Paradise Trinity Day

[Amigoo]

Re: Pi Are Square II design
"What goes around, comes around."

Same song - next verse:
aka "Simplicity Today - Complexity Tomorrow"
(after a late evening nap before bedtime)

But, it's just squared circle sqrt(2) symmetry
... and it's geometrically logical:

If sqrt(Pi) is the length of a side of a circle's square where D = 2.0,
then sqrt(Pi) is the hypotenuse of an isosceles right triangle whose sides
have length equal to a side of the square of the circle where D = sqrt(2).

Say what?! The pedantic precision of a Pi Corral.

Rod ... ... (off to lock the Corral "as is")

[Amigoo]

Re: Pi Are Square II design
"What goes around, comes around."

Same song - third verse:
aka "Simplicity Today - Complexity Tomorrow"

Now, a second red line! But its length is easy to calculate:
2(sqrt(1/Pi)) / sqrt(2)

Rod

[Amigoo]

Re: Pi Are Square II design
More geometric stuff added!

Now it's obvious that, of the two smaller light blue circles,
the inscribed square of the large circle (D = sqrt(2))
is the area square of the small circle (D = 2(sqrt(1/Pi))
HCIT!

Caveat: "some assembly required" Arc! Arc!
(only a portion of the lower circle is drawn)

What's the square? 1 One what?
One "Surely you jest!"

Rod

[Amigoo]

Re: Pi Are Square II design
More geometric stuff added! (again)

Who knew?! This geometry wants to show all of the recurring objects
in a neighborhood of circles tightly integrated and squared. And it prefers
to note the difference between an isosceles (two adjoining sides golden;
other two sides dark blue) and an irregular (four sides golden) trapezoid
... not to mention the large and influential scalene triangle.

Rod

[Amigoo]

Re: Yada Yada Yada design
Exploration of integration in Pi Are Square II

aka "Same song, next verse!"
aka "Been there! Done That! (again)"
aka "If you've seen one squared circle ..."
aka "YYY? (explore the obvious)"

Rod ... ... (off to find YYY? ring tone)
YYY? "Hello? Who's there? ... CYHMN?"

[Amigoo]

Re: Yada Yada Yada design

Hmmm ... CYHMN might be pronounced "Simon"
... as in "Simon Peter"?

Rod

[Amigoo]

Re: Yada Yada Yada design

HCIT! With more post-nap exploration, the red cross appeared!
Lines have length equal to a side of the square of the respective circle;
these lines were colored red before this internet inquiry:

From the internet ("What is the meaning of the red cross emblem?"):
"the red cross emblem continues to be an internationally recognized symbol
of protection and neutrality"

Rod

[Amigoo]

Re: Simply YYY design
"Lines and triangles and squares, oh Pi!"

About the red right triangle ...
(D = diameter, r = radius)

Given: Circle where D = 4(sqrt(1/Pi))
= 2.2567583341910251477923178062431..
and r = 2(sqrt(1/Pi))
= 1.1283791670955125738961589031215..

r^2 = 1.2732395447351626861510701069801..
= length of hypotenuse of red right triangle
whose long side has length = r.

Rod ... ... (on the yellow brick road)

[Amigoo]

Re: Simply YYY design
"Lines and triangles and squares, oh Pi!"

Note: angles of three-pointed star, inscribed in large circle,
all equal 27.597112635690604451732204752339.. degrees.
(whatever the value, the geometry proves they're equal)

Pop Quiz: What's the side length of the area square
of the large circle in YYY? It rhymes with "two".

Rod

[Amigoo]

Re: Simply YYY design
"Lines and triangles and squares, oh Pi!"

For several years, I have believed that geometry can contrast two similar squared circles in such a way that they're perfectly balanced in their overlapped positioning; any additional adjustment of lines in one circle prevents perfect squaring of the other. In other words, two similar circles so squared and contrasted can simultaneously exist in the Cartesian neighborhood only in perfect geometric balance.

The geometry of Simply YYY seems to have perfect balance
... but is a wee bit difficult to evaluate, albeit not "impossible".

Which are these circles in Simply YYY?
Those displayed as two green arcs. Arc! Arc!

Rod

[Amigoo]

Re: Simply YYY design
"Lines and triangles and squares, oh Pi!"

Regarding the Arc! Arc! circles ...

D = 2.5464790894703253723021402139602..
r = 1.2732395447351626861510701069801..
sqrt(r) = 1.1283791670955125738961589031215..

(circle-squaring geometry of rightmost and lower right triangles,
part of the three-points star inscribed in the center circle
where D = 2.2567583341910251477923178062431..
= 4(sqrt(1/Pi))

Rod ... ... (off to find the Arc! Arc! seal)

[Amigoo]

Re: Simply YYY II design
(exploration of "geometric balance")

This complex geometry may need more development (as design)
but seems sufficient to show the kind of "geometric balance"
required to prove overlapped and similar squared circles.

Rod ... ... (reflecting on the recent "good ol' days"
of Arc! Arc! simplicity in a Cartesian neighborhood)

[Amigoo]

Re: Simply YYY II design

Who knew?!
Squared circles have a unique frequency
that identifies the key for this "impossible"
geometric music of the spheres.

Regarding "tuning fork" in large circle:
(golden right triangle and inscribed red 'X')

Ratio of hypotenuse to long side = 2(sqrt(1/Pi))
= 1.1283791670955125738961589031215..

Length of short side of red 'X' = (2sqrt(1/Pi))^2
= 1.2732395447351626861510701069801..

Length of long side of red 'X' = (2sqrt(1/Pi))^2 x 2(sqrt(1/Pi))
= 1.4366969770013324935126558690215..

Ratio of long side to short side of red 'X' = 2(sqrt(1/Pi))
= 1.1283791670955125738961589031215..

Rod

[Sandy]

Just a little side note, Rod... this has nothing to do with your current geometry but as I was just now looking at some of your gorgeous geometric subjects on http://aitnaru.org/ , I rediscovered your Tree of Life Page http://aitnaru.org/homepage/treeoflife.html I had forgotten about that and in rereading the text again again "something" touched my heart (in a place that needed touched. ) Thank you!
love,
Sandy

[Amigoo]

Sandy,

Intriguing project (and important symbolism), that Tree of Life!

Seven years later, the "tree" is still alive (about 7" tall) but struggling
... as is the friend I had given it to (in rehab again; now for a broken hip).

The web page doesn't mention that I had noticed this leaf (next to its large pot in an office hallway)
two nights in a row as I left the office. Each night, I had expected the cleaning crew to discard it.
When it was still there the third night, I sensed a mission: appreciation of life force endurance.

Rod

[Sandy]

I LOVE that story!

Intriguing project (and important symbolism), that Tree of Life!

It is indeed!
Thank you! Now I'm off to feed the rabbits.
xxSandy

[Amigoo]

Re: Simply YYY II design
Ending on a sweet note, geometrically speaking

Of the two small circles ...
the diameter of the top circle has length equal to a side of the square of the bottom circle;
the side of the square of the top circle has half the length of the diameter of the bottom circle.
Indeed, a "signature key" of these circles, all squared.

Re: https://www.youtube.com/watch?v=I46IcoFAVdU
"The wheels on the bus go round and round ..."

Rod

[Amigoo]

Another midwayer prompt ...
What's with the 3:18 I've seen in recent weeks?

It's just 1/Pi, the starting point of 2(sqrt(1/Pi))
... and the reason for the four equal parts of
(2(sqrt(1/Pi))^2) / (1/Pi) = 4.0

Rod

[Amigoo]

Re: Some Assembly Required design

How did this "out of the box" geometry emerge from exploration of Simply YYY II?

After consideration of titles for the final design (selecting a title early gives guidance to development), Some Assembly Required best captured the essence of a squared circles toy box, once opened. Closing the box is difficult ("some assembly required"), but returning the newly discovered geometric objects back to the box may be impossible!

Perhaps the essence of "out of the box" thinking (and believing in the impossible): once your mind is out and about, returning to the box is no longer an option ... or desire.

Rod ... ... (still cruisin' out and about)

[Amigoo]

Re: Some Assembly Required design

In a Cartesian neighborhood of squared circles,
there's only one box to think outside of:
the box labled "Impossible!"

Why "Some Assembly Required"?
Once a mind ventures outside the box, returning is a bit difficult
("some assembly required" to reconstruct the confining box).

Rod ... ... Say what?! The box is a myth?!
(once perceived from the outside)

[Amigoo]

Re: Some Assembly Required design

The scalene triangle is the best focus for 2(sqrt(1/Pi)),
but those right triangles are not so obvious:
http://aitnaru.org/threepoints.html

Rod

[Amigoo]

Re: Some Assembly, Indeed! design

Who knew?!
Try to reassemble the "box" and it seems to disappear!
... confirming that such mind confinement is a myth.
Can it be ... "impossible" is often a popular myth?

Rod

[Amigoo]

Re: Waxing Waves design
(soon after "some assembly")

Exploring outside the box, once mythical?
Catch waxing waves with trapezoidal insight.

Did I draw left side of the golden isosceles right triangle?
When? Why? This line seems to have no purpose other than identifying
that triangle which fits so snuggly within large golden scalene
... and its hypotenuse length equals radius of largest circle.

Rod

[Amigoo]

Re: Waxing Waves design
(soon after "some assembly" ... twice)

Squared circle geometry keeps pushing back against "impossible"
... now defining a square inscribed in that scalene triangle.

Rod ... ... (off to review definition of "inscribed")

[Amigoo]

Re: Waxing Waves design
(soon after "some assembly" ... twice)

Geometric reflection in delight ...

When a certain scalene triangle appears with inscribed square,
it's time to acknowledge the Myth of Box, Once Confining
... but now confined within a Pi Corral.

Rod