Paradise Trinity Day

[Amigoo]

New wisdom ...

A creative masterpiece is not defined by days, weeks, or months,
but often by seasons of life lost while dedicated to completion
of the masterpiece envisioned and pursued.

Rod

[Amigoo]

Re: Circle the Square design

If the first object in a Cartesian neighborhood were an isosceles right triangle
having length of hypotenuse = 2.0, and the constant 2(sqrt(1/Pi)) were known
(or its angle complement: 27.597112635690604451732204752339.. degrees),
the rest of this geometry can be created ... with the circling of the square
being a later act of questionable precision.

Rod

[Amigoo]

Re: Circle the Square design
“Popular first survey of this Cartesian neighborhood
… and promotion of ‘Myth of Box, Once Confining’”

circling of the square being a later act of questionable precision

Hmmm ... which square is being circled later?
inscribed square or circle's square? or both?

Consistent display of geometric "mission creep"!
(simplicity easily evolves to complexity)

Rod ... ... (old wheels, still creeping)

[Amigoo]

Re: Circle the Square design
“Popular first survey of this Cartesian neighborhood
… and promotion of ‘Myth of Box, Once Confining’”

Squared circle geometry keeps pushing back against "impossible"

In this continuing evolution of simplicity to complexity,
two circle-squaring geometric objects are presented
as the Juxtaposition Of Uniquely Similar Trapezoids
(JOUST, aka "the 'impossible' unhorsing of Pi").

Diagonal of smaller trapezoid has length equal
to midpoint-to-midpoint length of larger trapezoid.

Rod

[Amigoo]

Re: JOUST design
(Juxtaposition of Uniquely Similar Trapezoids;
aka, "the courageous unhorsing of Pi")

With all of the geometry now to support such a tournament
(attempts to "unhorse Pi"), an annual geometry soiré is timely!

Rod

[Amigoo]

Re: Geometry of Pi design

A good reference to the geometry of Pi in a squared circle,
highlighting the presence of both 2(sqrt(1/Pi)) and Sqrt(Pi)
in the various line length ratios of the three-pointed star,
a right triangle inscribed in the large circle.

Rod

[Amigoo]

Re: Geometry of Pi design
"Long story short."

Line ratios of golden right triangle
(contains the three-pointed star):

hypotenuse to long side = 2(sqrt(1/Pi))
long diagonal to short diagonal = 2(sqrt(1/Pi))
upper red line to lower red line = sqrt(Pi)
diameter of leftmost circle = 2.0

Rod ... ... (off to check a gait )

[Amigoo]

Re: Squared Circle ZPE design

Perhaps, the zero point energy of
"How to square the circle."

... or metamorphosis of Flutterby Butterfly?
... or confirmation of the inaugural JOUST?
... or focus on other lines of interest?
... or new spelling of "impozzible"?

Rod

[Amigoo]

Re: Squared Circle ZPE design
(some assembly required)

I was certain that a 'Z' would appear in the final design
... but no! Obviously, the missing line is nearby and
associated with commentary about sqrt(2).

Rod ... ... (off for early Zzzz)

[Amigoo]

Re: Squared Circle ZPE design
Now you 'Z' it ... (after "some assembly")

Once I recognized the presence of sqrt(2) and 2(sqrt(1/Pi))
in the line length ratios, the 'Z' could be completed.

Why the "eye" in the golden scalene triangle?
Dunno - maybe the "Eye of Providence".

Rod

[Amigoo]

Re: Squared Circles ZPE design
Now you 'Z' it ... (after "some assembly")

Regarding the 3 squares, perfectly aligned
(and visible once "some assembly required"):

aka "geometers' eye candy"
aka "Now you 'Z' it!" (maybe)
aka "esoteric enlightenment"

Now the title's "ZPE" makes sense:
In a squared circles neighborhood, a Pi Corral,
geometric objects align with ground state energy.

Rod ... ... (Zzzz ... or a noisy muffler)

[Amigoo]

Re: Squared Circles ZPE design
Now you 'Z' it ... (after "some assembly" for Soiré)

If you're lacking in Cartesian direction (or just prefer an escort), visit the D&D Concierge upon arrival at the Squared Circles Soiré. A personal droid (floor cruisin' robot) or drone (winged mini droid) will be assigned to escort you to/from your seat in the ballroom whenever requested.

D&D escort "staff" return to their stations after performing the escort; tipping allowed - just insert your hotel guest key in the chip reader or wave your Soiré ZPE wand.

To avoid congestion, the number of D&D escort staff is limited, relative to ballroom occupancy. If you attend with your own D&D escort, allow your escort to first mingle in the D&D Lounge to acquaint her (him, whatever) with rules, regulations, and current D&D gossip.

Rod

[Amigoo]

Re: Ratios Illuminating Pi design
"RIP indeed!"

Wiggly numbers supporting ratios:

1.7724538509055160272981674833411.. sqrt(Pi)
/ 1.4142135623730950488016887242097.. sqrt(2)
= 1.2533141373155002512078826424052..

1.1283791670955125738961589031215.. 2(sqrt(1/Pi))
/ 1.4142135623730950488016887242097.. sqrt(2)
= 0.79788456080286535587989211986916..

1.0
/ 0.79788456080286535587989211986916..
= 1.2533141373155002512078826424052..

Rod

[Amigoo]

Re: Ratios Illuminating Pi design
"RIP indeed!"

Further evolution to complexity, a neighborhood inevitability,
presents visualization challenge: the two green right triangles,
plus a juxtaposed pair having their own similarity, fold in to
identify the inscribed square of a similar circle (both arcs).

Say what? Objects of squared circle geometry align perfectly
... even when repositioned by axial rotation on a point.

Say what? "Impossible" is but popular opinion
... most obvious when you 'Z' it in the neighborhood.

Rod ... ...
(practicing parallel parking by axial rotation)

[Amigoo]

Re: Ratios Illuminating Pi design
"RIP indeed!"

Actually ... (regarding the four juxtaposed right triangles)
Only one of each pair folds in (and out) ... and those two
triangles share a side (their hypotenuse*).

The ends of the golden scalene's rightmost diagonal
defines the two pivot points of the folding.

* a side of the largest circle's inscribed square ...
and diameter of one of the smaller twins. HCIT!

Rod

[Amigoo]

Re: RIP Squared X3 design

Further exploration of the RIP geometry presents impressive integration
of the three circles, all squared. But the four juxtaposed right triangles
are no longer visible . However, the center golden square with
its symmetric appendages emphasizes this tight integration

Rod

[Amigoo]

Re: RIP Squared X3 design

Further exploration (again) revealed another layer of complex geometry
well-integrated in this intriguing design. But the added geometry was
then reduced to a few line segments, segments which now symbolize
the Geometer's Mettle, a future award for squared circle aficionados
who discover the geometric fullness of that symmetric layer.

Rod

[Amigoo]

Re: RIP Squared X7 design (7 circles, all squared)

A neighborhood so complex that one must take a deep breath
just to pass through an intersection (of lines). If patterns
are not the perfect guidance for squared circles, guidance
does not exist ... in this local universe.

The Geometer's Mettle? Luckily, lurking.

Rod ... ... (luckily, lolling
... before design completion)

[Amigoo]

Re: RIP Squared X7 design (7 circles, all squared)
(aka "What's the point?!")

A squared circle geometer (who else would attempt such complexity?)
cannot draw this Cartesian neighborhood without asking many times
"What's the point?!" (what's the end point of a line or arc ...
or the center of a circle?). But with squared circle patterns,
identification of the points is as easy as 1,2,3,4.

Rod

[Amigoo]

Re: Rings-a-Pi design

A whimsical allusion to "Rings of Pi" ... probably inspired by whiplash
from running in so many squared circles ... especially in the very
complex neighborhood of RIP Squared X7.

Rings-a-Pi is a study of the two complementing right triangles that exist
in squared circles (along with the scalenes, trapezoids, and squares).

I'm reminded that line length ratios are true geometric indicators
of squared circles, but complementing angles also deserve attention.
Rings-a-Pi seemed to emphasize angles more than line length ratios
until I realized that the ratios are still present.

What are "rings"? Overlapping circular objects, all squared.

Rod

[Amigoo]

Re: Rings-a-Pi design

While not as complex as RIP Squared X7, this Rings geometry seemed a bit esoteric ...
until I noticed its message: long sides of the red right triangles form the longer sides
of the red scalene triangle. The is first time that I've seen this integration of the
circle-squaring scalene with those two complementing right triangles!

Rod ... ... (listening for Rings-a-Pi ring tone)

[Amigoo]

Re: Rings-a-Pi design
(repositioned to highlight large red scalene triangle)

Rings-a-Pi (aka Rings of Pi, Rings uP high*)
"Essential rights of a circle-squaring scalene."
* dial tone: 1.1283791670955125738961589031215..

Yada, yada, yada! (still more complexity)
... with whimsical esotericity (re: "dial tone")
... aka 2(sqrt(1/Pi))

3.1415926535897932384626433832795.. Pi
/ 1.1283791670955125738961589031215.. 2(sqrt(1/Pi))
= 1.5707963267948966192313216916405.. Pi/2

Rod

[Amigoo]

Re: Rings-a-Pi design (math corrected)

1.7724538509055160272981674833411.. sqrt(Pi)
/ 1.1283791670955125738961589031215.. 2(sqrt(1/Pi))
= 1.5707963267948966192313216916405.. Pi/2

Rod

[Amigoo]

Re: Rings-a-Pi design (final, final)

Squared circle geometry that "Rings uP high” ...
with scalene notes and trapezoidal resonance.

Sponsored by the bulldog who ate Pi ...
for days, weeks, months, and years.

Rod ... ... (Ring Ring! "No more Pi, please")

[Amigoo]

Re: "Believe iT or not" design

... ... (Ring Ring! "No more Pi, please")

Perhaps response from above to the "No Pi" petition,
"Believe iT" highlights the "Texas 'T'" as if to insist
that it's the geometric 'T' of the prompt years ago
... "Believe iT or not".

Rod