Paradise Trinity Day

[Amigoo]

Re: Quadrature of Vesica design

a shape that is the intersection of two squared circles with the same radius, intersecting in such a way that the center of each circle lies on the perimeter of the other, and the center of each square lies on the perimeter of the other square

Moving ever closer to proof of squared circle ...
Whereas the vesica piscis remains a vesica piscis in 360 degrees of rotation, the squares of those circles
align (as per definition) only at a few excruciatingly precise degrees around the circle.

Geometer's pop quiz: How many geometric postions does this include?

Rod

[Amigoo]

Re: Quadrature of Vesica design

the world's first Quadrature of Vesica

Well, it's the first term used to describe squared circles in a vesica piscis
... and the first time that such geometry has been proffered (IMO).

About the two largest red right triangles (one overlaps the other) ...
these triangles define the proportions: sqrt(Pi)/2 : 1 ~ 1 : 2(sqrt(1/Pi))

1 / 1.1283791670955125738961589031215..
= 0.8862269254527580136490837416705..

1 / 0.8862269254527580136490837416705..
= 1.1283791670955125738961589031215..

Rod

[Amigoo]

Re: Quadrature of Vesica design
Summary of the numbers ...

About the two largest red right triangles (one overlaps the other) ...
these triangles define the proportions: sqrt(Pi)/2 : 1 ~ 1 : 2(sqrt(1/Pi))
(adjacent side : hypotenuse ~ adjacent side : hypotenuse)

1 / 1.1283791670955125738961589031215..
= 0.8862269254527580136490837416705..

1 / 0.8862269254527580136490837416705..
= 1.1283791670955125738961589031215..

To calculate cosine angle (same for both triangles):

a = sqrt(Pi)/2, c = 1, cosine = adj/hyp
0.8862269254527580136490837416705.. / 1
= 0.8862269254527580136490837416705..
acos(0.8862269254527580136490837416705..)
= 27.597112635690604451732204752339.. degrees

a = 1, c = 2(sqrt(1/Pi)), cosine = adj/hyp
1 / 1.1283791670955125738961589031215..
= 0.8862269254527580136490837416705..
acos(0.8862269254527580136490837416705..)
= 27.597112635690604451732204752339.. degrees

Rod ... ...

[Amigoo]

Re: Quadrature of Vesica design

Summary of the numbers ("new & improved")
for the two largest red right triangles (one overlaps the other).

These triangles define the proportions: sqrt(Pi)/2 : 1 ~ 1 : 2(sqrt(1/Pi))
[ adjacent side : hypotenuse ~ adjacent side : hypotenuse ]
[ diameters of vesica piscis circles = 2(sqrt(1/Pi)) ]

To verify correspondence (means-extremes property)

1 / 1.1283791670955125738961589031215..
= 0.8862269254527580136490837416705..

1 / 0.8862269254527580136490837416705..
= 1.1283791670955125738961589031215..

To calculate cosine angle (same for both triangles):

a = sqrt(Pi)/2, c = 1; cosine = adj/hyp = a/c
0.8862269254527580136490837416705.. / 1
= 0.8862269254527580136490837416705..
acos(0.8862269254527580136490837416705..)
= 27.597112635690604451732204752339.. degrees

a = 1, c = 2(sqrt(1/Pi)); cosine = adj/hyp = a/c
1 / 1.1283791670955125738961589031215..
= 0.8862269254527580136490837416705..
acos(0.8862269254527580136490837416705..)
= 27.597112635690604451732204752339.. degrees

Conclusion: these related triangles appear to have
significance in the proof of nested squared circles.

Rod

[Amigoo]

Re: Pi Approximation Day, July 22

"Approximation" because the fraction 22⁄7 is an approximation of Pi,
accurate to 2 decimal places and originating at the time of Archimedes.

Such "truth in advertising" (Pi itself is an approximation, albeit
excruciatingly close now (billions of decimal digits and still counting).

Rod

[Amigoo]

Re: Scalene Entanglement (new design concept)

While roaming Google to find any previous use of this name, I found a poem with a few lines that well-describe the seeming chaos in this new Cartesian neighborhood of two nested circles, both squared ... and embellished with many geometric points of three and four:

Re: https://gauravvartak.wordpress.com/2015 ... anglement/

"Then comes the scalene with its mind unbalanced,
amid chaotic scenes it finds peaceful existance."

The dual concentricity (each of the squared circles contribute geometric objects to one of two sets, each having a unique center, and providing "chaotic scenes" in "peaceful existance") begs the question: "Are squared circles concentric by relationship* even when not aligned by geometric definition?"

* many lines in the objects of one set have twice the length of lines of similar objects in the other set.

Rod

[Amigoo]

Re: Scalene Entanglement design
"Dual concentricity via corresponding squares."

Relatively chaotic yet peaceful, with geometric entanglement
well within the boundaries of this Cartesian neighborhood.

Rod ... ...

[Amigoo]

Re: New Pi in_sight (new concept)

While contemplating the approaching Leftover Pi Day (Nov. 10 this year)
and the whimsical poem written for this celebration (next paragraph),
curiosity sent me back to the Cartesian neighborhood for a stroll.

"Simple Simon met a pie man, parting from the fair.
Said the pie man to Simple Simon: 'My pies are in repair
to lockers cold, thus wrapped - not sold; leftover fare is fair
for sweet repose, then new expose of pie beyond compare!'"

While strolling, I began to suspect that a new Pi (constant
or formula) might be discovered before this Pi Day.
So, what clues exist in geometry that would lead to this?

New Pi in_insight ("insight" or "in sight") displays an intriguing
relationship of the sides of two inscribed squares. The math*:

0.7071067811865475244008443621052..
/ 0.79788456080286535587989211986877..
= 0.88622692545275801364908374167057.. = sqrt(Pi)/2

* Length calculation of sides of inscribed squares:
[ diameters = 1, 2(sqrt(1/Pi)); divisor = sqrt(2) ]

1 / 1.4142135623730950488016887242097..
= 0.7071067811865475244008443621052..

1.1283791670955125738961589031215..
/ 1.4142135623730950488016887242097..
= 0.79788456080286535587989211986877..

Apparently, one side each of two inscribed squares
can be used to precisely calculate an increment of Pi.

What's next? Go find the pie man!
"pie beyond compare" may be baking already!

Rod

[Amigoo]

Re: New Pi in_sight (new concept)

Go find the pie man!
"pie beyond compare" may be baking already!

New Pi may be "in sight", but it's well-camouflaged by
the many objects in this Cartesian jungle (aka "neighborhood").

Not to worry - November 10th is four months away!
(enough time to get lost in the jungle 3.14.. times)

New Pi in_sight should be design after the next vine swing,
but may require many tree-top views to locate any New Pi.
Apparently, my interim destiny is to be a jungle swinger
... but those Plus size loincloths are a monkey's howl!

BTW: I may have spotted another version of the Texas 'T'
... or my bifocal lens got a wild squirt of fresh orange when
concocting the Evening Anchor (di Amore Quattro Orange).

Rod ... ...

[Amigoo]

Re: New Pi in_sight design
“… ere minds around its Cartesian IAM …”

"Been there! Done that!" (again)

About the small golden circle ...
geometer's humor for "completion"
(dotted the i's and crossed the t's).

Rod ... ... (gone shopping to Loincloths Plus
... and might select matching gloves for the vines).

[Sandy]

I love your clever sense of humour, Rod! Just so you know, I am hungry for "pie" now.
Have a great week end!

xxSandy

[Amigoo]

Re: New Pi Neighborhood design

Suspecting that New Pi in_sight was a salesman's perspective of the geometry,
I then talked with a geometer. New Pi Neighborhood is the geometry that
I expected when first strolling in this Cartesian neighborhood.

Diagonal line in center of red scalene triangle
calculates left and bottom side of scalene triangle:
(note: diameters = sqrt(pi) and 2)

1.2533141373155002512078826424055.. [ sqrt(Pi/2) ]
* 1.4142135623730950488016887242097.. [ sqrt(2) ]
= 1.7724538509055160272981674833395.. [ sqrt(Pi) ]
= diameter of circle for D = sqrt(Pi).

1.2533141373155002512078826424055.. [ sqrt(Pi/2) ]
x 1.1283791670955125738961589031215.. [ 2(sqrt(1/Pi)) ]
= 1.4142135623730950488016887242097.. [ sqrt(2) ]
= side of inscribed square for D = 2.

Rod

[Amigoo]

Re: New Pi Neighborhood design
"home of the circle-squaring scalene"

One more calculation (using the hypotenuse and side of the
isosceles right triangle that is part of the scalene triangle):

Perpendicular line in center of red scalene triangle
calculates left and bottom side of the scalene triangle:
(note: diameters = sqrt(pi) and 2)

1.2533141373155002512078826424055.. [ sqrt(Pi/2) ]
* 1.4142135623730950488016887242097.. [ sqrt(2) ]
= 1.7724538509055160272981674833395.. [ sqrt(Pi) ]
= diameter of circle for D = sqrt(Pi).

1.2533141373155002512078826424055.. [ sqrt(Pi/2) ]
x 1.1283791670955125738961589031215.. [ 2(sqrt(1/Pi)) ]
= 1.4142135623730950488016887242097.. [ sqrt(2) ]
= side of inscribed square for D = 2.

1.7724538509055160272981674833411.. [ sqrt(Pi) ]
/ 1.2533141373155002512078826424055.. [ sqrt(Pi/2) ]
= 1.4142135623730950488016887242097.. = sqrt(2)
= side of inscribed square for D =2

sqrt(Pi) / sqrt(Pi/2) = sqrt(2) (effectively defines the circle
squared by the hypotenuse of the isosceles right triangle).

Rod

[Amigoo]

Re: New Pi Neighborhood design
"home of the circle-squaring scalene"

sqrt(Pi) / sqrt(Pi/2) = sqrt(2) (effectively defines the circle
squared by the hypotenuse of the isosceles right triangle)

1.7724538509055160272981674833411.. [ sqrt(Pi ) ]
/ 1.2533141373155002512078826424055.. [ sqrt(Pi/2) ]
= 1.4142135623730950488016887242097.. = sqrt(2)
= side of inscribed square for D =2

LOL: sqrt(2) is the result for every isosceles right triangle.
Not a problem - just a technical note that research continues ...

Rod

[Amigoo]

Re: New Pi Neighborhood design
"home of the circle-squaring scalene"

Hmmm ... sqrt(Pi) / Pi = 0.56418958354775628694807945156077..
seems to have unique significance in this replicating geometry.

If 1.1283791670955125738961589031215.. were a diameter,
would its radius (0.56418958354775628694807945156077..)
have value in a New Pi calculation?

I'm in trouble! no apple pie a la mode for several years,
but the image (with aroma) just arrived in the neurons ...
and a foreign (another state) Blue Bell plant is testing!

Rod ... ... (off to the frozen foods aisle)

[Amigoo]

Re: New Pi Neighborhood design
"home of the circle-squaring scalene"

New observation of the replication ... whatever the value of Pi,
its increments/decrements are directly related to those of sqrt(2) ...
which says something about squares in "squared circles".

Therefore, if Pi is transcendental, sqrt(2) is transcendental
... or perhaps they're not!

Rod

[Amigoo]

Re: nPi Arc Arc design
"Lines and triangles and squares! Oh my!"

The only justification for this whimsical design name
is an impromptu "Lines and triangles and squares! Oh my!"
(such is the nature of "impossible" squared circle jungles)

Or perhaps anticipating a geometric "Wizard of Ahs".

Oh! the geometry?" A study of a circle with diameter = 2
and its area square with a side length = 1.

Rod

[Amigoo]

Re: nPi Arc Arc design
"Lines and triangles and squares! Oh my!"

Oh! the geometry?

This is better stated as ...
A study of a circle with diameter = 2
and a square with side length = 1.
(circle and square related indirectly)

Rod

[Amigoo]

Re: nPi Arc Arc design
"Lines and triangles and squares! Oh my!"

Hmmm ... Pi / (Pi/2) = 2 (maybe not)
So, 2 is transcendental ... sometimes?
Or Transcendental / Transcendental = NO T?

Maybe Pi / (Pi/2) is undefined since the total number of decimal digits is unkown.
So, I propose "Pi.xx" where xx = number of proven decimal digits. Therefore ...
even Pi.1,000,000 / (Pi.1,000,000 / 2), for example, still calculates as 2.

Obviously, the only transcendental Pi are those having more
than the current number of known decimal digits (IMO).

Or maybe only the odd decimal digits are transcendenta!l
(no, not a comment about gay marriage)

Talk about those "terrible twos"!
(still not a comment about gay marriage)

The only thing I know about "terrible twos" is that
most cartons of ice cream have but two servings:
the one eaten right way and the next one.

Rod ...

[Amigoo]

Re: Manifestation of 13 (new design concept)

13 overlapping circles, all squared.
Anticipating the storm before the calm?

Rod

[Amigoo]

Re: Manifestation of 13 design
"Perhaps, anticipating the storm before the calm."

A final 180 degree turn revealed a pyramid,
enhancing the symbolism of number 13.

Rod ... ...

[Amigoo]

Re: Manifestation of 13 design
"Perhaps, anticipating the storm before the calm."

Ratios of two similar right triangles
(side of pyramid = Pi; center line = 2(sqrt(Pi)):
[ long side : hypotenuse ~ long side : hypotenuse ]

sqrt(Pi) : 2 ~ Pi : 2(sqrt(Pi))
2(Pi) = sqrt(Pi) x 2(sqrt(Pi))
Pi = sqrt(Pi) x sqrt(Pi)
Pi = sqrt(Pi)^2

Does the line length 2 in this geometry
corral the transcendental nature of Pi?

Rod

[Amigoo]

Re: Manifestation of 13 design
Updated with an "eye for detail" ...

- four golden circles in center of this design have diameters = 2(sqrt(1/Pi)).
- four dark blue circles (2 above and 2 below golden circles) have squares whose side length = 2(sqrt(1/Pi)).
- ratio of hypotenuse of right triangle (half of pyramid; outer side) to triangle's long side = 2(sqrt(1/Pi)).

Logically, if the four dark blue arcs are accepted as circles, the two arcs of the "eye"
should be also (making 15 circles). But these two are not "overlapping circles, all squared"
(such is the "eye for detail").

The supporting fuzzy math (re: "warm and fuzzy"):

3.1415926535897932384626433832795..
/ 2.7841639984158539226424089910608..
= 1.1283791670955125738961589031215..
= 2(sqrt(1/Pi))

Pythagoras of the Greek island, Samos, knew a few things about geometric pyramids!
Hmmm ... pyramid power is real!

Rod

[Amigoo]

Re: Manifestation of 13 design

I was reminded that the geometric object is an isoceles triangle - not pyramid!

In defense, I would proffer that only two dimensional humans can step into
a two-dimensional pyramid (Cartesian triangle) and only the minds of
three-dimensional humans can comprehend a squared circle.

Rod ... ...

[Amigoo]

About Pi Approximation Day, July 22 ...
There may be more accurate approximation than 22/7.
Consider the enticing math clue of 1/Pi:

Pi = 3.1415926535897932384626433832795..

22 / 7 = 3.1428571428571428571428571428571..
7 / 22 = 0.31818181818181818181818181818182..

1 / Pi = 0.31830988618379067153776752674503..
= 1.59154943 / 5.0 converted to a fraction.

5 / 1.59154943 = 3.1415926554037344601983238434511..
= Pi approximation with 8 decimal digits of accuracy!

The Quest: What calendar day complements this fraction?
Discover the fraction and be famous! (or get free pie).

Rod