Paradise Trinity Day

[Amigoo]

Re: Points of Order design
"Conceptually, the Pythagorean ABCs of squared circles."

A final Points of Order for this Pythagorean playground:

Top (short side) of the large golden Pythagorean triangle = 1.
Top (short side) of magenta triangle to the right = (sqrt(Pi))/2.

So, "What's the point?"

Rod

[Amigoo]

Re: A Round Pi Square (new design)
Simple geometry that also "speaks for itself".

Contrasting sqrt(Pi) in two squared circles
where side of square = sqrt(Pi) [circle's diameter = 2]
and diameter = sqrt(Pi) [side of square = Pi/2].

Rod

[Amigoo]

Re: A Round Pi Square design

"A Round Pi Square with subtle 777 resonance
promoting pedagogical Pythagorean perspective."

A minor embellishment to show a circled square with side length = 2
quickly revealed a nested 777 (two red, one yellow) and persuading that
this geometry truly desires to speak for itself.

About the ".love" smilie ...
I tried to insert a .geek smilie but .love appeared.
So, I tried again to insert .geek and .love appeared.
Well ... Love must reign on high.

Rod

[Sandy]

Hi Rod,

Well ... Love must reign on high.

Amen to that!

xxSandy

[Amigoo]

Re: A Round Pi Square design
"A Round Pi Square illuminates 777 resonance,
promoting pedagogical Pythagorean perspective."

With "Twas the night before Christmas" a continuing theme, closure of this
Cartesian creativity was increasingly important yet elusive. The geometric "look & feel",
while satisfying, was not yet perfect (aka, not yet AGAIG). What could be missing?!

Who knew! This design incorporated a subtle, geometric palette sans thumbhole!
And it was not until the thumbhole (small blue circle) was discovered that another
"cPoP" became the essence of this composition: "coordinating Palette of Pi".

The small blue circle (diameter = 1) magically coordinates the plethora of Pi
on this Cartesian playground, illuminated with symbolic 777 resonance.

Say what? "Twas the night ..." of closure!

Rod ... ...

[Amigoo]

Re: A Round Pi Square design

Postscript ...

I was mystified that sqrt(Pi) was the center of attention for weeks,
with negligible geometric reference to sqrt(2)!

I finally noticed that the 777 resonance, when overlapped
with that little blue circle, highlighted a symbolic "707" ...
as in (sqrt(2))/2: 0.707106781186547524400844..

Rod

[Amigoo]

Re: concentric Patterns of Pi
(more cPoP geometry for a new design)

"Twas the night before Christmas ..."
and Santa's cookies were baking in the oven.
Say what? "These patterns have pleasing visual aroma".

Draw these concentric circles:

0.785398163397448309615660845819.. Pi/4
0.886226925452758013649083741670.. (sqrt(Pi))/2
1.0
1.128379167095512573896158903121..

1.570796326794896619231321691639.. Pi/2
1.772453850905516027298167483341.. sqrt(Pi)
2.0
2.256758334191025147792317806242..

3.141592653589793238462643383278.. Pi
3.544907701811032054596334966682.. 2(sqrt(Pi))
4.0
4.513516668382050295584635612484..

Draw this radius and straight lines
from the top of the circles to the radius
(creates Pythagorean triangles):

62.40288736430939554826779524767.. degrees

¡buen apetito! (design expected post-Santa)

Rod

[Amigoo]

Re: concentric Patterns of Pi
(more cPoP geometry for a new design)

1/Pi was missing from three diameters;
here's the restatement "for the record".

Also, the patterns suggest that diameters
1, 2 and 4 should have a Pi-qualified expression,
so x(Pi/Pi) is a reasonable placeholder.

"Twas the night before Christmas ..."
and Santa's cookies were baking in the oven.
Say what? "These patterns have pleasing visual aroma".

Draw these concentric circles:

0.785398163397448309615660845819.. Pi/4
0.886226925452758013649083741670.. (sqrt(Pi))/2
1.0 .......................................... 1(Pi/Pi)
1.128379167095512573896158903121.. 2(sqrt(1/Pi))

1.570796326794896619231321691639.. Pi/2
1.772453850905516027298167483341.. sqrt(Pi)
2.0 .......................................... 2(Pi/Pi)
2.256758334191025147792317806242.. 4(sqrt(1/Pi))

3.141592653589793238462643383278.. Pi
3.544907701811032054596334966682.. 2(sqrt(Pi))
4.0 .......................................... 4(Pi/Pi)
4.513516668382050295584635612484.. 8(sqrt(1/Pi))

Draw this radius, then straight lines
from the top of the circles to the radius
(creates Pythagorean triangles; x and y
axes should be drawn first).

62.40288736430939554826779524767.. degrees

¡buen apetito! (design expected post-Santa)

Rod

[Amigoo]

Re: concentric Patterns of Pi (cPoP)
aka "My Pi My Pi"
(humor en casa)
aka "My IT Pi"
(pronounced "Mighty Pi")
(meaning "My Irrational Transcendental Pi")

An early morning review of this geometry shouted "Seize the day!"
"Capture the obvious patterns and retain the impromptu colors."

Maybe, Santa has an IT Dept. with social media networks.
Apparently, for 2014, this cPoP is now declared AGAIG.

For geometric reflection:
As more decimal digits of Pi are discovered,
the diameters 1, 2, and 4 do not change.

...




..

Rod

[Amigoo]

Re: concentric Patterns of Pi (cPoP)
aka "My IT Pi" (pronounced "Mighty Pi")

The diameters, listed by concentric circles from largest to smallest,
and with Pi-qualification for diameters 1,2,4 (how diameter relates to Pi).

4.513516668382050295584635612484.. 8(sqrt(1/Pi))
4.0 .......................................... 8(sqrt(Pi)/2)(sqrt(1/Pi))
3.544907701811032054596334966682.. 2(sqrt(Pi))
3.141592653589793238462643383278.. Pi

2.256758334191025147792317806242.. 4(sqrt(1/Pi))
2.0 .......................................... 4(sqrt(Pi)/2)(sqrt(1/Pi))
1.772453850905516027298167483341.. sqrt(Pi)
1.570796326794896619231321691639.. Pi/2

1.128379167095512573896158903121.. 2(sqrt(1/Pi))
1.0 .......................................... 2(sqrt(Pi)/2)(sqrt(1/Pi))
0.886226925452758013649083741670.. (sqrt(Pi))/2
0.785398163397448309615660845819.. Pi/4

Note 1: [for D = 1,2,4] (sqrt(Pi)/2)(sqrt(1/Pi)) = .5
(sqrt(Pi)/2) = 0.88622692545275801364908374167057..
(sqrt(1/Pi)) = 0.56418958354775628694807945156077..

Note 2: In each group of 4 diameters, the smallest diameter
is multiplied by 2(sqrt(1/Pi)) giving the next larger diameter
[2(sqrt(1/Pi)) = 1.128379167095512573896158903121..]

Say what? "Conspicuous Sanitas Cyclometricus!"

Rod

[Amigoo]

Re: concentric Patterns of Pi (cPoP)

A concise summary of current research:
http://aitnaru.org/images/cPoP_5253.pdf

Rod ... ...

[Amigoo]

Re: Sonrise (new design)
"Long-term forecast"

"When squared circle geometry speaks for itself" ... in 2015.

Rod

[Amigoo]

Re: http://www.amazon.com/review/R6JGQY2FNOQZU

In 2015, let's bake more circle-squaring Pi.

Rod

[Amigoo]

Pythagorean Pi (new design)
( http://aitnaru.org/images/cPoP_5253.pdf )

Pythagorean Pi is the final design for 2014.

Rod ... ...

[Amigoo]

Balloonza Pi (new design)

Serious geometry with a whimsical name (for "Balloons of Pi"), highlighting the Pythagorean right triangles that define squared circles. The integrated four circles, all squared, on this Cartesian playground inspire out-of-the-box thinking about those "impossible" squares of circles.

Rod

[Amigoo]

Re: Sqrt(Pi)/2 (new design)
"Geographic center of a Cartesian playground."

The siblings of Pi (sqrt(Pi), sqrt(Pi)/2, Pi/2) seem to favor this playground
and often congregate within the circle of diameter 2.

Rod ... ...

[Amigoo]

Re: Sqrt(Pi)/2 design
"Geographic center of a Cartesian playground."

Time for Texas 'T' ...

As placed in this composition, the yellow Texas 'T' appears to unify the geometric objects as well as define two squared circles (small green, small blue). The vertical blue line has length of sqrt(Pi)/2 which is half the length of the vertical red line (within small blue circle), as confirmed by the mutually attached parallelogram.
Diameter of large green circle = 2.

Why "Texas 'T' "?
Look for similar geometry in the North Texas Tollway black-on-orange logo.

Definitely a Pythagorean Playground!

Rod

[Amigoo]

Re: Pythagorean Pi Corral (new design)
"SoS = SoIS(Sqrt(Pi/2)) = Sqrt(Pi)"

Easy as Pi.

Given: Diameter of circle = 2
SoS = Side of circle's square
SoIS = Side of circle's inscribed square

Rod

[Amigoo]

Re: Pythagorean Pi Corral design
"SoS = SoIS(Sqrt(Pi/2)) = Sqrt(Pi)"

A similar quadrilateral (red) with one right angle appears in every squared circle,
regardless of diameter and regardless of geometric integration in these Cartesian compositions.

Research is leading to further association with the circles' inscribed square.

"Running in squared circles" describes this beginning new year when "long winter's nap" was desired.

Rod

[Amigoo]

Re: Pythagorean Pi Corral design
"For D=2, SoS = SoIS(Sqrt(Pi/2)) = Sqrt(Pi)"

Correction/addition to previous comment:

A similar quadrilateral (red) with two right angles appears in every squared circle,
regardless of diameter and regardless of geometric integration in the composition:

One set of lines (forming a right angle) represents half of the inscribed square of the circle
and the other set contains a line with length equal to one side of the square of the circle.

Thus (in the equation), SoS is the calculation of multiplying the length of one side
of the inscribed square by Sqrt(Pi/2). It's as easy as Pi.

But the best identification of a squared circle is the scalene triangle formed when
a straight line is drawn from the vertex of one right angle (in the quadrilateral) to the
vertex of the other. This scalene has one side with length equal to the square of the circle,
a 45 degree angle, and one side with length equal to a side of the inscribed square.

... and this confirms inspiration several years ago that the geometry of a squared circle
must associate the square root of Pi with the square root of 2.

Rod

[Amigoo]

Re: Pythagorean Pi Corral design
"For D=2, SoS = SoIS(Sqrt(Pi/2)) = Sqrt(Pi)"

Enhanced geometry, revealing squared circles
as a dimension of Cartesian coordinates
as well as a dimension of mind.

Proposed T-Shirt layout:


For D=2, SoS = SoIS(Sqrt(Pi/2))

(center SoS geometry design here)

I've got a mind to square the circle.


Rod

[Amigoo]

Re: PPoSC (new design, pronounced "posse" \ˈpä-sē\ )
A Pythagorean perspective of squared circles.

In this Pi Corral, the PPoSC rules!
... trumping irrational and transcendental.

Rod ... ...

[Amigoo]

Re: http://aitnaru.org/images/cPoP_5253.pdf
A good plateau for this collection of geometry studies.

The PPoSC geometry is especially intriguing because it focuses on Pythagorean space (defined by Pythagorean triangles) both inside and outside the squared circle. The rectangular space (defined by magenta lines) to the left of the red quadrilateral has the same area as the rectangular space (similarly defined but as two Pythagorean triangles) to the right of the quadrilateral.

And within the quadrilateral, the two isosceles right triangles on the left have the same area as the isosceles right triangle on the right (re: Pythagorean Theorem);
a right triangle on one side overlaps a right triangle on the other side.

Whether or not this circle is squared, it is obviously Pythagorendered!

Rod

[Amigoo]

Re: PPoSC design

Leave a Pythagorean Pi Playground unsupervised for a day and squared circles replicate.
All three squared circles display a similar quadrilateral (one red, two yellow).

Diameters, from largest to smallest: 2,
1.7724538509055160272981674833411 - Sqrt(Pi),
1.4142135623730950488016887242097 - Sqrt(2).

Rod ... ...

[Amigoo]

Re: PPoSC design

I tried to improve the geometry of this design, but kept returning to its unique integration of three squared circles. In particular, the isosceles right triangle that squares the other two circles confirms the integration: triangle's hypotenuse of sqrt(Pi) squares the circle of diameter = 2; side of sqrt(Pi/2) squares the circle of diameter = sqrt(2).

Also, the circle enclosing this right triangle (diameter = sqrt(Pi)) is squared by a right triangle whose long side = Pi/2 and hypotenuse = sqrt(Pi). And three similar circle-squaring right triangles confirm that the long side of this triangle has length equal to the side of the square of its enclosing circle.

A Pythagorean Pi Playground extraordinaire!

Rod