (1 : (2/(sq rt Pi))

((sq rt Pi) : 2) : (1 : (2/(sq rt Pi)) reduces to 2 = 2
... if I understand the related algebra.

Say what? Notes for "What's the point?".

Rod

[Amigoo]

Re: Texas 'T' Prompt design
"Life everlasting journey of belief and faith."

Which right triangles are being compared?
The ones in the upper right quadrant of the circles.
(see red right triangle in Pythagorean Pi design)

Rod

[Amigoo]

Re: Texas 'T' Prompt design
"Life everlasting journey of belief and faith."

About the remarkable Pythagorean right triangle that defines both a circle and its square
(center of circle is defined where perpendicular to hypotenuse connects with long side):

Long side = Pi = 3.14159265358979323846264..
Hypotenuse = (square root of Pi) x 2
= 3.54490770181103205459633..
= side of circle's square.
Circle's diameter = 4.

Rod ... ...

[Amigoo]

Re: Texas 'T' Prompt design
"Life everlasting journey of belief and faith."

The scalene triangle mentioned previously is inherent in this Pythagorean Pi geometry.
The scalene's short side has length equal to one side of a square inscribed in the circle;
longest side forms a 45-degree angle with the hypotenuse of the right triangle.

One vertex of this Pythagorean triangle confirms the cosine angle that is complementary to Pi,
digit-for-digit, in these formulas that were created earlier in this project:

Area = ((Cos 27.597112635690604451732204752339..) x Diameter) Squared
Circum. = ((Cos 27.597112635690604451732204752339..) x square's side length) x 4

The more complex symbolism of the design depicts our mortal-to-morontia transition,
followed by continuing spiritual advancement in our local universe.

Rod

[Amigoo]

Re: Texas 'T' Prompt design
"Life everlasting journey of belief and faith."

While continuing review of this Pythagorean Pi geometry, I wondered:
Is this geometry "earth-shattering" or simply "door-opening"?

With awareness that it's possible to explore in many directions from the foundational right triangle,
I'm convinced that "door-opening" is the true essence - a new era has arrived!

Heading for a welcome respite, I'm curious about certain wiggly numbers:
Is the length relationship of the two sides (hypotenuse and long side) of the right triangle constant?

For if it is, the number of decimal digits of Pi become insignificant (after 40, for example);
the right triangle has the same angle at this vertex regardless of decimal digits.

Rod

[Amigoo]

Re: Texas 'T' Prompt design
"Life everlasting journey of belief and faith."

Is the length relationship of the two sides ... of the right triangle constant?

The answer to this question should have been intuitive, but I had to see it on a Cartesian plane.

Proportion of Pi to 2(square root of Pi) equals 1 : 1.12837916709551257389615890312..
... suggesting that every similar right triangle defines both a circle and its square.

Rod

[Amigoo]

Today's insight ...

every similar right triangle defines both a circle and its square

While the quote is true, non-similar right triangles also define a circle and its square. But that square is not necessarily the true square of the circle. So, the Pythagorean Pi geometry should only be described as "continuing exploration of squared circles", aka "this is what 'there' looks like".

Maybe the Cartesian plane is askew.

Rod

[Amigoo]

Here's a "promotional" summary of the large right triangle in the Texas 'T' Prompt design,
but these scalene triangles are not displayed in the design:

Demonstrate "Pythagorean Pi" with this right triangle that defines both a circle and its square. The center of the circle is located at the point where the perpendicular to the hypotenuse connects with the triangle's long side; circle's diameter = 4:

Long side = Pi = 3.14159265358979323846264..
Hypotenuse = (square root of Pi) x 2
= 3.54490770181103205459633..
= side length of circle's square.

Then draw the overlapping inscribed scalene triangles that are inherent in this Pythagorean geometry. The scalene's short side has length equal to one side of a square inscribed in the circle; the longest side forms a 45-degree angle with either end of the hypotenuse of the right triangle.

What's the point? Explore the fascinating geometry of a squared circle.

Rod

[Amigoo]

Plane Askew (new design)

Conversion of the CAD drawing reference for the previous post to a quick & easy design.
A small squared circle was included since the geometry already supported it.

Rod

[Amigoo]

Plane Askew design
"To many, squared circle geometry is plane askew
… but unites the square roots of Pi and 2."

Two-for-one geometry in this closure!
Diameters equal 4 and 2.

Rod ... ...

[Amigoo]

Plane Askew design
"To many, squared circle geometry is plane askew
… but unites the square roots of Pi and 2."

Contractors were already emptying the sandbox
when I scrawled one last doodle and noticed ...
a related vesica piscis looks promising!

Rod

[Amigoo]

Plane Askew design (more poetic)

"Squared circle geometry plane askew,
uniting the square roots of Pi and 2;
three points of scalene (Pythagoras’ two)
square the impossible (popular view).

Rod ... ...

[Amigoo]

Re: Plane Askew design (updated)

Expanding the Pythagorean magic ...
These similar right triangles (red) both define and square each
of the three circles (diameters = 4, 3.544907701811.., 2)

Rod

[Amigoo]

Re: Plane Askew design

Observation ...

If diameter = 2(sqrt of Pi) ...
and side of square = Pi, the circle is squared ...
precisely, including all decimal digits of Pi.

Who knew? Pi are square!

Rod

[Amigoo]

Re: Plane Askew design

More observation ...

What's the last decimal digit of Pi?
The one that is right.
What's the first decimal digit of Pi?
The one that is right.
What digit of Pi is left?
The whole one.
So, what's the point?
The one that is right of the whole
and left of the one that is right.

Rod ... ...

[Amigoo]

Re: Plane Askew design ("one more line")

About the dark blue circle and its square ...

Diameter of larger green circle = 2(sqrt of Pi);
diameter of dark blue circle = 2(sqrt of 2).
The dark blue circle is that which is squared
by the inscribed square of the green circle.

Say what?
A squared circles cross-reference
of the square roots of Pi and 2.

Say what?
"The one that is right of the whole
and left of the one that is right."

Rod