Paradise Trinity Day

[Amigoo]

Re: Cyan Seven Seven 16:1 design
(regarding circle-squaring right triangles)

2(sqrt(1/Pi)) = 1.1283791670955125738961589031215..
Circle's diameter = hypotenuse of right triangle

Diameter / 2(sqrt(1/Pi))
= long side of right triangle
= side of circle's square

Thus, 2(sqrt(1/Pi)) is a more robust "Pi" constant
because it highlights the Pythagorean geometry
inherent in squared circles.

Rod

[Amigoo]

Re: Cyan Seven Seven 16:1 design
(includes circle-squaring scalene triangle)

Quoth the ravin', "Furthermore ..."
(or at least once in a blue moon):

The long side (one side of the circle's square)
forms a scalene triangle having a 45 degree angle
and a side having length equal to a side of the
circle's inscribed square. HCIT!

Rod

[Amigoo]

Re: Cyan Seven Seven 16:1 design
Re: 2(sqrt(1/Pi)) = 1.1283791670955125738961589031215..

What's a good name for this "robust" Pi constant?

Try "rPi" (for "radial Pi") ... since this constant was first explored
using a circle's radius (not its diameter). Also, it sounds like "our Pi",
a growing perspective of squared circle aficionados.

Rod ... ...
(off to retrieve my radials from the repair shop $$$)

[Amigoo]

Re: Cyan Seven Seven 16:1 design
(upgraded to "Seven Seven Seven")

Since I find no relationship to "Euler's number"*,
the light blue 'E' must relate to "Esoteric".

* 2.7182818284590452353602874713527..
"'e' is used to compute the compound interest of a bank account
which is compounded continuously."

But how to compute compounded negative interest?
(new banking wizardry creeping into financial centers)

Rod ... (enjoying more Zzzzzzz these days)

[Amigoo]

Re: Squared, of Chords design
(sounds like "of Course")

What's most intriguing about patterns in this geometry
are the replicating isosceles right triangles (red) whose sides
become the side of a square of the respective circle.

And this geometry highlights the unique scalene triangles*
that square the circle (as do the right triangles).

* I believed only one until I see these two different scalenes,
tightly integrated with clearly visible juxtaposition.

Rod ... ... (off to confirm that this "juxta"
is not related to bold "chutzpah")

[Amigoo]

Re: Squared, of Chords design
(sounds like "of Course")

What about 2(sqrt(1/Pi)) ?
Disguised as diameter of large, dark blue circle where D = 4(sqrt(1/Pi))
and in the hypotenuse-to-long-side ratios of the red right triangles.

Ro (email typo long ago but whimsical on final page of PDF)

[Amigoo]

Re: Squared, of Chords design
“Lines and triangles and squares, oh Pi!”

There is only one unique scalene triangle,
containing these angles: 45.0 degrees
+ 62.40288736430939554826779524767..
+ 72.59711263569060445173220475233..
= 180 degrees

The seeming visual difference is geometric
Replication Integration Perturbation (RIP).

Rod ... ... (compass correction 180 degrees)

[Amigoo]

Re: Trapezoidal Transcendence design

This isosceles trapezoid can be visualized as one of many objects in the transition
from a square to an isoceles right triangle, with all objects inscribed in the circle.

The transition is easily identified by focusing on the midpoint-to-midpoint line
connecting the sides of the trapezoid and the attached (at both ends of midpoint line)
sides of squares inscribed in the circle (each side a part of an overlapping square).

What's the point?
This trapezoid contains two overlapping lines having length equal to the sides of the
circle's square with the midpoint line representing the side of an isoceles right triangle,
and where its hypotenuse has length equal to a side of the circle's square.

Say What?
If Pi is transcendental, sqrt(2) either disputes this popular claim
or also claims to be transcendental.

A Pi Corral with trapezoidal transcendence.

Rod

[Amigoo]

EEEEEEEEEEE! Laptop crashed! with backup two months old!
This is the 3rd (or 4th) computer crash in 8 years!

Starting over (yet again) with a new EEPC that was stored in a closet three years.
What a 3-day EFFORT to get hardware/software set up to resume daily activity.
Apparently, only significant loss was email and $150 for software.

Back to development of the Trapezoidal Transcendence design:

Possible squares of a circle from smallest (inscribed square)
to largest (inscribed circle). Within this range of objects,
the perfect square is sqrt(Pi). The --- value is the length
difference between SoCS values (Side of Circle's Square).

Given: Diameter = 2.0

SocS = 2.0
------------------- 2.0/sqrt(Pi) = 2(sqrt(1/Pi))
SoCS = sqrt(Pi)
------------------- sqrt(Pi)/sqrt(2)
SoCS = sqrt(2)

Relationships are shown by the vertical red trapezoid
at the right of the Trapezoidal Transcendence design.
(shows rightmost side of smallest and largest square)

Rod ... ... (if EEPC mini-bike is ready to ride)

[Amigoo]

Re: Trapezoidal Transcendence design:

Possible squares of a circle from smallest (inscribed square)
to largest (inscribed circle). Within this range of objects,
the perfect square is sqrt(Pi). The --- value is the length
difference between SoCS values (Side of Circle's Square).

Given: Diameter = 2.0

SocS = 2.0
------------------- = 2(sqrt(1/Pi))
SoCS = sqrt(Pi)
------------------- = sqrt(Pi)/sqrt(2)
SoCS = sqrt(2)

How do these three possible SoCS correlate?

2.0
/ 1.7724538509055160272981674833411.. sqrt(Pi)
= 1.1283791670955125738961589031215.. 2(sqrt(1/Pi))

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

1.1283791670955125738961589031215.. 2(sqrt(1/Pi))
x 1.2533141373155002512078826424055.. sqrt(Pi)/sqrt(2)
= 1.4142135623730950488016887242097.. sqrt(2)

sqrt(2)^2 = 2.0
Today's claim? Very tasty portions of Pi.

Rod ... ... (EEPC mini-bike seems fine
... but had to get a seat belt extension)

[Amigoo]

Re: Trapezoidal Transcendence design:

Current location:
http://aitnaru.org/images/Trapezoidal_Transcendence.pdf

Rod (re: mini-bike challenges)

[Amigoo]

Re: Trapezoidal Transcendence design:

Now with three circles all squared (D = 2, sqrt(2), 1).
Contrasts the similar trapezoid in each circle
(if you can spot them in this design busyness).

Rod

[Amigoo]

Re: Trapezoidal Transcendence II design


I. Analysis of Trapezoidal Transcendence

Given: Diameter = 2.0 (largest circle)

SocS = 2.0
---------------- 1.1283791670955125738961589031215.. 2(sqrt(1/Pi))
SoCS = sqrt(Pi)
---------------- 1.2533141373155002512078826424055.. sqrt(Pi)/sqrt(2)
SoCS = sqrt(2)

How do these three possible SoCS correlate?

2.0
/ 1.7724538509055160272981674833411.. sqrt(Pi)
= 1.1283791670955125738961589031215.. 2(sqrt(1/Pi))

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

1.1283791670955125738961589031215.. 2(sqrt(1/Pi))
x 1.2533141373155002512078826424055.. sqrt(Pi)/sqrt(2)
= 1.4142135623730950488016887242097.. sqrt(2)

Therefore, sqrt(2)^2 = 2.0


II. Derivation of Trapezoidal Transcendence II geometry
by analysis of 2(sqrt(1/Pi)) and sqrt(Pi)/sqrt(2)

1.1283791670955125738961589031215.. 2(sqrt(1/Pi))
/ 1.4142135623730950488016887242097.. sqrt(2)
= 0.79788456080286535587989211986838.. 2(sqrt(1/Pi))/sqrt(2)

1.2533141373155002512078826424055.. sqrt(Pi)/sqrt(2)
/ 1.4142135623730950488016887242097.. sqrt(2)
= 0.8862269254527580136490837416702.. sqrt(Pi)/2

I recognized 0.79788456.. as the side of an inscribed square,
then created that circle and its inscribed square, then followed
the clue of 0.88622692.. (sqrt(Pi)/2)

... ultimately giving two integrated circles squared
where D = 2(sqrt(1/Pi)) and 1.0


Rod

[Amigoo]

Re: Trapezoidal Transcendence II design

When squared circle geometry ...
... appears impossible to the naysayers
... is a serious challenge to the sayers
... and rates three by the universe.

The light blue lines* correlate the three trapezoids,
alternating as midpoint-to-midpoint and diagonal.
* like neutrino showers

Rod ... ... (aerating neurons with neutrinos)

[Amigoo]

Re: Trapezoidal Transcendence II design
2 / sqrt(2) / sqrt(2) : sqrt(Pi) / sqrt(2) / sqrt(2)

Say what?

When a circle (D=2) and its family are feted
by an isoceles right triangle and its family ...
giving circles integrated and squared.

Rod

[Amigoo]

Re: Trapezoidal Transcendence design
2 / sqrt(2) / sqrt(2) : sqrt(Pi) / sqrt(2) / sqrt(2)
Such geometric busyness

And who knew?!
Transcendence is evenly divisible by 4 ?

And who knew?!
Recent comments relate to Trapezoidal Transcendence
- not the Trapezoidal Transcendence II design.

Rod

[Amigoo]

Re: Trapezoidal Transcendence design
2 / sqrt(2) / sqrt(2) : sqrt(Pi) / sqrt(2) / sqrt(2)

About the "geometric busyness" ...
(re: white triangles, inscribed in a circle)

Since the circle-squaring right triangle for D = 1 has 1/4 the area
of the circle-squaring right triangle for D = 2, transcendental Pi
cannot exist ... unless this transcendence is divisible by 4.

Rod ... ...

[Amigoo]

Re: Trapezoidal Transcendence design
2 / sqrt(2) / sqrt(2) : sqrt(Pi) / sqrt(2) / sqrt(2)
(1:4 circle-squaring right triangles of D=1,D=2)

Simply "Squared Circles 101" (seems like 101 lines)

More lines were added to "highlight" (but only to squared circle geometers)
certain isosceles right triangles (left side of design) that correlate with sqrt(2)
downstepping of the large, circle-squaring, right triangle (white) ...
from diameters 2.0, to sqrt(2), to 1.0.

Geometry worthy of this month's 10/10/10 anniversary!

Rod

[Amigoo]

Re: Trapezoidal Transcendence design
2 / sqrt(2) / sqrt(2) : sqrt(Pi) / sqrt(2) / sqrt(2)
(1:4 circle-squaring right triangles of D=1,D=2)

The message of Trapezoidal Transcendence
(long story short):

Whatever the square of the circle with diameter = 2.0,
the square of the circle whose diameter = 1.0,
has 1/4 the area of the square of D = 2.0

Analysis: If non-transcendental Pi exists,
it's derived with the series 1/4,1/4,1/4, ...
and probably integrating 2(sqrt(1/Pi)).

Rod ... ... (off for 3/4 more petrol
for cruisin' on 10/10/10 anniversary)

[Amigoo]

Re: Trapezoidal Transcendence design
(1:4 circle-squaring right triangles of D=1,D=2)

About 2(sqrt(1/Pi)) and the 1/4 series ...

1.1283791670955125738961589031215..
0.28209479177387814347403972578025..
0.070523697943469535868509931445063..
0.017630924485867383967127482861266..
0.0044077311214668459917818707153164..
0.0011019327803667114979454676788291..
0.0002754831950916778744863669195..
0.000068870798772919468621591729875..
0.00001721769969322986715539793225..
0.0000043044249233074667888494830625..
0.00000107610623082686669721237075..
0.0000002690265577067166743030926875..

An obvious pattern of mathematical dilution!
... marginalizing decimal digits of Pi?

Rod

[Amigoo]

Re: Trapezoidal Transcendence design
(1:4 circle-squaring right triangles of D=1,D=2)

About sqrt(Pi) and its triangular lines ...

1.2533141373155002512078826424055.. sqrt(Pi)/sqrt(2)
x 1.1283791670955125738961589031215.. 2(sqrt(1/Pi))
= 1.4142135623730950488016887242097.. sqrt(2)

0.88622692545275801364908374167057.. sqrt(Pi)/2
x 1.1283791670955125738961589031215.. 2(sqrt(1/Pi))
= 1.0

Where the diameter = sqrt(2),
1.0 = side of inscribed square
in the series 2.0,sqrt(2),1.0

Rod

[Amigoo]

Re: Trapezoidal Transcendence design
(1:4 circle-squaring right triangles of D=1,D=2)

Wiggly Numbers of 2(sqrt(1/Pi)) x .25

D, 2(sqrt(1/Pi)) x .25, 2(sqrt(1/Pi)) / (2(sqrt(1/Pi)) x .25)
2, 0.28209479177387814347403972578039.., 4
sqrt(2), 0.070523697943469535868509931445097.., 16
1, 0.017630924485867383967127482861274.., 64
sqrt(2)/2, 0.0044077311214668459917818707153185.., 256
1/2, 0.0011019327803667114979454676788296.., 1024
sqrt(2)/4,
1/4,

Say what?! Dunno - MS CALC soon goes to e-
... and I start thinking z- (Zzzzzzz..)

Rod

[Amigoo]

Re: Trapezoidal Transcendence design
... and the Wiggly Numbers of 2(sqrt(1/Pi)) x .25

While trying to make sense of these expanding/contracting patterns, I suspect that this perspective relates to seemingly infinite Pi: As the decimal digits of Pi propagate into the universe (and beyond), these Wiggly Numbers suggest a counter-action of numeric essence.

Perhaps, in a Pi corral, a digital squeeze occurs (not related to the doctor's office).

Rod ... ...

[Amigoo]

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

Concentric but not on geometric centers.
A study of diameters and SoCS decrementing by half.
Using the foundational geometry of Trapezoidal Transcendence.

SoCS = Side of Circle's Square

Rod ... ... (scanning starry nights for "Y(es!)")

[Amigoo]

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

A squared circle is evidenced by the least value of Pi
as long as the 2(sqrt(1/Pi)) hypotenuse_to_long_side ratio
of the circle-squaring right triangle is maintained.

D, decrements of sqrt(Pi)
2, sqrt(Pi)
1, sqrt(Pi) /2
.5, sqrt(Pi) /4
.25, sqrt(Pi) /8 ...

Say what?! Even a pico Pi is nice ...
and geometrically tasty.

Rod