Paradise Trinity Day

[Geoff]

Well it just did it again here, its a big black page.

Geoff

[Sandy]

I've not noticed any flashes but then I am not very observant.
xxSandy

[Amigoo]

Re: iSibling Rights design

When the large circle's diameter equals 4, the seven inscribed squares that fit precisely in 360 degrees of this spiral are so intriguing that some of the lines indentifying the side-by-side right triangles were removed (for better display of the squares). Even more intriguing are the seven unintended 4s that appear on the perimeter of the spiral!

And the lines of this geometry (via the side-by-side right triangles) also create seven 7s along the perimeter!
Thus, 4s and 7s! What else is hidden in this geometry? seven 11s?

Rod

[Amigoo]

Re: iSibling Rights design

Intuitively, "4" seems to have unique value in relation to "7".

Consider: 1+2+3+4+5+6+7 = 28 / 7 = 4
Likewise: 4 is the center of the series 1234567

Rod

[Amigoo]

Re: Infinite Sibling Rights design
(name expanded for "product differentiation")

Who knew? Product differentiation is required!
since "i__" can refer to internet__, or imaginary__, or ...

Rod ... ...

[Amigoo]

Re: Infinite Sibling Rights design

Curiosity always delays bedtime ...

The spiral of squares and their enclosing circles is impressive.
But what about the area squares of the seven circles?
Not to worry! The wiggly numbers verify the spiral:

Diameter of Circle:
1. 40000000.0
2. 28284271.247461900976033774484194..
3. 20000000.0
4. 14142135.623730950488016887242097..
5. 10000000.0
6. 7071067.8118654752440084436210485..
7. 05000000.0

Square Root of Square:
1. 35449077.018110320545963349666823..
2. 25066282.746310005024157652848107..
3. 17724538.509055160272981674833411..
4. 12533141.373155002512078826424049..
5. 8862269.2545275801364908374167057..
6. 6266570.6865775012560394132120272..
7. 4431134.6272637900682454187083529..

Rod

[Amigoo]

Re: Vesica Pi (new design concept;
exploration of the Infinite Sibling Rights design)

Leading to this insight ...

If Vesica Piscis is the intersection of two circles with the same radius,
intersecting such that the center of each lies on the perimeter of the other,
then the two area squares of this Vesica Piscis must form a Vesica Pi
in which these two squares overlap on their diagonal.

Rod ... ...

[Amigoo]

Re: Vesica Pi design

Oops! The geometry for this design does not present two circles with their center on the perimeter of the other. But the way these squared circles interconnect suggest that this is the authentic Vesica Pi. At least, this will remain the "official" VP until more research proves otherwise.

So, the comment in the previous post should read:

"Vesica Piscis, intersection of two circles having the same radius
and intersecting with centers on the perimeter of the other,
is complemented by a Vesica Pi when these squared circles
interconnect on points where each circle meets its square."

Rod ...

[Amigoo]

Re: Vesica Pi design
Who knew? This geometry can wax nostalgic!

My, my, Miss Vesica Pi
how nostalgia draws us nigh
to hear another slice of Pie
yet dream of days not dry:

"It was a story about America"
http://www.whrc-wi.org/americanpie.htm
http://www.youtube.com/watch?v=VsZFiMo8TIc

Rod

[Amigoo]

Re: Scalene Moment (new design)

Like the Big Bang, there is just one scalene moment where the perfect triangle is created that squares the circle:

In this geometry, consider the movement of the green diagonal line on the left as it rotates (clockwise or counter clockwise) at the point where it meets the yellow line. The red isosceles right triangle pivots and is attached to the midpoint of the green line at the 90 degree vertex of the red triangle. The green diagonal line has length equal to one side of a square inscribed in the large circle.

Without a doubt, a Scalene Moment exists in this Cartesian landscape
where the yellow line has length equal to the square root of Pi
(large circle's diameter = 2).

Rod

[Amigoo]

Re: Scalene Moment (revised design)

Who knew? This geometry can wax monozygotic! Now debate begins about which is the authentic Vesica Pi:
Vesica Pi or this Scalene Moment? Fortunately, this dilemma can be resolved in a later generation.

"Later generation"? Of humans or of squared circle geometry? Best answer: "Yes".

Rod ... ...

[Amigoo]

Re: Scalene Moment (revised design - What! Again?)

In this House of Many Pi, if you can't find Pi,
the quest may be impossible.

But rotate the design and look again, just to be certain.
Note: computers are still searching for all of its digits.

Rod

[Amigoo]

Re: House of Many Pi design

The Scalene Moment design, revised again and renamed
because of its increased complexity and Pi-related components;
split-level to weather any controversy.

Most noteworthy:
The two largest circles do not form an authentic vesica piscis (center of circle on perimeter of other circle), but their CSC circles do when united by a third circle having the same diameter. Hmm ... a trinity.

Rod ... ...

[Amigoo]

Re: House of Many Pi design

Like a Christmas stocking needing a few more items, this House tolerated another "Many" revision.

Who knew?! As soon as Cartesian workspace is filled with straight lines, circles, squares, right triangles, scalene triangles, trapezoids, parallelograms, vesicae pisces, and even squared circles, the geometry begins to wax neuronic!

But one more object in this House of Many Pi and its self-appointed Archon Basileus
might develop symptoms of "second n-to-t" transformation.

Rod

[Amigoo]

HOMPi (new design)
"Geometric ovoviviparity?"

Further study of the geometry of the House of Many Pi.
aka "When squared circle geometry waxes maternal".

Rod

[Amigoo]

Re: Scalene Juxtaposition (new design)

Who knew? Squared circle geometry includes siamese trapezoids!

In this design, the circle-squaring scalene triangle of the larger circle in this CSC set is contrasted with the circle-squaring scalene triangle of the smaller circle. The two different but related isosceles trapezoids seem to provide important perspective on squared circle geometry.

Rod

[Amigoo]

Re: Scalene Juxtaposition design
and HH Galactic design

Who knew? Geometry "communicates" with other geometry!

I added two more lines to the Scalene design to highlight related line lengths*, then noticed that two red "H"s were formed. This hints of geometric relationship between the Scalene Juxtaposition and HH Galactic designs. At least, they're both colorful, complex, and esoteric.

I'll have to study this relationship over a belated summer break.

* The two vertical lines of each "H" have length equal to the side of
an inscribed square in that circle or to the radius of the circle.

Rod ... ...

[Amigoo]

Re: Scalene Juxtaposition design
"The two vertical lines of each "H" have length equal to the side of
an inscribed square in that circle or to the radius of the circle."

More observation (the value of patterns) ...

Each "H" is a defining characteristic of the scalene triangle that squares its enclosing circle. "H" is actually a butterfly shape (see diagonal green lines that connect the ends of the vertical red lines). Two pair of opposing angles are always present in this diagonal line set: 135 and 45 degrees.

Rod

[Amigoo]

Re: iSJ (infinite Scalene Juxtaposition, new design)

More late-night wandering into geometric complexity, motivated by curiosity:
How do these scalene "butterflies" appear in a CSC continuum?

Who knew? In circles squared by this scalene triangle, the geometric butterflies are concentric!
Replication of the pattern, both smaller and larger, indicates an infinite continuum.

These layers of integrated objects challenge the definition of "symmetry".

Rod

[Amigoo]

Re: "iPentagonal Ties" (new design)
The "i" in iPentagonal? irregular.

Who knew? The irregular pentagons in CSC squared circles wear ties!

As though Scalene Juxtaposition's layered nexus (including the design's concentric red "butterflies") that challenges the definition of symmetry is ineffective intimidation to diminish time-consuming exploration of squared circle geometry, now iPentagonal Ties tosses me a whimsical geometric correlation. Maybe humor will be the better carrier of the recently recurring message: "Get a life!".

Well, summer break is quickly approaching and I'll have no opportunity for multitudinous mouse manipulations in darkened rooms with faceless clocks. The iPentagonal suits are being tailored and the correlation Ties pressed for this research plateau's sultry catwalk.

Rod ... ...

[Amigoo]

Re: "iPentagonal Ties" design

Who knew? There are matching cufflinks for these ties!
(cufflinks not shown, but match conjoined bottoms of red ties)

This morning's habitual review of the previous evening's research was conducted in a complementary "morning fog" (sometimes reflective of alpha creativity). As I studied the conjoined bottoms of the red ties, red lines with no other lines nearby, intuition hinted that these were "transmission ties".

"What's a transmission tie?" I wondered. Long story short, after daring dabbles 'n doodles:

The lines of the ties, at their conjunction, contain sufficient geometric information to draw a circle and its matching square.
And the magical-mystical 62.402.. degree angle is present in two parallel sides of the circle's square!

The green arc instead of circle? Symbolic of "More work to be done!" and "Where's the Proof?"
aka, "The adventure continues."

Rod

[Amigoo]

Re: "HOMPi Squares" (new design concept)

This geometry concept now exists as "rough draft" but will have to reflect on its completion in a Cartesian cloister - summer break is promised, scheduled, and even mandated (or womandated).

Intuition informs me that this geometry is further evidence that proof of squared circles requires out-of-the-box thinking. The geometry of HOMPi offers a good starter set of objects for this study, but otherwise HOMPi is simply Home Pi (as in "No place like ...").

Rod

[Amigoo]

Evidence of out-of-the-box thinking
or hemispheric vacation
After you peel a banana, do you eat what's left or what's right?

Rod ... ...

[Amigoo]

Re: HOMPi Squares design

Curiosity (again) inspired late-night exploration of the geometry. What (which) is HOMPi Squares?

What I first sensed was not on target - the Squares are complex geometric sets, contrasting a circle-squaring scalene triangle with its smallest sibling. In fact, the 45 degree vertex of the largest scalene triangle (in one set) is the same vertex for the smallest triangle.

This design has such geometric intensity now that my little black mouse is refusing to point & click in this colorfully-convoluted Cartesian collage of "impossible" objects; at least, not without without a new contract guaranteeing stress-adjusted compensation.

Both mouse and man have agreed to rest a bit before completing the design, then quickly retire to a Cartesian cloister of vacation persuasion. Just where this design is going - at warp speed - is unknown, but the workspace and workmen are sizzling with rascally resonance. Distant observation of the collage from a calming cloister might clarify the adventure.

Rod

[Amigoo]

Re: HOMPi Squares design
"Can you hear me now?" (highlighted double-scalene "antenna")

When squared circle geometry waxes HOMPi, with colorfully-convoluted rascally resonance.
The scalene triangles in both sets of green squares are similar and their enclosing circles
are squared by area; small inscribed squares drawn last, after the geometric "DNA".

"HOMPi"? Home Pi (as in "No place like ..."), sung to the tune of Om.

Best guess for this design's composite symbolism:
Rascally resonance at the periphery of parallel universes, but circuits are inactive
until one planet in each universe "squares the circle".

Rod