Paradise Trinity Day

[Amigoo]

Re: Sqrt(2) Significances design

(off to complete the geometric spiral)

Been there! Done that!

Conjecture:
Since the geometry repeats after a 360 degree turn of the spiral,
Pi must express ALL of its decimal digits before the next turn
... but Pi cannot be transcendental if it does.

Is Pi/16 one of the Sqrt(2) Significances?
... or Pi/256 (= 16^2)?

Rod ... ...

[Amigoo]

Re: Sqrt(2) Significenter design
(detail of Sqrt(2) Significances)

aka "Next 360"

Rod

[Amigoo]

Re: Sqrt(2) Significenter design (better cropping;
spiral center of Sqrt(2) Significances)

Real Infinity of Pi

May there always be a Next 360
in the twists and turns of your life
... even inward to Paradise.

Rod ... ...
(still turning on the Paradise dime)

[Amigoo]

Regarding Sqrt(2) Significances design ...

Introducing Sqrt(2) Significenter II
"The less you see, the more you see."

First place setting with a Cartesian plate,
starring sqrt(Pi)/4 at this inaugural date.

Informal place setting for late evening trumpettes
(trumpet-shaped mini pastries)

Rod

[Amigoo]

Re: Sqrt(2) Significenter II design

In this Cartesian squared circles neighborhood, sqrt(Pi) may have a value anywhere from the circle's diameter to a side of the circle's inscribed square. Regardless of value, a "Mini Pi" lurks nearby with dimensions of its objects (relative to the main Pi) representing a precise division by 4.

Serving Sqrt(Pi)? First serve the symbolic Mini Pi hors d'oeuvre

Incidentally, this geometry shouts that the decimal digits of Pi
must be evenly divisible by 4 ... or 16 ... or 256 ... or

Rod ... ... (off to find a larger Mini Belt after so much Pi)

[Amigoo]

Re: Sqrt(2) Significenter II design

More about the recipe ...

Sqrt(2) is the geometric foundation of this counter-clockwise spiral (dimensions of similar objects keep decreasing by sqrt(2)). This sqrt(2) precision appears to be a "corral" for the seeming infinity of Pi's decimal digits (i.e., Pi may not be transcendental ). Since 2, 4, 16, 256, etc. (a geometric progression) are all factors in the geometry of this inward-progressing spiral, this hints that the total decimal digits of Pi might indeed be known by advanced calculation!

Home Chef's Translation: For every Pi there's a precise Mini Pi inherent in the recipe!
(at a Squared Circles Soirée, serve the Mini Pi hors d'oeuvre before the featured Pi)

Rod

[Amigoo]

Re: Scalene Interlocution design

Advanced example of squared circle "geometry that speaks for itself",
with intentional focus on the circle-squaring scalene triangle.

If showing "the circle is squared", then "scalene interlocution" might be
defined as "precise conversation between two or more people".

... otherwise, "impossible conversation between two or more people"
might be the definition until this unique geometry is perfected.

Rod ... ... (off to discover salient points)

[Amigoo]

Re: Scalene Interlocution design
(salient points discovered)

aka "Circular Scalene Interlocution"
aka "CSI of transcendental Pi"

Rod

[Amigoo]

Re: Scalene Salience design

A geometrically impressive composition of three circles squared,
all with similar triangular objects (right, isosceles, scalene).

If triangles don't square the circle,
Pythagoras was a horizontal plumb bob.

Rod ... ... (off to find a circular plum bob)

[Amigoo]

Re: Scalene Salience design

Tonight's first choice for theme song:
https://www.youtube.com/watch?v=6E2hYDIFDIU

Rod

[Amigoo]

Re: Scalenity Octavo design

Eight overlapping scalene triangles with the sum of their 45-degree angles equal to 360 degrees,
and featuring "Who knew?!" esoteric points creating an " impossible" squared circle in the center.

Geometric symmetry that invites a Pythagorean perspective of mathematical studies.
See also: http://www.2012-spiritual-growth-prophe ... ngs-2.html

"Numerology meanings under Pythagoras breaks everything in the universe down into numbers.
Mathematics is the basis of all things, and geometry the highest form of mathematical studies."

Rod

[Amigoo]

Numerology meanings under Pythagoras breaks everything in the universe down into numbers.
Mathematics is the basis of all things, and geometry the highest form of mathematical studies.

... since it reaches into depths of mind beyond advanced mathematics
where universe patterns communicate enlightenment.

Rod ... ...

[Amigoo]

Re: Transcendental Spiral design

Obviously (apparently ), it may take millennia to discover geometry
that "squares the circle" according to the Greek rules (paper, pencil, etc.),
but geometry that proves sqrt(2) overcontrol of "transcendental" exists!

Transcendental Spiral proffers more evidence that Sqrt(2) Rules!
these Cartesian Neighborhoods of "impossible" squared circles.

Rod

[Amigoo]

Re: Transcendental Spiral design

Who knew?! Geometry of Transcendental Spiral
reveals the iGR ("impossible" Golden Ratio)
of the circle-squaring scalene triangle:

In largest circle where Diameter = 1,
Side of Circle's Square (SoCS) = sqrt(Pi)/2,
one side of circle-squaring scalene triangle
provides these a and b dimensions:

a = 32757056.412..
b = 62665706.866..
a+b = 95422763.278..

62665706.866.. b
/ 32757056.412.. a
= 1.9130445079626710873950184043784.. b/a

95422763.278.. a+b
/ 32757056.412.. a
= 2.9130445079626710873950184043784.. (a+b)/a

the revealed iGR
1.9130445079626710873950184043784.. b/a
2.9130445079626710873950184043784.. (a+b)/a


Rod

[Amigoo]

Re: Transcendental Spiral design

Geometry of Transcendental Spiral reveals the iGR ("impossible" Golden Ratio)

Truth in advertising (re: "impossible") ... since the real GR is a > b and (a+b)/a = a/b
... but a and b are reversed in the Transcendental Spiral conjecture. Say what?!

And what's the 1.913.. and 2.913.. ? A co-inkydink

Rod ... ...
(don't know my a from my b_rain)

[Amigoo]

Re: Transcendental Spiral design

Geometry of Transcendental Spiral reveals the iGR ("impossible" Golden Ratio)

Incidentally, the conjecture describes this relationship:
b > a and (a+b)/a = (b/a) + 1

Rod

[Amigoo]

Re: Transcendental Spiral design and iGR

What fun this faux "golden ratio" of a and b, but 2(sqrt(1/Pi))
remains the premier "golden ratio" of squared circles. Interestingly,
a/b can be defined as the hypotenuse (a) and long side (b)
of a circle-squaring right triangle.

The iGR of squared circles must be 2(sqrt(1/Pi).

2.0 / 1.7724538509055160272981674833411.. 2/sqrt(Pi)
= 1.1283791670955125738961589031215.. the iGR

Rod

[Amigoo]

Re: http://aitnaru.org/images/Scalene_Salience.pdf

Recent geometry featuring the circle-squaring scalene triangle.

"iDiMw" is an acronym for the title of the theme song,
with aesthetic balance of upper/lower case.

Rod ... ...

[Amigoo]

Re: Correlation of Squares design

As simple as squared circle geometry can be!
... and a continuing message:

Sqrt(2) rules this Cartesian neighborhood, featuring the "golden" 2(sqrt(1/Pi)) ratio;
whatever the Pi value, sqrt(2) is the controlling increment across CSC diameters.

Rod ... ... (time for an ealy Spring Break?)

[Amigoo]

Re: Correlation of Squares design

HCIT! Geometers' tip:

The two similar angles (one yellow line, one green line shared) appear to define equilibrium when the two adjacent sides of the similar right triangles are imagined to pivot at either end of the green line (diameter of largest circle), geometrically contrasting the short and long sides (not hypotenuse) of the circle-squaring right triangle.

Say what?!
Be there (in your Cartesian mind) and be square (with quadrature insight) ...
then run off to create the world's first geometric proof of a circle squared.

Rod

[Amigoo]

Re: Correlation of Squares design

From the internet ..

"January 29th is National Puzzle Day (U.S.). If you want to enter in the midst of puzzles and forget yourself by solving puzzles, you can have all the fun on National Puzzle Day. It is a day of entertainment. People can enter into a big area of entertainment and fun on national puzzle day to solve a wide range of puzzles."

Since it's National Puzzle Day, let's say that the geometric proof of a circle squared
is due by Leftover Pi Day, November 10, 2017 (this year's Julian 314).

And since it's National Puzzle Day ...

Here's an upgrade to the geometry, nicknamed Euler's Nose
(Euler knows now about this 2(sqrt(1/Pi)) constant)

Rod ... ...

[Amigoo]

Re: Correlation of Squares design (Euler's Nose)

Bottom line (IMO): This geometry is the first (ever) to present opportunity for geometric proof of a squared circle.

Geometers' Tip: Expand the two light blue arcs into circles to visualize the pivoting of the two perpendicular sides (yellow) of the right triangle, where the opposite ends of the lines follow that circle's circumference. When the angles created by the yellow and shared green lines are identical, the circle is squared.

Euler knows.

Rod

[Amigoo]

Re: Correlation of Squares design (aka "Euler's Nose")

When squared circle geometry has the last laugh.
(while exploring the similar line length ratios,
Euler's Nose became well-defined)

Rod

[Amigoo]

Re: Correlation of Squares design (aka "Euler's Nose")

similar line length ratios

About those "identical angles" ...
these line length ratios may be more definitive
of squared circles; they have "golden ratio" essence.
What new "essence"? 1.91306703016223193..

Rod

[Amigoo]

Re: Correlation of Squares design
"Stop the presses!" (more accurate ratio)

Regarding Pythagorean Theorem: a^2 + b^2 = c^2
Given: Diameter = 2.0 = c, SoCS = sqrt(Pi) = b
where SoCS = Side of Circle's Square

Therefore, c^2 - b^2 = a^2
4 - 3.1415926535897932384626433832795..
= 0.8584073464102067615373566167205.. = a^2

sqrt(0.8584073464102067615373566167205..) a^2
= 0.92650275035220848584275966758914.. = a

sqrt(3.1415926535897932384626433832795..) b^2
= 1.7724538509055160272981674833411.. = b

Ratio of long side (b) to short side (a):

1.7724538509055160272981674833411..
/ 0.92650275035220848584275966758914..
= 1.9130583802711007947403078280205..

In the Correlation of Squares geometry, yellow line segments reflect this ratio
in at least three sets of line segments, with one set right under Euler's Nose.

Rod