Paradise Trinity Day

[Amigoo]

Re: Boxa Scalenity,
added to: http://aitnaru.org/images/Tasty_Pi.pdf

"Geometry is like a box of chocolates -
one point always leads to another!"

Yum! All Boxa lines = sqrt(2)/2


OMG! Is the box open or closed?!
I keep sensing "Yes!" (and "Impossible!")

"A mind is like a box of chocolates -
full, empty, or betwixt and between."


Ro ... ...

[Amigoo]

Re: Boxa Scalenity, "The bell tolls for thee"
* updated in: http://aitnaru.org/images/Tasty_Pi.pdf

Pythagorean Quadrature is like a box of chocolates
on a national holiday - memorable when you savor the points!

While the circle-squaring scalene triangles can be subdivided into 4 similar scalenes,
the large scalene-enclosing isosceles trapezoid can also be subdivided into 4 trapezoids
(one trapezoid is divided into two parts, not adjacent). HCIT

"Lines and Triangles and Squares, Oh My!"

Ro ... ...

[Amigoo]

Re: Boxed Pi design,
added to: http://aitnaru.org/images/Tasty_Pi.pdf

"That's a wrap!" - sqrt(2)

Ro ... ...

[Amigoo]

Re: Finite Fourths of Pi, showing Pi divisible by sqrt(2)
added to: http://aitnaru.org/images/Tasty_Pi.pdf
"A Pi divided against itself cannot stand!"

Numbas of the 7 sqrt(2)-nested circles ...

D = 4.0
C = 12.566370614359172953850573533118..
A = 12.566370614359172953850573533118..
SoCS = 3.5449077018110320545963349666823..
c = C/4 = 3.1415926535897932384626433832795..
d = D/sqrt(2) = 2.8284271247461900976033774484194..
c/d = 1.1107207345395915617539702475152..

D = 2.8284271247461900976033774484194..
C = 8.8857658763167324940317619801214..
A = 6.283185307179586476925286766559..
SoCS = 2.506628274631000502415765284811..
c = C/4 = 2.2214414690791831235079404950304..
d = D/sqrt(2) = 2.0
c/d = 1.1107207345395915617539702475152..

D = 2.0
C = 6.283185307179586476925286766559..
A = 3.1415926535897932384626433832795..
SoCS = 1.7724538509055160272981674833411..
c = C/4 = 1.5707963267948966192313216916398..
d = D/sqrt(2) = 1.4142135623730950488016887242097..
c/d = 1.1107207345395915617539702475152..

D = 1.4142135623730950488016887242097..
C = 4.4428829381583662470158809900607..
A = 1.5707963267948966192313216916398..
SoCS = 1.2533141373155002512078826424055..
c = C/4 = 1.1107207345395915617539702475152..
d = D/sqrt(2) = 1.0
c/d = 1.1107207345395915617539702475152..

D = 1.0
C = 3.1415926535897932384626433832795..
A = 0.78539816339744830961566084581988..
SoCS = 0.88622692545275801364908374167057..
c = C/4 = 0.78539816339744830961566084581988..
d = D/sqrt(2) = 0.70710678118654752440084436210485..
c/d = 1.1107207345395915617539702475152..

D = 0.70710678118654752440084436210485..
C = 2.2214414690791831235079404950303..
A = 0.39269908169872415480783042290994..
SoCS = 0.62665706865775012560394132120276..
c = C/4 = 0.55536036726979578087698512375759..
d = D/sqrt(2) = 0.5
c/d = 1.1107207345395915617539702475152..

D = 0.5
C = 1.5707963267948966192313216916398..
A = 0.19634954084936207740391521145497..
SoCS = 0.44311346272637900682454187083529..
c = C/4 = 0.39269908169872415480783042290995..
d = D/sqrt(2) = 0.35355339059327376220042218105242..
c/d = 1.1107207345395915617539702475152..


How to calculate sqrt(2)-relative Pi constant
(values total closer to Pi with more calculations)

Pi = 3.1415926535897932384626433832795..

\ 2 = 1.5707963267948966192313216916398..
\ 4 = 0.78539816339744830961566084581988..
\ 8 = 0.39269908169872415480783042290994..
\ 16 = 0.19634954084936207740391521145497..
\ 32 = 0.09817477042468103870195760572748..
\ 64 = 0.04908738521234051935097880286374..
\ 128 = 0.02454369260617025967548940143187..
\ 256 = 0.01227184630308512983774470071594..
\ 512 = 0.00613592315154256491887235035797..
\ 1024 = 0.00306796157577128245943617517899..
\ 2048 = 0.0015339807878856412297180875895..
\ 4096 = 0.00076699039394282061485904379474597..
\ 8192 = 0.00038349519697141030742952189737299..
\ 16384 = 0.00019174759848570515371476094868649..
\ 32768 = 0.000095873799242852576857380474343247..
\ 65536 = 0.000047936899621426288428690237171623..
\ 131072 = 0.000023968449810713144214345118585812..
\ 262144 = 0.000011984224905356572107172559292906..
\ 524288 = 0.0000059921124526782860535862796464529..
\ 1048576 = 0.0000029960562263391430267931398232265..
Total: 3.14158366542111422103356300386..

Quick analysis: Significant digits of PI, relative to sqrt(2),
are not counted in the 100s (certainly not in the 1000s).


Ro ... ...

[Amigoo]

Re: New Millennium Pi,
added to: http://aitnaru.org/images/Tasty_Pi.pdf
"Tasty sqrt(2) refreshment"

About geometric foundation of calculations:
When Diameter = 2, Circumference = 2Pi, therefore
1/4 Circumference = Pi/2 and associates directly
with one side of circle's inscribed square
which has length = sqrt(2).

They say, the regular octagon (Diameter = 2) created by this star is a clue
to calculating a New Millennium Pi, relative to sqrt(2).

They say, divide 1/4 Circumference (Pi/2) by 1,2,4.. increments of sqrt(2),
where total of line length segments drift toward Pi/2 (= 1/4 C).

They say, this octagon has direct relationship to circle's enclosing/inscribed squares.

Go ahead Calculate New Millennium Pi and be famous! (or infamous).
They say, a geometric proof is required for your infamy (dissing old Pi).

Try a calculator: https://keisan.casio.com/exec/system/1223432608
Let radius = 1, then calculate segment lengths with increments doubling (start with 4096),
then multiply segment length by increment, then divide by 4 (keeps approaching Pi/2).
Somewhere out there in the numbas a sqrt(2)-relative Pi exists (they say).

Just super curious ...

4000000000 segments
x 0.00000000157079632679489661907.. segment length
= 1.570796326794896619 07.. New Pi (18 similar digits)
= 1.570796326794896619 2313216916398.. Pi/2

Quick analysis: 16 Pi digits seem practical limit
(more digits require LARGE increase of segments)
and correlate with NASA's "15 digits":

Re: https://www.wired.com/story/how-much-pi ... ally-need/
"How Much Pi Do You Really Need?"

"1 digit is not enough, and 15 digits is good enough for everything you can imagine.
It's even good enough for NASA."

Of course, need and want are two different perspectives. After all,
Pi is free and infinite (apparently) ... and helps fatten articles about Pi.
Pi sufficient for sqrt(2) in a Cartesian Neighborhood is reasonable.


Ro ... ...

[Amigoo]

About Coffee Table Pi (nerd bites)
The "Pi has no length!" moment in math history?

Incidentally, that "Pi has no length!" is communicated in Pi's definition:
"Ratio of Circumference to Diameter". "Ratio" refers to relative length
(Circumference and Diameter have length - not numeric Pi, a ratio)

This concept of Circumference as line segments highlights why Pi is infinite!
... and "proves" that at least three categories of Pi exist:

1. Mathematical Pi - infinite decimal digits, not 0.0000000015707963..
but smaller - even smaller than 0.0000000000000000000000000000001
Mathematical Pi defines "points" (on Circumference) without length!

Who knew Mathematical Pi alone cannot be used
to define Circumference length (only point-to-point has length
and this defines Circumference as connected line segments).

2. Geometric Pi - Pi that has value in relation to sqrt(2).
Geometric Pi is essentially "mathematical Pi" because
both Pi and sqrt(2) can define sub-atomic lengths.

3. Cartesian Pi - NASA's Pi (15 digits) still "mathematical"
but not "sub-atomic"; relative to Cartesian representation.
Cartesian Pi is Pi as fat points, relative to "size matters".

Then geometry junior asks the professor
"So, what's your preferred point, fatty?!"


Postscript ...

No matter how many segments Diameter is divided into,
the complementing Pi segment (on Circumference)
is always 3.14.. times longer. Say what?!

This shouts that Pi is indeed infinite, however
in Cartesian space, sqrt(2) rules! Pi is infinite
but only to the boundaries of sqrt(2) ...
if this Pi is in Cartesian space.

New pizza term: "Pi a la Cartesia"
("Square pizza , please!")


Rod

[Amigoo]

Re: Millennium Pi Segments,
added to: http://aitnaru.org/images/Tasty_Pi.pdf
"The New Art of Pi"

"What's it for?" Timely new direction.
Hey kids! What time is it?

Code: Select all

Millennium Pi Numbas for D = 2
(angle, chord, segments, circumference)

90.0           1.414213562373095..      4  5.6568542494923804..   
45.0           0.76536686473018..       8  6.12293491784144..
22.5           0.390180644032257..     16  6.242890304516112..
11.25          0.19603428065912..      32  6.27309698109184..
5.625          0.098135348654836..     64  6.280662313909504..
2.8125         0.049082457045822..    128  6.282554501865216..
1.40625        0.024543076571446..    256  6.283027602290176..
.703125        0.01227176929832..     512  6.28314588073984..
.3515625       0.006135913525919..   1024  6.283175450541056..
.17578125      0.00306796037261..    2048  6.28318284310528..
.087890625     0.001533980637466..   4096  6.283184691060736..
.0439453125    0.000766990374929..   8192  6.283185151418368..
.02197265625   0.0003834951942..    16384  6.2831852617728..
.010986328125  0.000191747597761..  32768  6.283185283432448..

Who knew
In Cartesian space, Pi must be precisely divisible by 4
in order to host circle's enclosing and inscribed squares.

"Be there AND be square!"

Next update ...
Better visual to consider Pi as point-to-point segments,
where segment length doubles precisely from D=2 to D=4,
and begging "Whence the transcendence of PI?"

Apparent answer: Number of segments desired,
begging "Of what value is atomic measurement?"
especially in Cartesian space!

Who knew
Because the circle has an enclosing and an inscribed square,
Pi must be divisible by 8 (when squares have similar alignment,
the 4 corners and 4 midpoints are points on the circumference)
to provide the 8 equidistant points needed by the squares.

Who knew
Because angles can be divided in half precisely, Pi needs
to be divisible by more than 8 (squares can rotate
according to angle divisions - without those points
the squares cannot rotate).

"Be there AND be square!"


Yum! "PB&J Transcendence" (Pi Bites Juxtaposed)

This Pi Segments design with 128 segments per circle
indicates each circle must have 128 equidistant points
(Pi must be divisible by 128 to provide these points!).

Thus, geometric symmetry (with equidistant points
relative to circle's enclosing and inscribed squares)
suggests a necessary 4,8,16.. doubling of segments
(with decreasing segment length) to approach
the Pi ratio of circumference to diameter.

Note: These points are created by dividing
angles into two parts ... precisely!
(as suggested by the design)

"Say what?! Pi is not related to Diameter?"
Related via its enclosing/inscribed squares.


Long story short ...

Better visual to associate portion of diameter with related arc of Pi where diameter = 4.
Tip: 32 of 128 segments / Pi = 10.18.. segments (you can count them!).

In Cartesian space, where Pi relates to both circles and squares, Pi's transcendence is tempered by sqrt(2),
host of Cartesian space. This geometry shows that Pi is the ratio of arc length to side of inscribed square
(4 segments per circle). As number of segments increase, the ratio moves in the 1:1 direction (millions+).
Thus, this variable number of segments (selected) seems the direct cause of Pi's transcendence.

Right triangle (upper left) encloses 8 segments of Pi
and triangle's ratio of long side to short side = sqrt(2) + 1,
more evidence that sqrt(2) is host of this Pi Corral.

They say, some comprehension of nuclear fusion helps
to conceptualize Pi and sqrt(2) united in Cartesian space.


Speaking of "Pi Corral" ...

Design now shows two Pythagorean right triangles
with long side to short side ratio of sqrt(2) + 1,
precisely enclosing an 8-segment arc of Pi.

Relative side lengths: 1, sqrt(2)+1, sqrt(2(sqrt(2))+4)
where a = 2.4142135623730950488016887242097..
b = 1.0, c = 2.6131259297527530557132863468544..

Pythagorean confirmation: a^2 + b^2 = c^2
5.8284271247461900976033774484194.. + 1
= 6.8284271247461900976033774484194..
sqrt( = 2.6131259297527530557132863468544..

"Either sqrt(2) is transcendental or Pi is not."


"It's a wrap (ped sandwich) !" ...

A segments perspective of length of a circle's circumference indicates that
Pi's points (re: decimal digits) do not directly control this length. Each point
is the connection of two segments and every point has an opposite point
on the circumference, the pair having exact distance of circle's diameter.

Thus, only segment length change affects segmented-arc length.
Since segment is a straight line, more segments must be used
to get ever closer to the exact length of the circumference.
The variable number of segments (from 4 to infinity)
is the apparent source of Pi's transcendental essence.
Pythagorean geometry "as easy as Pi !" (they say)

In this arc of Pi (D = 4), 8/32 segments = .25(Pi)
= 0.78539816339744830961566084581988..
and creates this Pi Sandwich, aka "Pi Corral":

0.8284268537483.. triangle's hypotenuse
0.7853981633974.. .25(Pi)
0.7653668573824.. triangle's long side


Re: MPi Pythagorean design,
added to: http://aitnaru.org/images/Tasty_Pi.pdf
"9-point Pi Wedgie"


Ro ... ...

[Amigoo]

Re: MPi Pythagorean design,
added to: http://aitnaru.org/images/Tasty_Pi.pdf
"9-point Pi Wedgie"

This geometry forecasts a new Pi, integrated with sqrt(2),
the natural host of squares in Cartesian space

Discover this new constant and be famous!
(or infamous, considering Pi's sacredness)

Voted best nickname for constant: "The People's Pi",
somehow derived from "Be there AND be square!"


Re: Texas Downstep design,
added to: http://aitnaru.org/images/Tasty_Pi.pdf
"Big Texas Downstep on Little Pi Plateau"

New boot-scootin' dance step from MPi Pythagorean
where large diameter = 4, 90 degree arc = Pi

Beginning Pythagorean right triangle:
Sides = 1, sqrt(2)+1, sqrt[2(sqrt(2))+4)]
= 1.0, 2.4142135623730950488016887242097..,
2.6131259297527530557132863468544..

Shows geometric pattern for replicating (downstepping)
right triangles to complement smaller and smaller segments.

Half of hypotenuse gives value of next smaller long side,
but next smaller hypotenuse must be adjusted by
a decrease in the previous angle by half.
"It's as easy as Pi" (they say)

Tip: The 1:1 length ratio approaches
as segment length gets closer to Pi arc length
(both lengths decrease during downstep)

"Say what?!" Dunno?! Get down, turn around
and see what's new on the Cartesian plain.


Ro ... ...

[Amigoo]

Re: Texas Downstep design, "final, final"
updated in: http://aitnaru.org/images/Tasty_Pi.pdf
"Big Texas Downstep on Little Pi Plateau"
A Pi divisible by 2, then 2 ...

"Not so fast with that 'final, final' Pythagoras texted. "Where's the Texas Toothpick
The brown sword that guides the geometry from one triangle to the next smaller.
Keep exploring your Cartesian sandbox of Quadraturial persuasion. It's there!"

Been there! Done That!
('heel toe, docie doe' for hours!)


Pattern proves the next division of Pi by 2
and should replicate (downstep) to infinity.

Angular reference to infinite division of Pi by 2 ...
45, 22.5, 11.25, 5.625, 2.8125, 1.40625, 0.703125,
0.3515625, 0.17578125, 0.087890625, 0.0439453125,
0.02197265625, 0.010986328125, 0.0054931640625
x 8192 = 45


Geometry exists, but pencil gets microscopic!
And requires LOTSA segments in the circumference
to see (with a microscope) what's happenin'.

"Get down, turn around, go to town!" ...
and watch sqrt(2) stay synchronized with Pi.
Pickin' all the way to the celestial county line


About the math,
where long side / short side = sqrt(2)+1,
starting with largest right triangle (blue step):

long side = 2(sqrt(2))
= 2.8284271247461900976033774484194..
^2 = 8
short side = 2(sqrt(2)) / sqrt(2)+1
= 1.1715728752538099023966225515806..
^2 = 1.3725830020304792191729804126448..
hypotenuse = sqrt(8 + (8/(2(sqrt(2))+3))
= 3.0614674589207181738276798722432..
^2 = 9.3725830020304792191729804126448..

= 8 + 1.3725830020304792191729804126448..
Who knew Boot scootin' Pythagoras?


Segment and Pi arc length approach 1:1
in this comparison of those lengths where
Pi is divided by 2 and segments measured:

32 segments
2.8284271247461900976033774484194.. = chord
3.1415926535897932384626433832795.. = Pi
24 segments
2.2222808129794.. = chord
2.3561944901923449288469825374596.. = .75(Pi)
16 segments
1.5307337199084.. = chord
1.5707963267948966192313216916398.. = .5(Pi)
12 segments
1.1611386898623.. = chord
1.1780972450961724644234912687298.. = .375(Pi)
8 segments
0.7803614729239.. = chord
0.78539816339744830961566084581988.. = .25(Pi)
4 segments
0.3920679590009.. = chord
0.39269908169872415480783042290994.. = .125(Pi)
2 segments
0.1962710275834.. = chord
0.19634954084936207740391521145497.. = .0625(Pi)
1 segment
0.0981650863549.. = chord
0.09817477042468103870195760572748.. = .03125(Pi)


It's getting 2EZ now for seasoned geometers (if they're familiar with right triangle bisection).
Apparently, no more clues needed to compose math that downsteps largest right triangle (blue)
into smaller right triangles that escort Pi-arc / circumference-segment ratio to it's 1:1 destiny
... in Cartesian infinity.


Makes sense (to a Texas Downstepper): Those right triangles
maintain their long side to short side length ratio of sqrt(2)+1.
An Essence of singularity in Matrix of Cartesian quadrature?
Strangely, E = MC^2 comes to mind.

Long story short: http://aitnaru.org/images/Texas_Downstep.pdf
Pi is presented as a 90-degree arc (D = 4) enclosed by sqrt(2)
and Pythagorean right triangles in this Cartesian "Pi Corral".


Emerging non sequitur of geometers:
"I knew that squaring the circle was impossible,
but I didn't know that it would be so difficult!"


Ro ... ...

[Amigoo]

Re: Texas Downstep III design,
added to: http://aitnaru.org/images/Tasty_Pi.pdf
"Quizzical Quavers of Pi"

So esoteric, this rendition of the Texas Downstep,
that I'll need to review my 8 notes to comprehend it.


It's apparent that Pythagorean sqrt(2)+1 ratio (long side to short side),
as reflected in these right triangles (and in this isosceles trapezoid),
is key to the "impossible" geometric association of Pi and sqrt(2).
Go explore! and try Diameter = 4.

Why D = 4? Circumference = 4(Pi), therefore 1/4 Circumference = Pi,
each side of trapezoid associates with Pi, top line + bottom associate with 2(Pi).
Who knew?! Pythagoras?

Length ratio of bottom to top line?
sqrt(2)+1 Who knew?!


Emerging non sequitur of geometers:
"I knew that squaring the circle was impossible,
but I didn't know that it would be so difficult!"

Ro ... ...

[Amigoo]

Re: 44 Quavers of Pi design*,
added to: http://aitnaru.org/images/Tasty_Pi.pdf

* Renamed to acknowledge the abstract '44' (gold) that appears
in this geometry that highlights relationship of sqrt(2) and Pi.

Line length ratios of right triangle (upper left)
dividing Pi by 2 (Pi = 1/4 of Circumference for D = 4):

hypotenuse to inside diagonal (gold): sqrt(sqrt(2)+2)
long side to short side (red): sqrt(2)+1

Also, bottom line to top of trapezoid (green): sqrt(2)+1

Why D = 4? because Circumference = 4(Pi), thus 1/4 Circumference = Pi,
each side of trapezoid associates with Pi, top line + bottom associate with 2(Pi).
Who knew?! Pythagoras?


Emerging non sequitur of geometers:
"I knew that squaring the circle was impossible,
but I didn't know that it would be so difficult!"


Ro ... ...

[Amigoo]

Re: 44 Quavers of Pi design,
updated in: http://aitnaru.org/images/Tasty_Pi.pdf

Proves total length of sqrt(2)-defined segments
of Circumference approaches Pi-calculated Circumference
as segments double and segment lengths decrease.


About the "Pythagorean slice of Pi"
(estimated by sqrt(2) in this design):
D1 = 4 and D2 = 4(sqrt(2)), an arc
angle a = 33.75 degrees
side a = 1.7453753606131697..
side b = 2.612138822708026..
side c = 3.1415926535897932..

More on that slice of Pi
(as reported in The Quibbler)

"Sqrt(2), via two overlapping circles
where D = 4 and D = 4(sqrt(2)) an arc,
effectively defines a right triangle having
hypotenuse with length = Pi."

Is this possible?! Or just a Pi cameo?
Or does it matter ... in The Quibbler?

Must be sensationalism in The Quibbler
(actual geometry is not the a,b,c calculations).
However, the right triangle's sqrt(2)+1 ratio
(long side to short side) remains the key
to constructing this geometry.


To clarify, '44' is abstract design that appears in this geometry.
'4' is the minimum number of quavers (sides of inscribed square)
in this conceptual association of sqrt(2) and Pi (a Circumference).

The circle (D = 4, quavers = 8 ) and arc (D = 4(sqrt(2)),
quavers = 8 and 16, not all displayed) are the geometry
complementary to the sqrt(2) calculations.

The point Straight line segments in the Circumference
represent direct sqrt(2) relationship to Pi when those segments
maintain sqrt(2) increments (doubling of initial 4 quavers).

Why Because sqrt(2) - not Pi - is host of Cartesian space
and best describes (?) Pi's relationship to Diameter.

"Why 'quavers'?" You be asking about eighths of Pi?
(design adapted for this quizzical quibbling in The Quibbler)
And for quizzical quibbling, a subtle slice of Pythagorean Pi
(off the menu, with complementary delivery fee).

Geometric Haiku, oft perceived on an eighth note?
Myth until proven otherwise: Sqrt(2) knows length of Pi,
gracias a Pythagorean Theorem (per: The Quibbler).


Square Rooted 2s of Segmented Pi
If you're already cipherin' smaller segments of Circumference,
these patterns associate with 8 and 16 segments respectively:
( SoIS = Side of Inscribed Square )

sqrt(sqrt(2)+2)
= 1.8477590650225735122563663787936..
sqrt(sqrt(sqrt(2)+2)+2)
= 1.9615705608064608982523644722685..

Example 1: D = 4 with 8 segments,
SoIS / (sqrt(sqrt(2)+2)) = length of 1 of 8 segments.

4/sqrt(2) / sqrt(sqrt(2)+2)
= 1.5307337294603590869138399361216..
x 8 = 12.245869835682872695310719488973..
C = 12.566370614359172953850573533118.. 4(Pi)

Example 2a: D = 4(sqrt(2)) with 8 segments,
SoIS / sqrt(sqrt(2)+2) = length of 1 of 8 segments:

4 / sqrt(sqrt(2)+2)
= 2.1647844005847879375988928214656..
x 8 = 17.318275204678303500791142571725..
C = 17.771531752633464988063523960243.. 4(sqrt(2))(Pi)

Example 2b: D = 4(sqrt(2)) with 16 segments,
length of 1 of 8 segments / sqrt(sqrt(sqrt(2)+2)+2)
= length of 1 of 16 segments:

2.1647844005847879375988928214656..
/ sqrt(sqrt(sqrt(2)+2)+2)
= 1.1035975171317720493438302546775..
x 16 = 17.65756027410835278950128407484..
C = 17.771531752633464988063523960243.. 4(sqrt(2))(Pi)


Segmentation Salience of Circumstantial Pi
Review of respective arcs of Circumference of two Diameters
associated by sqrt(2) proves Pi is indeed divisible by sqrt(2):
(SoIS = Side of Inscribed Square)

D1 = 4
C1 = 4(Pi)
= 12.566370614359172953850573533118..
/ 4 = 3.1415926535897932384626433832795.. C1/4, SoIS, 2 of 8 segments

D2 = 4(2sqrt(Pi))
C2 = Pi(4(2sqrt(Pi)))
= 17.771531752633464988063523960243..
/ 4 = 4.4428829381583662470158809900608.. C2/4, SoIS, 4 of 16 segments

4.4428829381583662470158809900608.. C2/4
/ 3.1415926535897932384626433832795.. C1/4
= 1.4142135623730950488016887242097.. sqrt(2)


Emerging non sequitur of geometers:
"I knew that squaring the circle was impossible,
but I didn't know that it would be so difficult!"

Ro ... ...

[Amigoo]

Re: Eternity of Change design,
added to: http://aitnaru.org/images/Tasty_Pi.pdf
(w/ NSEW alignment for sqrt(2) association)

"NSEW? Say What?!"
Think "ïnscribed square", sqrt(2) foundation.

More exploration of sqrt(2) in the Pi Corral,
with sqrt(2)+1 defining relationship of the two circles:

D = 4, C = 12.566370614359172953850573533118..
TSeg = 12.245869835682872695310719488972..
C/TSeg = 1.0261721529770308888714677808729

D = 1.656854249492380195206.. = 4/(sqrt(2)+1)
C = 5.2051611382742920342105788487618..
TSeg = 5.1717796650761424..
C/TSeg = 1.0064545427995641747492963456598..

Ratios indicate total segment length approaches
Pi-calculated circumference as total segments increase
with complementing decrease in segment length.


"Change is eternal ... until it's not."
Who knew?! The Book of Change?

("until it's not" alludes to those
who choose not eternal life)


Ro ... ...

[Amigoo]

Re: Salience of Pi design,
added to: http://aitnaru.org/images/Tasty_Pi.pdf
"A Pi divided against itself is transcendental."

Concentricity for the next Squared Circles Soirée
with three circles squared, gracias a sqrt(2);
concentricity that proves Pi is divisible by 2
... thus by sqrt(2).

The two triangles definitive of Quadrature (right, scalene)
are clearly identified in this Cartesian composition.

Tip: The right triangle's hypotenuse/long side ratio,
2/sqrt(Pi), is ultimate proof that a circle is squared
(a ratio that defines both circle and its square).

2/sqrt(Pi) = 1.1283791670955125738961589031216..
4/Pi = 1.2732395447351626861510701069801..
= (2/sqrt(Pi))^2

Ro ... ...

[Amigoo]

Re: Equidistant Points of Pi design,
added to: http://aitnaru.org/images/Tasty_Pi.pdf
(a study of those scalene triangles, overlapped)

Q: What's the point?
A: What's the point!
Q: ?
A: !

The geometry of Quadrature for one circle
applies to all circles, especially to circles nested
gracias a sqrt(2).

Geometer's secret ...
An irregular pentagon is the premier objet d'art
in all four of these squared circles, an object
hosting Quadrature's proven impossibility.

"A Pi divided against itself is transcendental."


Ro ... ...

[Amigoo]

Re: Geometric Points of Pi design,
added to: http://aitnaru.org/images/Tasty_Pi.pdf

Q: ?
A: !

They say, further simplification of "impossible" Quadrature
would undoubtedly beg a metaphysical "What's the point?!"

Ro ... ...

[Amigoo]

Re: Geometric Points of Pi^77 design,
added to: http://aitnaru.org/images/Tasty_Pi.pdf

"The bear went over the mountain to see what he could see.
And all that he could see was "impossible" Quadrature ...
identified by a certain right and scalene triangle."

Suddenly, fractalicious describes a potential
of Quadrature's "impossible" geometry.

Who knew
"Impossible" Quadrature even wafts Biblical
with its "seventy times and seven" (UB, 139:2.5)

Naturally, any Geometry Junior of Quadraturial persuasion
could see from simplification that a circle-squaring right triangle
(defines both circle and square) can be rotated 45 degrees twice
to create two more circles similarly squared (plus more).

Those red '7'? Dunno! The days they rest?

Ro ... ...

[Amigoo]

Re: Arc! Arc! design,
added to: http://aitnaru.org/images/Tasty_Pi.pdf

A geometric Pi sandwich, aka "Pi Corral",
where D = 4, (2(sqrt(2)))-2, (2(sqrt(2)))-2 / sqrt(2)
and arc length = Pi.

(2(sqrt(2)))-2
= 0.8284271247461900976033774484194..
(2(sqrt(2)))-2 / sqrt(2)
= 0.5857864376269049511983112757903..

"Either sqrt(2) is transcendental or Pi is not!"

Analysis: Mathematical Pi is inherently transcendental -
Geometric Pi is transcendentally constrained by sqrt(2).
Then sqrt(2) commented "Nice design, but get real!" ...
sqrt(2) has its own infinity in Cartesian space.


Translation for non-geometers ...
The arc whose length is Pi is enclosed in this sqrt(2)-defined isosceles right triangle
which prevents expansion of Pi in any direction. That is, Pi's decimal digits
are limited by sqrt(2) and cannot be mo bettah.

How a Hawaiian dude might compare his date to a friend's date:
"Dis beach mo bettah."

Postscript ...
Sqrt(2) is such a showboat! Second arc length = Pi/2
(even Geometry Juniors can see this!)

Pi / 2(sqrt(2)) = 1.1107207345395915617539702475152..
reveals the constant that can be multiplied by the circle's
Side of Inscribed Square (SoIS), then by 4
to obtain the circle's Circumference:

For Diameter = 4, SoIS = 4/sqrt(2)
= 2.8284271247461900976033774484194..
x 1.1107207345395915617539702475152..
= 3.1415926535897932384626433832796..
x 4 = 12.566370614359172953850573533118..

C = Pi(D) = Pi(4)
= 12.566370614359172953850573533118..

SoIS effectively highlights that sqrt(2)
is the host of this Cartesian space.


Ro ... ...

[Amigoo]

Re: !crA !crA design,
added to: http://aitnaru.org/images/Tasty_Pi.pdf

More sqrt(2) showboatin'.
Circumferences have fixed length (C = 8, C = 4)
and the squares reflect the Pi values. HCIT

Both concepts (Pi-flavored arcs in Arc! Arc! design
and Pi-flavored squares in !crA !crA design,
positioned for contrast as dual presentation.

"Which is which?"
Sqrt(2) says they're conceptually identical in Cartesian space.
"What about the two missing green bubbles in !crA? !crA"
Pi in the sky?

Ro ... ...

[Amigoo]

Re: Arc! !crA design,
added to: http://aitnaru.org/images/Tasty_Pi.pdf

Long story short ...

Whether Pi reflects as circle or as square,
sqrt(2) is host of this Cartesian space
where C = 4(Pi), C = 6

Ro ... ...

[Amigoo]

Re: Arc! Arc! de Triangles design,
added to: http://aitnaru.org/images/Tasty_Pi.pdf

Total segments length approaches Pi arc length as segments increase
relative to powers of 2 and decrease in individual length.

Wisdom: A million sqrt(2) mini bites of Pi do not comprise one Pi,
but you'll think you've eaten the whole Pi. Bon Appétit

How to calculate sqrt(2)-relative segments of Pi
(aka "what geometry nerds eat for breakfast")

Cartesian composition where large Diameter = 4
and 1/4 of Circumference = Pi (provides starting
values for the right triangle in the pattern):

a = (2(sqrt(2)) - 2) / sqrt(2)
a = 0.5857864376269049511983112757903..
^2 = 0.34314575050761980479324510316121..

b = sqrt(2)
b = 1.4142135623730950488016887242097..
^2 = 2

c = sqrt(a^2 + b^2) (Pythagorean Theorem)
= sqrt(2.3431457505076198047932451031612..)
c = 1.5307337294603590869138399361216..

The rest is a piece of Pi ! (they say)


About the "predictable geometry" ...

Tip 1: Half the hypotenuse in the beginning triangle
becomes the long side in the downstep triangle;
half of angle A becomes angle A in the downstep;
beginning angle A = 22.5 degrees.

Tip 2: Multiply downstep side c by 4 (next power of 2),
then by 4 to calculate total segments length relative to Pi.

Tip 3: A computer can be programmed to downstep
to tiny segments whose total length gets very close to Pi.
This is the Tasty Pi relative to sqrt(2)

Tip 4: This a new way of calculating Pi,
using only sqrt(2) values!


Decreasing sqrt(2) segments lead to Pi
(calculations using trigonometry calculator
where angle A = 45 /2 /2 ..
side b = previous side c /2 )

A = 22.5
b = 1.4142135623730950488016887242097..
c = 1.5307337268917396..
x 2 = 3.0614674537834792..

A = 11.25
b = 0.7653668634458698..
c = 0.7803612832416639..
x 4 = 3.1214451329666556..

A = 5.625
b = 0.39018064162083195..
c = 0.3920685572664755..
x 8 = 3.136548458131804..

A = 2.8125
b = 0.19603427863323775..
c = 0.196270686637696..
x 16 = 3.140330986203136..

A = 1.40625
b = 0.098135343318848..
c = 0.09816490543420567..
x 32 = 3.14127697389458144..

A = 0.703125
b = 0.049082452717102835..
c = 0.04908614609652455..
x 64 = 3.1415133501775712..

A = 0.3515625
b = 0.024543073048262275..
c = 0.024543535073374..
x 128 = 3.141572489391872..

A = 0.17578125
b = 0.012271767536687..
c = 0.012271825290234..
x 256 = 3.141587274299904..

A = 0.087890625
b = 0.006135912645117..
c = 0.0061359198643231..
x 512 = 3.1415909705334272..

A = 0.0439453125
b = 0.00306795993216155..
c = 0.0030679608345627..
x 1024 = 3.1415918945922048..

A = 0.02197265625
b = 0.00153398041728135..
c = 0.0015339805300815..
x 2048 = 3.141592125606912..

A = 0.010986328125
b = 0.00076699026504075..
c = 0.00076699027914077..
x 4096 = 3.14159218336059392..

A = 0.0054931640625
b = 0.000383495139570385..
c = 0.00038349514133289..
x 8192 = 3.14159219779903488..

A = 0.00274658203125
b = 0.000191747570666445..
c = 0.00038349514177352..
x 16384 = 3.14159220140867584..

Pi = 3.1415926535897932384626433..


This calculation emphasizes why so few decimal digits
of Pi are needed, especially for Cartesian exploration
where sqrt(2) is host of Cartesian space

Re: https://www.jpl.nasa.gov/edu/news/2016/ ... ally-need/
"How Many Decimals of Pi Do We Really Need?"

"For JPL's highest accuracy calculations, for interplanetary navigation,
we use 3.141592653589793." (i.e., traditional Pi is abstract - not real)


For a proper slice of Pi (they say)
when Diameter = 4, Circumference = 4(Pi)

For sqrt(2) association to segments of Pi,
Pi digits must be evenly divisible by 4
(relates to circle's inscribed square)

3141592
/ 4 = 785398
31415926535897932
/ 4 = 7853981633974483
31415926535897932384
/ 4 = 7853981633974483096
314159265358979323846264
/ 4 = 78539816339744830961566
31415926535897932384626433832
/ 4 = 7853981633974483096156608458
31415926535897932384626433832795028
/ 4 = 7853981633974483096156608458198757
314159265358979323846264338327950288
/ 4 = 78539816339744830961566084581987572
3141592653589793238462643383279502884
/ 4 = 785398163397448309615660845819875721
314159265358979323846264338327950288419716
/ 4 = 78539816339744830961566084581987572104929
3141592653589793238462643383279502884197169399375105820
/ 4 = 785398163397448309615660845819875721049292349843776455

"What kind of genius math is that ?!"
The same where 1/4 of Circumference = one Pi.


sqrt(2) Circumference segments
for 16 decimal dgits of Pi where D = 4,
C = 12.5663706143591728 = 4(Pi)

3.1415926535897932 (divisible by 4)
/ 10 = 0.00000000031415926535897932

10 billion segments per 1/4 Circumference
with length = 0.00000000031415926535897932
total 3.1415926535897932


Long story short ...
Sides of beginning right triangle
(all three are sqrt(2) values)

1.4142135623730950488016887242097..
- 1 = .4142135623730950488016887242097..
x 2 = 0.8284271247461900976033774484194..
/ 1.4142135623730950488016887242097..
= 0.5857864376269049511983112757903.. = a

2(1.4142135623730950488016887242097..)
= 2.8284271247461900976033774484194..
/ 2 = 1.4142135623730950488016887242097.. = b

Pythagorean Theorem calculates side c

0.5857864376269049511983112757903.. = a
^2 = 0.34314575050761980479324510316121..
1.4142135623730950488016887242097.. = b
^2 = 2
sqrt(2.34314575050761980479324510316121..)
= 1.5307337294603590869138399361216.. = c


Ro ... ...

[Amigoo]

Re: Three Pennies Solution design,
added to: http://aitnaru.org/images/Tasty_Pi.pdf
... then replaced by Three Pennies PivoT

33 degree angle (11+11+11), trisected and simplified(?)

Create the geometric proof and be famous ...
or infamous (trisection was proven impossible in 1837).

Given: 33 degree angle with arbitrary circle near the vertex.
Construct geometry to define location of 3 pennies on an arc,
circles that confirm(?) the trisection of that angle.

How to trisect the angle? Ask a real geometer ...
since trisection is confirmed only by a geometric proof -
not by a visually convincing design.


Postscript days later (30 degree angle) ...
Design "reverse engineered" to display geometry of a trisected angle
if three pennies (circles) on an arc could confirm angle trisection.

Geometry is not as easy as it looks!
The detail requires geometric "deciphering".


Postscript weeks later (20 degree angle) ...
Three Pennies PivoTA design added.

Geometry is less easier than it looks!
Just draw a circle in the vertex to start.
Bon Appétit, geometrically speaking.

The circles may be geometric red herring
since straight lines rule and the upper arc
is then drawn down below to reveal trisection
(so they say).


Ro ... ...

[Amigoo]

Re: Py Day design,
added to: http://aitnaru.org/images/Tasty_Pi.pdf
along with http://aitnaru.org/images/Py_Day.pdf

"Irrational balance in Cartesian space"
where a right triangle squares the circle.

"Pi that goes around comes around
... in a sqrt(2) spiral."

"Lines and triangles and squares, oh my!"
Geometry which squares the circle squares its sqrt(2) sibling.
Suggests "Either sqrt(2) is transcendental or Pi is not."

Tip: A similar quadrilateral of Quadraturial
essence exists in every squared circle.


About Py Day (Pythagorean perspective)
c = 2.0 hypotenuse, circle's diameter
b = 1.7724538509055160272981674833411.. sqrt(Pi), long side
a = 0.92650275035220848584275966758914.. sqrt(4-Pi), short side
a^2 + b^2 = c^2 where (4-Pi) + Pi = 4

Geometer's secret (squares not shown):
Each side of the circle-squaring right triangle
is side of a square with areas Pi + (4-Pi) = 4
(long side, short side, hypotenuse).

Regarding sides as diagonals of squares:
If the area of the circle's inscribed square = 2,
combined areas of squares representing Pi = 2,
then combined areas = a rational number*
and the circle's area (Pi) = area of a square
whose sides have length = sqrt(Pi).

* Re: https://byjus.com/maths/irrational-numbers/
"addition or multiplication of two irrational numbers may be rational"

A/Pi = r^2, thus D =2 Who knew?

Is Pi a Schrödinger's Cat (irrational and rational),
depending on its mathematical operations
Is the circle squared, depending on ...


Ro ... ...

[Amigoo]

Re: Py Day II design,
added to: http://aitnaru.org/images/Tasty_Pi.pdf
along with http://aitnaru.org/images/Py_Day.pdf

Circle's diameter = 2,
octagon's perimeter = 8.
Who knew?

"Pi Corral" comes to mind.


Re: Py Day III design,

Circle and one isosceles right triangle defines other triangle;
both triangles define the circle. And circle with either triangle
defines the circle-squaring right triangle. HCIT?

"Pi Corral" comes to mind ... again.

OMG! A certain scalene triangle is the root of Quadrature,
defining circle, all of its squares, and enclosing octagon.
Tip: Save some for Leftover Py Day (Julian 314).


[D = Diameter, SoCS = Side of Circle's Square]
In Quadrature's marriage of sqrt(Pi) and sqrt(2),
D/SoCS and SoCS/D ratios remain identical
as values increase/decrease by sqrt(2).

Example:
sqrt(Pi)/2 = 0.88622692545275801364908374167055..
2/sqrt(Pi) = 1.1283791670955125738961589031215..
/ (1.1283791670955125738961589031215..)^2
= 0.88622692545275801364908374167055..

S= 3.5449077018110320545963349666822.. ^2= 4(Pi)
D= 4.0 ^2= 16

S= 2.506628274631000502415765284811.. ^2= 2(Pi)
D= 2.8284271247461900976033774484194.. ^2= 8

S= 1.7724538509055160272981674833411.. ^2= Pi
D= 2.0 ^2= 4

S= 1.2533141373155002512078826424055.. ^2= Pi/2
D= 1.4142135623730950488016887242097.. ^2= 2

S= 0.88622692545275801364908374167057.. ^2= Pi/4
D= 1.0 ^2= 1

S= 0.62665706865775012560394132120276.. ^2= Pi/8
D= 0.70710678118654752440084436210485.. ^2= .5

S= 0.44311346272637900682454187083528.. ^2= Pi/16
D= 0.5 ^2= .25

Values (Diameter, Area, SoCS) increase/decrease by sqrt(2),
proving Quadrature's Pi is precisely divisible by increments of 2.

D = 4
A = 3.1415926535897932384626433832796.. Pi
S = 3.5449077018110320545963349666823.. 2(sqrt(Pi))

D = 2.8284271247461900976033774484194.. 2(sqrt(2))
A = 2.2214414690791831235079404950303.. Pi/sqrt(2)
S = 2.5066282746310005024157652848111.. sqrt(Pi)(sqrt(2))

D = 2
A = 1.5707963267948966192313216916398.. Pi/2
S = 1.7724538509055160272981674833411.. sqrt(Pi)

D = 1.4142135623730950488016887242097.. sqrt(2)
A = 1.1107207345395915617539702475152.. Pi/sqrt(2)
S = 1.2533141373155002512078826424055.. (sqrt(Pi)(sqrt(2))/2

D = 1
A = 0.78539816339744830961566084581988.. Pi/4
S = 0.88622692545275801364908374167058.. sqrt(Pi)/2

Good T-shirt sub-titles for Py Day IV design:

Morbus Cyclometricus, in deed!

Lines and triangles and squares, oh my!

I knew that Quadrature was impossible,
but I didn't know that it was so difficult!

Quadrature is like Artificial Intelligence
and Quadrature was proven "impossible".


Re: Py Day V, VI designs
"Juxtapositional scalenity, Quadraturially asymmetric"
(if TMI Py just save for Leftover Py Day)

This geometry proves that sqrt(Pi) and sqrt(2) are directly related,
suggesting mathematicians need to re-evaluate the Pi calculation. However,
"squaring the circle" still seems impossible despite the Cartesian reality that
squared circles exist ... somehow. Go figure! (and be famous or infamous)

Re: https://www.perplexity.ai/

Q: How does sqrt(2) relate to sqrt(Pi) in geometry?

"In geometry, the relationship between sqrt(2) and sqrt(Pi) is not direct. However, there are interesting mathematical connections between these two numbers. sqrt(2) is related to the diagonal of a square with side length 1, while sqrt(Pi) is related to the radius of a circle. One interesting aspect is that sqrt(2) appears in quadrilaterals, while Pi is more commonly associated with circles. The connection between these two numbers in geometry lies in their fundamental properties within different geometric shapes rather than a direct relationship between them."

Interestingly, squares are inherent in circles and vice versa, suggesting
direct relationship of circles and squares via the sqrt(2) Cartesian host.


Wisdom: In this New Py Day millennium,
Quadrature is all about thinking outside the box
of anything claimed to be "impossible".


Rod

[Amigoo]

Who knew? Texas Sqrt(2) Step geometry anticipates
the uniting of heaven (circle) and circle's square (earth).

In this sqrt(2)-style calculation of Pi and Cartesian Quadrature of nested circles,
2/sqrt(Pi) remains the same ratio of every Diameter/SoCS (Side of Circle's Square)
and suggests that Quadrature assumes this ratio as diameters change by sqrt(2).
Also, this Pi calculation confirms direct relationship of Pi to sqrt(2).

Suddenly, it's obvious that Pi is transcendental because its formulas
are inherently transcendental, but Perplexity shouts "Au contraire!" (see below).
This explanation hints that all formulas that calculate Pi's digits incorporate
Pi's inherent mathematical nature, effectively proving that the formulas
are still inherently transcendental, IMO (then Perplexity agrees!).

Re: https://myscienceshop.com/product/book/82426
"Sciencia: Mathematics, Physics, Chemistry, Biology and Astronomy for All"

p.14, sqrt(2)-style calculation of Pi for n=6 (2^6 = 64)
2^n x sqrt(2-(sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2)))))))

1.4142135623730950488016887242097..
+ 2 = 3.4142135623730950488016887242097..
sqrt = 1.8477590650225735122563663787936..
+ 2 = 3.8477590650225735122563663787936..
sqrt = 1.9615705608064608982523644722685..
+ 2 =3.9615705608064608982523644722685..
sqrt = 1.990369453344393772489673906219..
+ 2 = 3.990369453344393772489673906219..
sqrt = 1.9975909124103447854295432095182..

2 - 1.9975909124103447854295432095182..
= 0.0024090875896552145704567904818..
sqrt = 0.04908245704582457606346905891858..
x 64 = 3.1412772509327728680620197707891..

It's time to write the song
and choreograph the dance!


Re: https://www.perplexity.ai/

Q: Are the formulas that calculate Pi the reason that Pi is transcendental?

"... In summary, Pi is transcendental due to its inherent mathematical nature, not because of the formulas used to compute its digits.

The key points are:

- Pi is an irrational number, meaning it cannot be expressed exactly as a ratio of two integers. This was proven by Johann Lambert in 1761.
- Pi is also a transcendental number, meaning it is not the root of any polynomial equation with rational coefficients. This was proven by Ferdinand von Lindemann in 1882.
- The transcendence of Pi implies that it is impossible to 'square the circle' using only a compass and straightedge, as this would require constructing a square with the same area as a given circle, which is not possible with a transcendental number like Pi.
- The formulas and methods used to calculate Pi digits do not determine its mathematical properties. Pi's irrationality and transcendence are fundamental characteristics, independent of how it is calculated.

In summary, Pi is transcendental due to its inherent mathematical nature, not because of the formulas used to compute its digits. The ability to calculate Pi to trillions of digits is an impressive computational feat, but does not change the underlying reason why Pi is transcendental."

(Q: What is the formula for calculating Pi using only 2 and sqrt(2)?

"The formulas used to calculate Pi do not determine whether Pi is transcendental. Pi is transcendental because it cannot be the root of any polynomial equation with rational coefficients, as proven by Ferdinand von Lindemann in 1882 ... Pi's transcendence is a fundamental characteristic proven through rigorous mathematical proofs, not a consequence of the computational methods used to find its digits."

Q: Since formulas that calculate Pi's digits reflect Pi's inherent mathematical nature, are these formulas inherently transcendental?

"... the formulas used to calculate Pi's digits are inherently transcendental, reflecting the fundamental mathematical properties of this famous constant."

I seem to recall that Pi did not exist until a series of calculations
were createsd, then followed by a generalized formula.


Re: https://www.perplexity.ai/

Q: How does a formula relate to a series of calculations?

"... In summary, a formula provides a concise, generalized way to represent a series of calculations,
allowing you to easily compute a desired quantity by plugging in the relevant variables."

Apparently, Pi's digits were first a series of calculations which were then generalized as a formula.
If the calculations were inherently transcendental, then the formula would inherit that essence


Rod