Paradise Trinity Day

[Amigoo]

Re: POM Indeed! design

The relationship of the larger circles to the overlapping smaller circles
was too mysterious to ignore! Regarding this POM geometry ...

When the ratio of the diameters of a large and small circle
equal sqrt(Pi), the small circle effectively squares the large.
(see: largest blue circle and topmost orange circle)

D = 9.027033336764100591169271224972..
d = 5.092958178940650744604280427920..
side of square of circle D = 8.

Rod

[Amigoo]

Re: POMscript design (POM + script from "postcript")
About the POMscript geometric proportions ...

2(sqrt(1/Pi)) : 1 ~ 2 : sqrt(Pi)
means/extremes: 2(sqrt(1/Pi)) x sqrt(Pi) = 2
1.1283791670955125738961589031215..
x 1.7724538509055160272981674833411.. = 2

So, all decimal digits of Pi are located within
the boundaries of this Pi corral ... apparently.

Rod

[Amigoo]

Re: POMscript design (POM + "script" from "postscript")

"Who can tell?"

The Texas 'T' now appears as golden lines within a small circle
at the top of this design. Authentication of the geometry?

Rod ... ...

[Amigoo]

Re: POMscript design (more embellishment)
'T' might be for "Texas", but 'T' also reflects "Trident" and "Trinity".

As the design changed, POM (meaning "flower") began to suggest
approaching announcement of the "flowering of an age". Say what

Possible translation: the unseen forces of the universe are announcing
their presence and proclaiming the true destiny of this inhabited planet.

Rod

[Amigoo]

Re: Pythagorean Perpetuity design

In Pythagorean Perpetuity ...
ask not "Is the circle squared?"
ask "When is the circle not squared?"

Rod ... ...

[Amigoo]

Re: Trinities of Cartesian Dualities (new design concept)

Geometry incorporating increments of 4(sqrt(1/Pi))
in overlapping circular triads. Say what?

Squared circle geometry that speaks for itself
(or prefers to remain silent for an age).

Rod

[Amigoo]

Re: Trinities of Cartesian Dualities design
"Triangulation in the multipolar universes."

Another selection from the bottomless, squared-circles toy box.

Rod

[Amigoo]

Re: SCG101 design
“denoting the elementary or basic facts”

Rod

[Amigoo]

Re: SCG101 design
“denoting the elementary or basic facts”

Another geometry prompt confirmed:
11:28 (relates to 2(sqrt(1/Pi))
= 1.1283791670955125738961589031215..

= length of diagonal lines in magenta triangle.
= twice diameter of largest dark blue circle.

D = 2 in largest green circle;
side of its square = sqrt(Pi).

Analysis: This Pi is good!

Rod

[Amigoo]

Re: SCG101 design
“denoting the elementary or basic facts”

Numeric cousins (see smallest set of dark blue circles):
2(sqrt(1/Pi)) = 1.1283791670955125738961589031215..
(2(sqrt(1/Pi)))^2 = 1.2732395447351626861510701069801..

Multiply diameter of smaller circle by (2(sqrt(1/Pi)))^2
to get diameter of larger circle. This increment applies
to all three sets of these circles in this design.

Increment between circle sets: Pi/2
= 1.5707963267948966192313216916398..
Diameter of largest circle in each set: .5, 1, 2

What's the point?
Sanitas Cyclometricus.

Rod ... ...

[Amigoo]

Re: SCG101 design
“denoting elementary or basic facts”
A squared circle kundalini of enlightenment?

This design kept evolving, but is finally stable! With this recoloring,
the flame of each symbolic candle represents the Pi/2 increment*
(multiplier or divider) between the two-circle sets.

* Pi/2 = 1.5707963267948966192313216916398..

Rod

[Amigoo]

Re: SCG101 design
“denoting elementary or basic facts”

Increments of Pi in this geometry hint that its resonance is detectable
... and that the geometry might qualify as "sacred".

Notes:
1. diameter of large green circle = 2.
2. side of this circle's square = sqrt(Pi). (golden diagonals)
3. flame of each symbolic candle represents a Pi/2 increment
(multiplier or divider) between the two-circle sets.
4. diameter of large magenta circle = 4(sqrt(1/Pi)).
5. diameters of green circles = 2, 1, .5, .25.

Rod

[Amigoo]

Re: Encircled Insuperables design

The geometry of SCG101, highlighting a vertical Pi "cantilever truss" and two irregular quadrilaterals of unique, squared-circle salience:

The lengths of the horizontal and top diagonal sides have a ratio of sqrt(Pi). This is the first time (in this Cartesian neighborhood) that the short sides of corresponding, circle-squaring right triangles are known to have Pi-defined relationship ... and probably why the SCG101 design appears so geometrically balanced.

Say what? You can always truss encircled insuperables.

Rod

[Amigoo]

Re: Encircled Insuperables design
"Irregular quadrilaterals of squared-circle salience. "

A review of the Pi increments hints that this geometry
has good potential for: 1) proving that the circle is squared.
2) creating a new "Pi beyond compare".

Rod ... ...

[Amigoo]

Re: Encircled Insuperables design
"Irregular quadrilaterals of squared-circle salience. "

In the total right triangle (red), the ratio of the lengths
of the top horizontal line to the bottom horizontal line = Pi
= 3.1415926535897932384626433832795..

Rod

[Amigoo]

Re: This Way Up design
"Bon voyage!"

In the Post-R universe, Paradisiacal guidance assists navigation,
making directional inquiries unnecessary but still welcome.

Rod ... ...

[Amigoo]

Re: these Cartesian neighborhoods

Without a doubt ...
these are three interelated core values
that define a Pythagorean right triangle
that squares the circle.

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

sqrt(Pi) x 2(sqrt(1/Pi)) = 2
= hypotenuse of right triangle
= diameter of circle

Rod

[Amigoo]

Re: these Cartesian neighborhoods
Pi/2 is also a member of this interrelationship:

2.0
1.7724538509055160272981674833411.. = sqrt(Pi)
1.5707963267948966192313216916398.. = Pi/2
1.1283791670955125738961589031215.. = 2(sqrt(1/Pi))

2 * 2(sqrt(1/Pi))
= 2.256758334191025147792317806242
= diameter of circle whose side of square = 2

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

sqrt(Pi) x 2(sqrt(1/Pi)) = 2

1.7724538509055160272981674833411..
x 1.1283791670955125738961589031215.. = 2
= hypotenuse of right triangle
= diameter of circle

sqrt(Pi) / 2(sqrt(1/Pi)) = Pi/2

1.7724538509055160272981674833411..
/ 1.1283791670955125738961589031215..
= 1.5707963267948966192313216916398.. = Pi/2
= side of square for diameter = sqrt(Pi)

Perhaps, the four values are paired:
2 & Pi/2, sqrt(Pi) & 2(sqrt(1/Pi))

OMG! Which way is up?
(see "This Way Up" design)

Rod ... ...

[Amigoo]

Re: these Cartesian neighborhoods
Pi/2 is also a member of this interrelationship:

The addendum design, This Way UP IV, highlights the geometric object (red)
that incorporates all of these interrelated, circle-squaring values:

2.0
1.7724538509055160272981674833411.. = sqrt(Pi)
1.5707963267948966192313216916398.. = Pi/2
1.1283791670955125738961589031215.. = 2(sqrt(1/Pi))

Rod

[Amigoo]

Re: This Way UP IV design (renamed to "IV League")
After redrawing, This Way Up IV no longer appeared as an
addendum of This Way Up, so it claimed independence!

IV League (IV = Roman numeral 4 but is pronounced "I V")
presents new geometric perspective on Pi's "transcendental".

IV league of circle-squaring values:
2.0, sqrt(Pi), Pi/2, 2(sqrt(1/Pi))

Means/Extremes Property:
1 : sqrt(Pi) ~ 2(sqrt(1/Pi)) : 2
sqrt(Pi) x 2(sqrt(1/Pi)) = 2

Rod

[Amigoo]

Re: Veritas design

A concise summary of squared circle geometry.
Say what? "Vincit omnia veritas!"

Rod ... ...

[Amigoo]

Say what?
About the 2-for-1 Daily Special*
at the Squared Circles Cafe ...

Ingredients:

Pi = 3.1415926535897932384626433832795..
Pi/2 = 1.5707963267948966192313216916398..
sqrt(Pi) = 1.7724538509055160272981674833411..
sqrt(Pi)/2 = 0.88622692545275801364908374167057..

1/Pi = 0.31830988618379067153776752674503..
sqrt(1/Pi) = 0.56418958354775628694807945156077..
2(sqrt(1/Pi)) = 1.1283791670955125738961589031215..

Directions (Part A):

1 / 2(sqrt(1/Pi)) = sqrt(Pi)/2
1 / sqrt(Pi)/2 = 2(sqrt(1/Pi))
2(sqrt(1/Pi)) x sqrt(Pi)/2 = 1

1.0 / 1.1283791670955125738961589031215..
= 0.88622692545275801364908374167057..

1.0 / 0.88622692545275801364908374167057..
= 1.1283791670955125738961589031215..

1.1283791670955125738961589031215..
x 0.88622692545275801364908374167057.. = 1

Directions (Part B):

2 / 2(sqrt(1/Pi)) = sqrt(Pi)
2 / sqrt(Pi) = 2(sqrt(1/Pi))
sqrt(Pi) x 2(sqrt(1/Pi)) = 2

2 / 1.1283791670955125738961589031215..
= 1.7724538509055160272981674833411..

2 / 1.7724538509055160272981674833411..
= 1.1283791670955125738961589031215..

1.7724538509055160272981674833411..
x 1.1283791670955125738961589031215.. = 2

The Daily Special ...

Item #1: 2(sqrt(1/Pi)) x sqrt(Pi)/2 = 1
Item #2: sqrt(Pi) x 2(sqrt(1/Pi)) = 2

The 2-for-1: sqrt(Pi) x 2(sqrt(1/Pi))
= 2(2(sqrt(1/Pi)) x sqrt(Pi)/2)

Most popular order ...

Waiter: "What portion of Pi would you like today?"
Customer: "I'll just have a glass of water."
Waiter: "And no Pi?"
Customer: "Yes, I'll have no Pi."
Waiter: "What portion of no Pi would you like today?"
Customer "Just bring me the Daily Special."
Waiter: "Good choice! One or two servings?"
Customer: "A small glass, please."

* Refers to the integrated line lengths of the
circle-squaring, Pythagorean right triangle
in the IV League and Veritas designs.

Rod ... ...

[Amigoo]

Re: Veritas 27597 design

The common ratios (0.564..) and angles (27.597..) created by
paired lines in the circle-squaring, Pythagorean right triangle.
(color pairs = red, yellow, green)

Red
1.1283791670955125738961589031215.. / 2
= 0.56418958354775628694807945156077..

Yellow
0.52272320087706331513679711195272..
/ 0.92650275035220848584275966758914..
= 0.56418958354775628694807945156077..

Green
1 / 1.7724538509055160272981674833411..
= 0.56418958354775628694807945156077..

Rod ... ... (off to see the lights)

[Amigoo]

Re: Veritas 27597 design

The Veritas 27597 extended geometry that shows
why the circle is squared (IMO):

http://aitnaru.org/images/Veritas_27597e.pdf

Rod

[Amigoo]

Re: Veritas 27597e design

After more exploration (and circular design changes),
a geometer might ask: Is this serious geometry?
or another Smile of Pythagoras*?
... or both?

* turn design sideways to see "smiling face"

Rod