Paradise Trinity Day

[Amigoo]

Re: iQuadrature design

Há Pi New year !!

To some geometers, "iQuadrature" means "impossible Quadrature";
to other geometers, it means "integrated Quadrature", but to me it means:
"i can't believe you write 'finale' then create another design!"

Rod (just connecting points in the Neighborhood)

[Amigoo]

Re: iiQuadrature design*
"impossible Quadrature" or "integrated Quadrature"
* http://aitnaru.org/images/Yin_Yang_Pi.pdf

Regarding circle-squaring scalene triangle where D = 2.0
(shares SoCS with circle-squaring right triangle) ...

sqrt(Pi) = 1.7724538509055160272981674833411.. Side of Circle's area Square
sqrt(4-Pi) = 0.92650275035220848584275966758914.. short side of right triangle

1.7724538509055160272981674833411..
/ 1.4142135623730950488016887242097..
= 1.2533141373155002512078826424055..

0.92650275035220848584275966758914..
/ 1.4142135623730950488016887242097..
= 0.65513637756203355309393588562466..

1.2533141373155002512078826424055..
+ 0.65513637756203355309393588562466..
= 1.9084505148775338043018185280302.. scalene long side
1.7724538509055160272981674833411.. scalene middle side
1.4142135623730950488016887242097.. scalene short side

"What squared circle contains this scalene!" Yes!

Rod (just connecting points in the Neighborhood)

[Amigoo]

Re: iiQ Three design
("three" refers to the triangle object)

Composition highlights circle-squaring right triangle
(sides have length = sqrt(Pi), sqrt(4-Pi), 2.0) that defines
circle's inscribed circle-squaring trapezoid which contains
three overlapping circle-squaring scalene triangles.

Rod (cruisin' in the Neighborhood's Threes)

[Amigoo]

Re: iiQ Three design
("three" refers to the triangle object)

Embellished for another rewarding year of Morbus Cyclometricus.

Rod ... ... (still cruisin' in the Neighborhood's Threes)

[Amigoo]

Re: iiQ Three design
("three" refers to the triangle object)

Who knew
It's suddenly Ball Three in Morbus Cyclometricus redux!
What happens next

Rod

[Amigoo]

Re: iiQ Ball Three design

What happens next

Obviously, with all this geometric "eye candy",
what a good geometer does next is Eat Pi

Rod ... ...

[Amigoo]

Re: iiQ Ball Three design

What happens next

Rod

[Amigoo]

Re: Inscribable Scalenity design

What happens next

Geometers' secret ...
Objects of "impossible" Quadrature replicate easily ...
when found in the sqrt(2)-dominated Neighborhoods.

Rod

[Amigoo]

Re: Inscribable Scalenity design
(updated in: http://aitnaru.org/images/Yin_Yang_Pi.pdf )

Inscribably delicious! (geometrically speaking)

Rod

[Amigoo]

Re: Inscribable Scalenity design

Inscribably delicious! (geometrically speaking)

Now geometrically symplified!
(with symmetry and color, otherwise more esoteric)

Rod ... ...

[Amigoo]

Re: Inscribable Scalenity design
aka "Cartesian Scalenity Inscribable" (CSI)

Wherewith the transcendence of Pi ?

Rod

[Amigoo]

Re: Inscribable Scalenity design

Wherewith the transcendence of Pi ?

A good time to declare this increasingly esoteric design "final, final"
since circle-squaring right triangle in large circle has precisely 4 times the area
of a similar circle-squaring right triangle in the rightmost smaller circle.

Aromatic Pi precisely divisible by 2 is indeed Cartesian de light.

Rod

[Amigoo]

Re: Inscribable Continuum design (IC ic)

Wherewith the transcendence of Pi ?

One more (in the spirit of Morbus Cyclometricus)

Rod ... ...

[Amigoo]

Re: Inscribable Continuum design (IC ic)

Wherewith the transcendence of Pi ?

Setting the Quadrature bar higher (or out of reach ; - ) for crop circle makers,
human or ? ... and redefining Morbus Cyclometricus for this new millennium!

Rod

[Amigoo]

Re: Inscribable Continuum design (IC ic)

Geometer's secret ...
Distance between centers of two larger golden 'X' objects
is precisely the diameter of the two smallest circles:
D = sqrt(2)/2 = 0.7071067811865475244..

Rod ... ...

[Amigoo]

Re: Encubation of Quadrature design

Setting the Quadrature bar higher (or out of reach)
for crop circle makers, human or

Suddenly, "out of reach" comes to mind ...
after the cube appeared in the Neighborhood,
a cube revealing the transcendental sqrt(2).

Geometer's secret ...
Every line of this cube has length = sqrt(2)/2
= 0.70710678118654752440084436210485..

Rod

[Amigoo]

Re: Phi of Pi design (redux)

Probably, imaginary (or impossible) Cartesian precision.

Rod

[Amigoo]

Re: Phi of Pi design
(added to: http://aitnaru.org/images/Yin_Yang_Pi.pdf )

Probably, imaginary (or impossible) Cartesian precision
... since Phi of Pi = sqrt(Pi) / sqrt(4-Pi)
= 1.9130583802711007947403..

Rod

[Amigoo]

Re: Phi of Pi design

A slowly developing myth is that Quadrature required decades of Morbus Cyclometricus simply because
geometers would not dare to publicly conceptualize POPS LOIN (Phi Of Pi Spiral and Lines Of Interest Noted).
Now (and because of the internet), if you've seen one set circles squared you've seen them all!

Rod ... ... (Oops! Mythed the red light.)

[Amigoo]

Re: Thrice Phi of Pi design
(highlights three Phi spirals)

To recap ...
Phi of Pi = sqrt(Pi) / sqrt(4-Pi)
= 1.9130583802711007947403078280204..
where a circle-squaring right triangle defines Quadrature:
diameter = 2.0, long side = sqrt(Pi), short side = sqrt(4-Pi)
and diameter = triangle's hypotenuse.

Rod

[Amigoo]

Re: Thrice Phi of Pi design
(highlights three Phi spirals)

Now contrasts 360 degrees of a spiral (for astute geometers).
EZ geometry clue: ((Phi of Pi)^2)^2

Rod ... ...

[Amigoo]

Re: Thrice Phi of Pi design
(highlights three Phi spirals)

((Phi of Pi)^2)^2
= 1.7724538509055160272981674833411..
/ 0.92650275035220848584275966758914..
= 1.9130583802711007947403078280204..
^2)^2 = 13.394080164614312715066..

May the Phorce of Quadrature enlighten!

Rod

[Amigoo]

Re: Phi Four of Pi design
aka "Quadrature's Three-Fingered Phi"

Geometer's secret ...
There's more Phi of Pi when D = 4

Rod

[Amigoo]

Re: Phi Four of Pi design

There's more Phi of Pi when D = 4

Best guess about this more esoteric composition
with phantom "fingers": Playing of precision instruments
requires celestial participation from above.

Rod ... ...

[Amigoo]

Re: Thrice Phi of Pi design

Geometer's secret ...
"sqrt(2) = sqrt(Pi)/sqrt(2) x 2/sqrt(Pi)" refers to one golden and two red lines
in lower circle (light blue) where red lines form an isosceles right triangle;
intersecting golden line represents a Side of Circle's Square (SoCS)

Rod