Paradise Trinity Day

[Amigoo]

Re: Son-iT design

After realizing that I was pronouncing "Son-iT" like "sonnet",
I found this: http://en.wikipedia.org/wiki/Sonnet

... leading to this caution:
aficionados of Son-iT geometry might be labeled "sonneteers",
with such labeling reminiscent of "morbus cyclometricus".

About the red "T" shape in the center of the design:
If a diagonal line were drawn in the largest green trapezoid, from upper left to lower right, it would define the scalene triangle (left portion of trapezoid) that was being studied when the smoke alarm's false signal occurred months ago. I had just drawn the top of the red "T" when the alarm occurred. The vertical red line was drawn next.

What does this mean? That right triangles are probably components of squared circles
(and maybe that a certain Midwayer likes right triangles). "Who can tell?"

Rod

[Amigoo]

Re: iTrapezoid design (analysis of patterns)
A bit technical, but patterns also have numbers.

Beginning with the largest circle (let d = 1 x 10,000,000),
note that the longest line of each inscribed trapezoid
(longest line has length = side of circle's square)
is also an hypotenuse of a right triangle.

Consider the geometry of this CSCSCSC continuum:
(d = diameter, h = hypotenuse, s = side of right triangle)

d: 10000000.000
h: 8862269.255
s: 6266570.686

d: 7071067.812
h: 6266569.060
s: 4431135.678

d: 5000000.000
h: 4431136.550
s: 3133284.852

d: 3535533.906
h: 3133287.022
s: 2215568.271

The reason that Pi is irrational and transcendental
must be hidden in the symmetry of this CSCSCSC continuum.

Rod ... ...

[Amigoo]

Re: iTrapezoid design (analysis of patterns)

Sees 'em suggested that the numbers be copied
from the CAD drawing and not from notes:

d: 10000000.000
h: 8862269.255
s: 6266570.687

d: 7071067.812
h: 6266570.687
s: 4431134.627

d: 5000000.000
h: 4431134.627
s: 3133285.343

d: 3535533.906
h: 3133285.343
s: 2215567.314

Rod ...

[Amigoo]

Re: Mysty Vision (new design)

Again (and again) I seem to be lectured that this research is all about patterns. My role is to promote patterns - not solutions to geometry challenges from antiquity. While such solutions might be found with the guidance of patterns, I seem destined to find only patterns ... some with impressive geometry, some esoteric as design, some whimsical, and some mysterious (thus "Mysty").

Exploration of CSC geometry this morning (instead of attending to chores) resulted in a CSC set with an embedded vesica piscis appearing as eye glasses ... with windshield wipers! What?! Is this serious geometry? Or a good example of "Mysty" patterns?

"Mysty" will have to wait in the toy box while chores take priority for a week
... despite the "eye glasses" being circles squared by trapezoids!

Rod

[Amigoo]

Re: Son-IAM (new design)

Those rascals! I was convinced that I was ready to break for chores, so briefly reviewed world news online before heading to "the mines". The trend of the news was still "much trouble ahead". And this sent me back to the geometry for a "spirit refresher".

Long story short: A quick study of the circle-squaring trapezoid revealed a simple yet powerfully symbolic pattern. A message from our sponsor seems to be its essence: "All is well - attend to immediate priorities."

Rod

[Amigoo]

Re: CSC-iT (new design)

One tough day of landscape maintenance in the early heat of summer (aka "paying for negligence") convinces me that the geometry will have much less priority for a month or two (house painting is next).

So, the Mysty Vision design can wait in the toy box, but this CSC-iT design (pronounced "Sees it!" or "Seize it!") is so intriguing geometrically that it was rushed to the "portfolio". The correspondence between the two trapezoids (at 45 degrees offset) includes such integrated symmetry that the irrational and transcendental Pi must certainly lurk within (or nearby).

Rod

[Amigoo]

Re: CSC-iT design (the story)

There's something on the menu for every geometer at the Squared Circles Cafe ...

Some notice the out-of-the-box geometry and CSC-iT ("seize it")
while others grin but otherwise just CSC-iT ("sees it") ...
and most ask for the 1882 menu.

Rod ... ...

[Amigoo]

Re: "Wag the dog" (new design, CSC-iT with another circle)
I wonder what is really being communicated by some geometry designs and their titles?

At the Cartesian sandbox level, this design is just CSC-iT with one more circle added (essentially CSCSC) to contrast a circle with one whose diameter* is equal to the larger circle's radius. In this comparison, the geometric similarities (especially of the inscribed trapezoids) emphasize the circle-squaring power of a unique scalene triangle.

At a higher level but still close to the surface of the earth, "Wag the dog" seems to hint of pending performance perpetration. Maybe the four sets of "wiggly numbers" previously reported caused this sensitivity to other wagging tongues and tales.

* Diameters in focus: 20,000,000 and 10,000,000 units.

Rod

[Amigoo]

Re: "/4 " (new design, emphasis on division by 4)
Apparently, CSC-iT was the necessary pathway to conceptualization of the power of this scalene triangle.

In /4, the comparison of a circle having twice the diameter of another (a CscsC continuum) needs minimal discussion. The scalene triangle in the larger circle is subdivided into four similar triangles with one of those triangles being the inscribed scalene in the smaller circle. The square of the smaller circle is obviously (per geometry) 1/4 the area of the square of the larger circle.

Some wiggly numbers:
Area of circle = Pi x radius squared

Area of D=20,000,000 units
= 314159265358979.32384626433832795

Area of D=10,000,000 units
= 78539816339744.830961566084581988
= 314159265358979.32384626433832795 / 4

Thus, D10 has 1/4 the area of D20

Rod

[Amigoo]

Re: "/4 " design
"Crème de la crème of squared circle geometry."

Having noticed the 46 pages of this topic and the many recent x:46 prompts
(including the symbolic 6:46 yesterday), a project plateau seems certain!

"Who can tell?"

Rod

[Sandy]

Having noticed the 46 pages of this topic and the many recent x:46 prompts
(including the symbolic 6:46 yesterday), a project plateau seems certain!

We see 46 pages of awesome aspirations, deductions, inspirations and gorgeous geometric designs!
Its good to be allowed to share this journey with you, Rod!
xxSandy

[Amigoo]

Re: "Slash Flash" (new design)

What fun! A title that will provide much entertainment when pronounced rapidly
... even without adult refreshments!

But the title's derivation is logical:
I was studying the /4 geometry for several hours (as usual, after midnight) when the "balancing" pattern appeared.
Long story short: when the two red lines have equal length, the two right triangles are correctly positioned,
allowing the scalene triangle to be created which squares the circle.

At least, this is what appears in this Cartesian creativity sandbox, post-midnight.
The "Flash" means it's news to me; a sudden burst of enlightenment!

Rod ...

[Amigoo]

Re: "Slash Flash" design

The numbers in units:
D = 20000000 (large circle)
D = 14142135.623.. (small circle)
red lines = 13494774.691..

Geometry "candy" for a lifetime! (or more)

Rod

[Amigoo]

Re: "Slash Flash" design

Curiosity about the seeming dissymmetry of the two red lines in this geometric composition lured me into more exploration. An hour of perusal became an hour of perusal - no progress. But when I noticed that the lines were attached to the side of an inscribed square (three sides not displayed), I added the squares to the CAD drawing, one square per circle.

This visual reference was sufficient to recall the Free Will design from several years ago.

In Free Will, two sets of overlapping squares in a CSCSC continuum identified the squares of those circles (the area squares). The integration of the two red lines then became obvious: their symmetry exists within a composition of overlapping inscribed squares, overlapping inscribed scalene triangle sets in a trapezoid, and all hosted by nested circles in a CSC continuum.

This explains why "squaring the circle" has been such a challenge! A simple solution is impossible - a squared circle is not a simple geometric object. But this elusive geometric creature can be tamed in a Pi corral ... once it's captured.

Rod

[Amigoo]

Re: "Slash Flash" design

The green squares* were potentially present in the previous version of this design, but their actual presence visually explains the balancing role of the two red lines. In fact, if only the two green squares were present in this Cartesian workspace, they would allow the rest of the geometry to be created: pivot one corner each of the green squares on a single point until the red line distances exist.

* each square represents the inscribed square of one circle in the CSC continuum.

Who knew? Pi are square!

Rod

[Amigoo]

Re: "Slash Flash" design

After a 3:00 am snack and hopes of early bedtime (relative to 6:00 am), I reviewed the Flash geometry one more time.
I was uncertain - which is more descriptive: "Pi are square." or "To square the circle, one must circle the square."

I checked the computer clock: 3:14 am
Hmmm ...

Rod

[Amigoo]

Re: "Slash Flash" design
Another day, another decimal point ...

"Reverse engineering" of the angles and lines to confirm the concept was not going smoothly this morning. "Close, but no cigar!" (from old county fairs) was the penetrating message. Indeed, the unsettling question remains: "What happens after three decimal points of length?"

This circle may seem squared visually, but is not according to the numbers!
Time to kick sand in this creativity litter box ... some other day.
Nevertheless, "close" is still close.

To Do List: Reconsider the reliability of those two red lines.

Rod

[Amigoo]

Re: "Slash Flash" design

Even the CAD software is protesting today:
"What are you trying to do? Square a circle?!"

But I persisted and finally reconstructed the current geometric objects without the two red lines ... and found opportunity for an isosceles right triangle whose sides have length equal to half the square root of Pi. These sides will be the "new & improved" red lines.

CAD conversion to design later today or tomorrow.

Rod

[Amigoo]

Re: "Slash Flash" design

I'll need to lower the volume of my TV ...
The questionable rumors about unaccompanied minors having ease of entry somehow influenced the geometric objects in the creativity sandbox. When I rationalized that now was a better time to convert the CAD drawing than after a day of yardwork, more objects rushed to the Cartesian workspace.

I was a bit overwhelmed but finally guided them into meaningful areas of the composition. It's certainly livelier now but will facilitate better study of the geometry of squared circles. "Who can tell?" what solutions now lurk within this integrated family of geometric objects?

Rod

[Amigoo]

Re: "Slash Flash" design
Expecting simplicity is habitual ...

The revised Flash design proves that "simplicity" is not the geometric essence of squared circles.
The "two red lines" confirmation was simplistic - accurate confirmation is provided by:

- isosceles right triangle with red sides (length = half square root of Pi).
- horizontal yellow line of scalene triangle (length = square root of Pi).
- two green squares representing inscribed squares of CSC circles.

Green squares are positioned by isosceles right triangle and horizontal scalene line.

About the extra objects (added then removed) ...
The Final Review Board insisted that virtual objects have the same value.
(points exist in the composition for these to be added if desired)

Rod ... ...

[Amigoo]

Re: "Slash Flash" design
Who knew? Pi are square ... twice!

Such a roller coaster ride this week! In less than 48 hours, the "two red lines" were discovered, touted, then disgraced. And now a different set of "two red lines" appears in the geometry! In this latest set, both lines are diagonal with one line twice the length of the other.

The selection of squares that integrate with a CSC continuum provided unexpected revelatory depth to this complex, squared circles geometry (both CSC circles are squared). The geometry might seem a bit chaotic but I've learned to think of the objects as overlays, especially with an overlay order influenced by the color selected for the object (some colors appear better in the background).

Slash Flash is now both colorful and loquacious, geometrically speaking. I should be able to enjoy a few days of rest while it tries to explain the conceptual simplicty of this set of squared circles.

Rod

[Amigoo]

Re: "Hypotenuse Harmony" (new design)

Reflecting on the Slash Flash design ...
Who knew? Squared circles wear brightly colored square pants!
Or maybe it's beach wear: Slash Flash Splash!
(Now serving refreshing Pi_ña Coladas!)

Well ... a few Pi_ña Coladas, virtually speaking, were sufficient to blur the Cartesian landscape; "simplicity" came to mind.

And this is how Hypotenuse Harmony emerged as a subset of Slash Flash. The "harmony" is sponsored by the hypotenuse of the red isosceles right triangle that connects the midpoints of the hypotenuses of the green isosceles right triangles. The geometric significance is obvious to geometers.

This composition offers the definitive geometry of squared circles when studied as a CSC continuum.
Refreshing Pi_ña Coladas might become the preferred libation for study of these CSCs.

Rod

[Amigoo]

Re: "HH Galactic" (new design)
"When squared circle geometry waxes interstellar."

I couldn't resist exploring the Hypotenuse Harmony geometry again, perhaps to discover the real secret of squared circles. A slippery slope again: no solution - just colorful, esoteric design ... without the creativity benefit of several Pi_ña Coladas!

HH Galactic ("HH" from Hypotenuse Harmony) might be another lecture about promoting patterns - not solutions.

Rod

[Amigoo]

Re: "HH Galactic" design
Back from the design shop with reconfigured energy circuits:

The three identical concentric circles sets in the upper right represent lines of energy* connectors, allowing the HH Galactic craft to be guided forward, as if by gravitational pull. The similarly colored concentric circles at the rear of the craft collect circulating space energy, allowing it to be stored for future needs of the craft as well as for propulsion.

The two circles to either side of the primary energy processor (largest set of colored concentric circles) represent resonators that condense space energy for storage or process it for craft propulsion. The outer dark blue, electromagnetic ring of the craft as well as the two half-circle nozzles at the rear are both components of this craft's propulsion system.

This is just conjecture about the configuration of this craft's energy circuits.

* "three of the universally distributed lines of energy,
each having a clear space velocity of 186,280 miles per second" (39:3.9)

Rod

[Amigoo]

Re: "rPi" (new design)

When Pi is the square of a circle whose diameter equals 2,
the radius of that circle is the diameter of a circle
that squares the circle enclosed by Pi.

Such simple geometry to be so profound!
... if true.

Rod