Paradise Trinity Day

[Amigoo]

Re: Jointed Quadrature 62.403 design

The long-forgotten myth of the Jointed Quadrature (specifically, Jointed Quadrature 62.403) is quite entertaining! This apparent mathematical prototype of Pythagoras (or crafted during his era) had been misplaced and, being perceived a new hand combat weapon for a Greek infantryman, was "returned" to a general, then in transit to regional headquarters.

The general, not wishing to reveal that he had never seen such a "weapon", thanked the peasant, but said no more. Well, the army spent few days evaluating this device and even enjoyed much merriment over the benign design. The two-jointed wooden Quadrature soon became effective kindling for a roaring fire.

Lost to history was the Quadrature's circle-squaring geometric simplicity. For when this instrument was unfolded and the tip of the shortest segment (yellow) was placed at the center of the longest segment (red), a right triangle was effectively created, having a precise vertex of 62.403 degrees. Such a right triangle was an impressive approximation of the theorized triangle that effectively - and precisely - squares every circle!

Dimensions of the Jointed Quadrature (design specifications* - not prototype):

Segment 1 (red) - 1.7724538509055160272981674833411.. (sqrt(P))
Segment 2 (green) - 1.5707963267948966192313216916398.. (Pi/2)
Segment 3 (yellow) - 0.88622692545275801364908374167057.. (sqrt(Pi)/2)

* "Pi" is more recent terminology for these values.

Rod ...

[Amigoo]

Re: Jointed Quadrature 62.403 design

Two more dark blue lines were added to note a comment in the Specifications:
these lines are the geometric "lock" that proves the geometry of the Quadrature.

"Say what?" Lock your joints when running in squared circles!

Rod ... ... (driving with locked joints)
Not to worry! It's on Cruise Control.

[Amigoo]

Re: Jointed Quadrature 62.403 design

Colored lines (arcs) now identify points in this Cartesian neighborhood
where whiplash might occur when you're driving with locked joints.

These 90-degree turns on the circumference of a squared circle are
always surprising - you'll roll your eyes and turn your head!

Rod

[Amigoo]

Re: PythagoRooter (new design concept)

"PythagoRooter" ( my nickname for this since its name was not recorded on the Ionian parchment fragment) may represent design specifications for a geometry square root table. The table seems to provide a pattern by which it can be expanded to "calculate" the square root of many line lengths (geometry - not math). I plan to add a few more convincing entries this week, then briefly describe its apparent operation.

Tempting clues to whet one's appetite for geometry "revelation" (perhaps):
sqrt(2) is represented as a square in the lower left of this table with isosceles right triangles for reference line lengths.

Rod

[Amigoo]

Re: PythagoRooter (new design concept)

After this late night "first draft" of the concept, I might be convinced that this is a device for scraping scales off fish ... with no other functionality! Obviously, it shows line lengths and square roots, but I'll need to explore the geometry at length (fun pun) to confirm that it associates ("calculates") a length with its square root.

Rod

[Amigoo]

Re: PythagoRooter (new design concept)

Whatever the purpose of the supposed geometry square root table,
random line lengths do not function in this table. "PythagoRooter" is
redefined to "a squared circle geometer who roots for Pythagoras".

Rod (running in squared circles with locked joints)

[Amigoo]

Re: PythagoRooters (fan club and
squared circle PR Reference design)

The hours spent exploring the "squared circles table" finally reminded me
that the 27.59711263569060445173220475233.. angle (half of the ASR)
is circle-squaring precise!

Diameters = 2(sqrt(Pi)), 2, 1
Squares = Pi, sqrt(Pi), sqrt(Pi)/2

Rod

[Amigoo]

Re: PR Reference design (another circle added).
The golden circle and its square is a good example of the
circle-squaring ability of this unique right triangle:

Side of circle's square = Pi - sqrt(Pi)

= 3.1415926535897932384626433832795..
- 1.7724538509055160272981674833411..
= 1.3691388026842772111644758999385..

A = (1.3691388026842772111644758999385..)^2
= 1.874541061015736166727498418917..

To calculate circle (diameter) of the square,
with restated formula for radius of a circle:

A = Pi(r^2)
A/Pi = r^2
sqrt(A/Pi) = r
2r = Diameter

sqrt(A/Pi) = r
sqrt(1.874541061015736166727498418917.. / Pi)
= sqrt(0.5966849517787611838663084165969..)
= 0.77245385090551602729816748334094..
= radius

2r = 1.5449077018110320545963349666819..
= Diameter

Rod

[Amigoo]

Re: PR Reference design (another circle added ... again).

Squared circle geometry is so loquacious and needs precise boundaries!
But you know it's time to put lipstick on the thing (end its development)
... when the Smile of Pythagoras appears!

Rod

[Amigoo]

Re: PR 2(sqrt(1/Pi)) design (geometer's perspective)
(DOI = Diameter Of Interest)

Dimensions are listed for circles from left to right.
Note the cross-correlation of the two sets
(geometric proof of squared circles?)

Rod ... ...

[Amigoo]

Re: PR 1128 design
About Simple Simon's "pie beyond compare" ...
Is this math the foundation of the new recipe

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

Proportions (long_side:hypotenuse) of right triangles
in the two sets of squared circles in PR 1128 geometry:
sqrt(Pi) : 2 ~ 1 : 2(sqrt(1/Pi))

Means/Extremes math property:
sqrt(Pi) : 2 ~ 1 : 2(sqrt(1/Pi))
sqrt(Pi) x 2(sqrt(1/Pi)) = 2
1.7724538509055160272981674833411..
x 1.1283791670955125738961589031215.. = 2

So, what happens to "irrational" and "transcendental"?
A previously unknown quality of the number 2?

Rod

[Amigoo]

Re: PR 1128 design (all the numbers)
About Simple Simon's "pie beyond compare" ...
Is this math the foundation of the new recipe

Every geometer knows this:
Pi/2 = 1.5707963267948966192313216916398..
sqrt(Pi) = 1.7724538509055160272981674833411..
sqrt(Pi)/2 = 0.88622692545275801364908374167057..
2(sqrt(1/Pi)) = 1.1283791670955125738961589031215..

I. Proportions between the two-circle sets:

Proportions (long_side : hypotenuse) of right triangles
in two sets of squared circles in PR 1128 geometry:
sqrt(Pi) : 2 ~ 1 : 2(sqrt(1/Pi))

Means/Extremes math property:
sqrt(Pi) : 2 ~ 1 : 2(sqrt(1/Pi))
sqrt(Pi) x 2(sqrt(1/Pi)) = 2
1.7724538509055160272981674833411..
x 1.1283791670955125738961589031215.. = 2

II. Proportions within each two-circle set:

Pi/2 : sqrt(Pi) ~ sqrt(Pi) : 2
2 x 1.5707963267948966192313216916398..
= 1.7724538509055160272981674833411..
x 1.7724538509055160272981674833411..
Pi = Pi

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

Rod

[Amigoo]

Re: PR 1128 design
Separate page is included for all the numbers (proportions).

Must be new math! All this just to learn that:
Pi = Pi, Pi/2 = Pi/2, 2 = 2, 1 = 1

BTW: sqrt(2) < sqrt(Pi) < 2 is the "Pi corral"

Rod ... ...

[Amigoo]

Re: Triangular Pi design
(addendum for PR 1128 geometry)

Isosceles right triangle whose hypotenuse = sqrt(Pi)
has side length = sqrt(Pi)/2 and squares two circles:
1) sqrt(Pi) is square of circle D = 2;
2) sqrt(Pi)/2 is square of circle D = sqrt(2)
(side of square inscribed in circle D = 2)

sqrt(Pi) / sqrt(2) = sqrt(Pi)/2
(1.7724538509055160272981674833411..
/ 1.4142135623730950488016887242097..)
= 1.2533141373155002512078826424061..

Rod

[Amigoo]

Re: Triangular Pi 1010 design (more geometry development and "1010" added to name)

The Triangular Pi design seemed to resist closure even though it had an obvious geometric signature: the isosceles right triangle whose hypotenuse squares the large circle (D = 2) and whose side of triangle squares the circle whose diameter (D = sqrt(2)) has length equal to the side of a square inscribed in the large circle.

Apparently, there was more to learn about this juxtaposition of objects (three squared circles with arcs defining the line that is the square of that circle). Upon awakening from an afternoon nap, "10:10" (the mental sound more than the time stamp) encouraged more exploration. Besides, I was not conscious of any direct association between 10:10 (Midwayer time stamp) and the geometry.

Today's internet search offered some possibilities: 10-10 "transmission completed", 1010 "something asked about may be revealed", and 10-10 "fight in progress" (Old Pi vs. New Pi?). What might be revealed

Interim guess: a New Pi constant will relate to the inscribed squares (or isoceles right triangles therein) of squared circles.

Rod

[Amigoo]

Re: Progression of Pi (new design concept)
Crunching numbers for breakfast ...
2(sqrt(1/Pi)) = 1.1283791670955125738961589031215..

A fascinating geometric progression:
Pi/2, sqrt(Pi), 2, 4(sqrt(1/Pi)) =

1.5707963267948966192313216916398..,
1.7724538509055160272981674833411..,
2.0,
2.2567583341910251477923178062431..

Since each value is multiplied by 2(sqrt(1/Pi)) to get
the next higher value, what happens in the concentric
geometry relative to the known decimal digits of Pi?

Assuming that the number 2 remains in this progression
and that the other values remain relative to the decimal
digits of Pi, does number 2 corral transcendental Pi?

Rod ... ...

[Amigoo]

Re: Progression of Pi design
"increments of 2(sqrt(Pi))"

Red right triangle displays a convincing pattern whose segments*
relate to the values in this progression: Pi/2, sqrt(Pi), 2, 4(sqrt(1/Pi))
(diameter of golden circle = 2)

* in each right triangle: long side and hypotenuse

Rod

[Amigoo]

Re: Progression of Pi design (more intrigue)
"increments of 2(sqrt(Pi))"

Notice the 1,2,4,8 progression of these numbers,
identified as "geometric anchors" (no decimal digits):

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

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

2.2567583341910251477923178062431.. = 2(2(sqrt(1/Pi)))
2.5464790894703253723021402139602..

3.1415926535897932384626433832795.. = 2(Pi/2)
3.5449077018110320545963349666809.. = 2(sqrt(Pi)
4.0 ........................................... = 2(2)

4.513516668382050295584635612484... = 4(2(sqrt(1/Pi)))
5.0929581789406507446042804279155..

6.2831853071795864769252867665651.. = 4(Pi/2)
7.089815403622064109192669933368... = 4(sqrt(Pi)
8.0 ........................................... = 4(2)

Rod ... ...

[Amigoo]

Re: Progression of 1248 design
"DOI = 1, 2, 4, 8"

This geometry highlights the 1,2,4,8 Diameters of Interest
and progression pattern of these "geometric anchors".

Rod

[Amigoo]

Re: Progression of Pi design
(regarding the 1,2,4,8 progression of numbers)

1.23370055013616982735431137498.. is another increment:

2.5464790894703253723021402139602..
x 1.23370055013616982735431137498.. =
3.1415926535897932384626433832795.. = 2(Pi/2)

5.0929581789406507446042804279155..
x 1.23370055013616982735431137498.. =
6.2831853071795864769252867665651.. = 4(Pi/2)

Rod ... ...

[Amigoo]

Re: Progression of Pi design
"increments of 2(sqrt(Pi))"

Table updated to show both increments:
( 1.1283791670955125738961589031215.. 2(sqrt(1/Pi))
and 1.23370055013616982735431137498.. )

1.0

1.1283791670955125738961589031215.. = 2(sqrt(1/Pi))
1.2732395447351626861510701069789..
x 1.23370055013616982735431137498.. =

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

2.2567583341910251477923178062431.. = 2(2(sqrt(1/Pi)))
2.5464790894703253723021402139602..
x 1.23370055013616982735431137498.. =

3.1415926535897932384626433832795.. = 2(Pi/2)
3.5449077018110320545963349666809.. = 2(sqrt(Pi)
4.0

4.513516668382050295584635612484... = 4(2(sqrt(1/Pi)))
5.0929581789406507446042804279155..
x 1.23370055013616982735431137498.. =

6.2831853071795864769252867665651.. = 4(Pi/2)
7.089815403622064109192669933368... = 4(sqrt(Pi)
8.0

Rod

[Amigoo]

Re: Progression of Pi design
"increments of 2(sqrt(Pi))"

About the other increment, 1.23370055013616982735..
Geometers must know (and those who chase wiggly numbers
while running in squared circles ... on a hot day in Texas)

Pi/2 = side of square inscribed in a circle =
1.5707963267948966192313216916398.. Pi/2
x 1.4142135623730950488016887242097.. sqrt(2)
= 2.2214414690791831235079404950303.. diameter
/ 2 = 1.1107207345395915617539702475152.. radius
^2 = 1.2337005501361698273543113749845.. increment

Finally! sqrt(2) reveals itself in this progression.

Rod

[Amigoo]

Re: Pi, Oh My! (new design concept)

Impressive geometric concentricity, highlighting geometry of squared circles
with increments of Pi and sqrt(2) and including the "geometric anchor" 2
... all supported by the foundational Pythagorean right triangle!

Rod

[Amigoo]

Re: Pi, Oh My! design

I must have been complaining about the difficulty of chasing increments of Pi
and sqrt(2) around this Cartesian neighborhood because the entire project
was suddenly halted for an important team meeting!

When the project resumed, the focus was on squared circle symmetry
... with only coincidental reference to increments of Pi and sqrt(2).

An intriguing design, suggesting dimensional transformation!

Rod

[Amigoo]

Re: POM Indeed! design (increments of Pi and sqrt(2))

An intriguing design, suggesting dimensional transformation!

"The final decimal digit awaits discovery
in this impossible Cartesian neighborhood."

Rod ... ...