Paradise Trinity Day

[Amigoo]

Re: Mitosis CQ design
"geometrically telophasic w/CQ"

Geometers' secret ...
Short side of circle-squaring right triangle in the 3 CSCSC circles
downsizes by sqrt(2) from the largest to the smallest circle.

Rod

[Amigoo]

Re: Mitosis CQ design*
"geometrically telophasic w/CQ"

Geometers' secret ...
Short side of circle-squaring right triangle in the 3 CSCSC circles
downsizes by sqrt(2) from the largest to the smallest circle.

Geometers' secret (astute geometers know this) ...
Center of Quadrature is a common point on the long side.

* updated in: http://aitnaru.org/images/Quinale!.pdf

Rod ... ...

[Amigoo]

Re: Mitosis CQ design
"geometrically telophasic w/CQ"

Geometer's conclusion ...
Having chased these circle-squaring geometric objects all over the Neighborhood
(for several weeks), the conclusion is obvious: Pi cannot exist without sqrt(2)

"As Above, So Below" (the math and the geometry must agree):
"Either sqrt(2) is transcendental or Pi is not" Period! End of story!

Rod

[Amigoo]

Re: Mitosis CQ design
"geometrically telophasic w/CQ"

Geometers' secret ...

Center of Quadrature includes 2 sets of opposite angles:
45 degrees and 135 degrees (90 + 270 = 360).
(geometric "strong force" = 135/45 = 3)

Rod

[Amigoo]

Re: Holometabolous design

Revealing 7th architectural type of life design in Quadrature
... and the belated antithesis of morbus cyclometricus?
... and evidence of constellation Q~Lepidoptera?

Rod

[Amigoo]

Re: Holometabolous design,
updated in: http://aitnaru.org/images/Quinale!.pdf

Re: https://www.youtube.com/watch?v=e5hRPy-pDsY

There's so much Quadrature packed into this Cartesian composition that
"Holy Metabolous!" becomes another Batman exclamation!

Rod ... ...

[Amigoo]

Re: Holometabolous design

Fascinating geometric construction, this circle-squaring scalene in the largest circle
with two overlapping right triangles on the left that pivot inward on their right angle,
creating a smaller circle-squaring scalene ... with the short sides of both scalenes
having obvious sqrt(2) line length relationship.

"Holy Metabolous!"

Rod

[Amigoo]

Re: Holometabolous design

Speaking of "right triangles" and "pivot" ...
3 circle-squaring right triangles also pivot on their right angle, creating a sqrt(2) sibling
where respective sides of each triangle have sqrt(2) relationship, and largest/smallest
of the 3 triangles differ by 2.0, precisely! Thus, Pi is evenly divisible by 2.

"Holy Metabolous!"

Rod

[Amigoo]

Re: Holometabolous design

Regarding the direct relationship of Pi and 2.0, sqrt(Pi) and sqrt(2),
Salient Seams of Means 'n Extremes is a table of those numbers.

"Holy Metabolous!"

Rod ... ...

[Amigoo]

Re: Holometabolous design

Now, the geometry is hinting that there are two centers

"Holy Metabolous!"

Rod

[Amigoo]

Re: Holometabolous design
"Holy Metabolous!"

Either '77' is geometrically esoteric ...
or the number of times this design has changed!

Rod ... ...

[Amigoo]

Re: Holometabolous design,
updated in: http://aitnaru.org/images/Quinale!.pdf

Geometers' secret ...
The tip of each adjoined "scalpel" identifies one of the geometric centers
with "scalpel" hinting the precision of this "impossible" Quadrature.

Rod

[Amigoo]

Re: Holometabolous design

Relative numbers (line lengths) ...
"Say what?!" Go figure!

1.4142135623730950488016887242097.. SoIS
1.2533141373155002512078826424055.. SoCS/SoIS
1.7724538509055160272981674833411.. SoCS
1.1283791670955125738961589031215.. Diam/SoCS
2.0 Diameter

1.2533141373155002512078826424055.. SoCS/SoIS
x 1.1283791670955125738961589031215.. Diam/SoCS
= 1.4142135623730950488016887242097.. Sqrt(2)

"SoIS, SoCS, Diam?!" Sides 'n Center

Rod

[Amigoo]

Re: Holometabolous design

Geometers' secret ...

The largest circle's inscribed isosceles trapezoid
(two overlapping circle-squaring scalene triangles)
has a sqrt(2) sibling with a similar trapezoid.

And that circle would have a sqrt(2) sibling.

Rod ... ...

[Amigoo]

Re: Holometabolous design

Geometers' Neighborhood secret ...

The inner trapezoid (dark blue) has precisely one-fourth the area
of the larger, enclosing trapezoid (proves with simple geometry).
At least, "simple" for good geometers.

Who can tell
This Cartesian Neighborhood design has enough stuff (geometric objects, etc.)
to be the foundation of a new geometry-teaching board game.

"Whatsit called?"
How would I know "Holy Metabolous!"

Rod

[Amigoo]

Re: Holometabolous design

Given concentric squares with areas = 4.0, Pi, 2.0

4 / 3.1415926535897932384626433832795..
= 1.2732395447351626861510701069801..

3.1415926535897932384626433832795..
/ 2.0 = 1.5707963267948966192313216916398..

1.2732395447351626861510701069801.. (2/sqrt(Pi))^2
x 1.5707963267948966192313216916398.. Pi/2
= 2.0

Pi has a precise location between 4.0 and 2.0
... but is not "precisely between".

Rod ... ... (usually in a precise location
... between Here and There)

[Amigoo]

Re: Peace of Pi design

The salient concentricity of Quadrature.

Rod

[Amigoo]

Re: Peace of Pi design*

The salient concentricity of Quadrature.

Geometers' secret ...
Side length of circle's square = sqrt(Pi)
Side length of inner square = 2/sqrt(Pi)
sqrt(Pi) / (2/sqrt(Pi)) = Pi/2, D = 2.0

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

* added to: http://aitnaru.org/images/Quinale!.pdf

Rod

[Amigoo]

Re: Peace of Pi design

Who knew
There's a Dunkin on the ER Bridge!

Regarding the inner circle ...
Diam = 1.2732395447351626861510701069801.. (2/sqrt(Pi))^2
SoCs = 1.1283791670955125738961589031215.. 2/sqrt(Pi)

Rod ... ... (prefer Cartesian Plain w/ "impossible" sprinkles)

[Amigoo]

Re: Peace of Pi design

Geometers' "impossible" secret ...
2/sqrt(Pi) defines both the circle (Diameter)
and its square (Side of Circle's Square, SoCS)

1.1283791670955125738961589031215.. 2/sqrt(Pi), SoCS
^2 = 1.2732395447351626861510701069801.. Diameter

And in the circle-squaring right triangle,
ratio of Diameter to Long Side = 2/sqrt(Pi)

1.2732395447351626861510701069801.. Diameter
/ 1.1283791670955125738961589031215.. Long Side
= 1.1283791670955125738961589031215.. 2/sqrt(Pi)

Rod

[Amigoo]

Re: Peace of Pi design

Quadrature is like a box of round chocolates -
there's a Peace of Pi in every one!

Rod ... ... (shoppin' around the rounds ...
but usually get around or three)

[Amigoo]

Re: Peace of Pi design

Quadrature is like a box of round chocolates -
there's a Peace of Pi in every one!

Regarding the circle-squaring right triangle ...

ACOS (SoCS/Diameter) = ACOS (sqrt(Pi)/2)
= ACOS ( (2/sqrt(Pi)) / (2/sqrt(Pi))^2 )
= ACOS (0.88622692545275801364908374167057..)
= 27.597112635690604451732204752339.. degrees

27.597112635690604451732204752339..
+ 62.402887364309395548267795247661..
+ 90 = 180 degrees

Rod

[Amigoo]

Re: Peace of Pi design
... and its circle-squaring triangles

Of "impossible" Quadrature,
geometer-believers are always asked "What's your angle?!"

Angles of the circle-squaring scalene triangle:
45.0 , 62.402887364309395548267795247661..
72.597112635690604451732204752339.. degrees

Angles of the circle-squaring right triangle:
90.0 , 62.402887364309395548267795247661..
27.597112635690604451732204752339.. degrees

Pop Quiz:
Having driven in a straight line but wish to return to your starting point,
how many turns are required if each turn is 179 degrees
The geometry hints that distance traveled is important.

Rod ... ... (known to execute U-Turns)

[Amigoo]

Pop's Quiz:
Having driven in a straight line but wish to return to your starting point,
how many turns are required if each turn is 179 degrees
The geometry hints that distance traveled is important.

Pop's Riddle: Not all turns have to be all left or all right,
but the best turns will be alright if none are left.

Rod

[Amigoo]

Re: Pi Fork 'n BC design*
"And that's the truth!"

aka "Wildly Outrageous 'n Esoteric!" (WOE!)

* added to: http://aitnaru.org/images/Quinale!.pdf

Rod ... ...