Paradise Trinity Day

[Amigoo]

Re: Impossible Endorsement design

Perceived Stroke Order (vertical alignment):
Horizontal strokes, drawn from top to bottom,
receive numeric counts of 1, 3, 5, 7.

Say what? "The simplified version."
Say what? "Focus on the seventh stroke."

Say what?
"The seventh stroke may signalize multiaxial integration,
facilitating equitable exchange of Cartesian values."
aka, WYSIWYG (geometrically speaking)

Rod " "

[Amigoo]

Re: Oracle of Pi (new design concept)

A study of a state of Pi, announced by a CAD software anomaly:
a line drawn as a diameter duplicated itself about ten times
as if to announce arrival of a geometric impossibility.

"Who can tell?"
Leisurely development seems best, considering other states of Pi:

Re: http://en.wikipedia.org/wiki/Pi_(state)
"In the Shang Dynasty (1600–1046 BC), Pi was at war with the imperial court
in order to resist their attempts to govern the state."

Hmmm ... a state of Pi with Texas' attitude.
"Giddyap! ... but take your time."

Rod

[Amigoo]

Re: Oracle of Pi design

"Taint paint." ...

Imagine! Just past the 11-mile marker (approx.) on another 4-legged ride across the Cartesian plane, in search of squared circles meaning and value for this state of Pi, I met a full-whiskered Oracle Wannabe (IMHO). "Probably couldn't get a ticket to Burning Man." I conjectured, according to his state of dress and obvious preparation.

"What's tied to your saddle?" he asked. "A bag of bones?" he grinned. "Hmmm ..." I agreed ... "Those geometric objects might be considered a bag of bones - a bag of Oracle bones." [pyromancy, a form of divination using oracle bones was popular in China during the late Shang dynasty]

"Lemme try 'em!" he insisted. "No, they've never been used for that." I protested. "Well, you try 'em! ... and I'll read them bones!"

Intrigued by the suggestion, I tossed the geometric "bones" onto the ground in a squared circle (IMHO). O. Wannabe stuck a weed in his mouth, chewed for 11 minutes (approx.), then spoke slowly with a soft, rhythmic voice: "Taint taint ... taint taint ... taint bad! ... taint good! ... taint taint."

"Say what?!" I mumbled. But O. Wannabe was lost in thought, chewing his weed.

"Taint paint!" I suddenly blurted, remembering that I had promised the city to complete my outside house maintenance by next week! And now, taint rain, taint snow, taint but the show - I'm outta excuses to procrastinate!

About the Oracle bones (aka, Oracle of Pi) ...

Sum of squares of two small circles equals square of large circle.
Sides of squares = sqrt(Pi) [= 1.7724538..], sqrt(2(Pi)) [= 2.5066282..]
Diameters = 2, 2(sqrt(2)) [= 2.8284271..]

Say what? "Sorry, taint paint."

Rod " "

[Amigoo]

Re: http://aitnaru.org/images/cPoP_5253.pdf

The whole bag of bones.

Rod ... ...

[Amigoo]

Re: Taint Paint, Indeed (new design concept)

Fortunately, the "whole bag of bones" is a modern plastic flex-bag because the geometric objects in Taint Paint, Indeed are many! The foundational geometry of Oracle of Pi is being explored and replicated to larger circles and squares.

The Taint Paint title is a reminder to the explorer that the house painting deadline still aprproaches. In his defense, such work is now daily - geometry exploration occurs at night after his feedbag is empty and he enjoys a few hours of mindless mouse manipulation ( hopefully, "mindless" at alpha level).

Rod

[Amigoo]

Re: Taint Paint, Indeed design

I tried to take legible notes, but O. Wannabe was talkin' through his weed and I was scribblin' fast because of my "Taint Paint" dalliance.

Later, a first draft of the design (O. Wannabe's vision of the oracle "bones") was created from the notes. I suspect that his vision alludes to a Burning Man ligneous temple where man's attempts to "square the circle" during the year are honored and preserved (via the smoke "signals") in the fervid fiery finale.

"Who can tell?" I'll revist the design post Taint Paint.

Re: http://burningman.org/
"A culture of possibility. A network of dreamers and doers."

Encouraging evolution of the concept!

Rod

[Amigoo]

Re: The Pi Pointer (new design)

"O. Wannabe’s perfect state of Pi"
(when golden lines have equal length)

The whole bag of bones, indeed!

Rod ... ...

[Amigoo]

Re: The Pi Pointer design

(when golden lines have equal length)

"But wait, there's more!" A perfect state of Pi ...
when golden lines have equal length and only as drawn.

Rod " "

[Amigoo]

Re: The Pi Pointers (new design concept)

In this geometry, two sets of pointers (circle inscribed in square inscribed in circle)
show that the inner circle's scalene triangle can move into another position and prove
that each of its three lines is a side of an isosceles right triangle associated with the
larger squared circle.

Rod

[Amigoo]

Re: The Pi Pointers design
"(SIP 62.4028873643093955.. degrees)"

SIP = Shift Inner Pointer

Try a mental SIP to see how this scalene triangle fits
in the geometric composition (62.403 is close enough)

Rod ... ...

[Amigoo]

Re: The Pi Pointers design

A contrasting set* of two circles and their squares will be required to help prove geometrically that the circles are squared. The Pi Pointers geometry now includes a simple marker (one more green line) that shows a radius of the circle for the square whose side length = 1.

* in this design, largest (dark blue) circle's diameter = 2, smallest (golden) square's side length = 1 (only one side is shown).

Say what? The integrated geometry of a squared circle and circle squared.
Say what? Both circles have a similar circle-squaring Pythagorean triangle,
as well as similar circle-squaring scalene triangle.

Rod

[Amigoo]

Re: The Pi Pointers design

geometry now includes a simple marker (one more green line) that shows a radius of the circle for the square whose side length = 1

The color of the marker radius was changed from green to dark blue for better design presentation
... and the square (magenta) associated with the golden line was added.

I'm still uncertain why this radius is available but see that it is
... and defines the circle for the square whose side length = 1,
A parallelogram near the bottom of this square is intriguing!

Rod ... ...

[Amigoo]

Re: The Pi Pointers design

I'm still uncertain why this radius is available but see that it is
... and defines the circle for the square whose side length = 1

My curiosity was intense! the mystery is solved:

The radius is part of an object set (not all lines shown) that defines two back-to-back Pythagorean triangles, each having an hypotenuse equal to sqrt(Pi), and the shared side equal to Pi/2. This triangle set defines a circle whose diameter equals 2 (the largest circle in the design). Interestingly, the "parallelogram", viewed as a diamond, became the perfect pointer for solving this mystery!

Rod

[Amigoo]

Re: Concise Summary of Pi design

This Concise Summary of Pi (CSOP*) benefits from careful selection
of contrasting dimensions for circles and their squares.

* also related to another CSOP acronym:
"Command Standard Operating Procedure"
(for precise squared circle geometry)

Dimensions from smallest diameter to largest:
(D = diameter, SoCS = Side of Circle's Square)

If D = 2.0000000000000000000000000000000.. = 2,
SoCS = 1.7724538509055160272981674833411.. = sqrt(Pi)

If D = 2.828427124746190097603377448419.. = 2(sqrt(2)),
SoCS = 2.506628274631000502415765284811.. = 2(sqrt(Pi/2))

If D = 3.544907701811032054596334966682.. = 2(sqrt(Pi)),
SoCS = 3.141592653589793238462643383279.. = Pi

According to O. Wannabe, "Summary" means "fini".
Perhaps, but "Who can tell?"

Rod

[Amigoo]

Re: Concise Summary of Pi design

I was persuaded that one more squared circle (center of design) creates a better Concise Summary of Pi.
But I suspect that it's just payback for daring to think that a "Concise Summary" could be created!

Here are the revised numbers:

This Concise Summary of Pi (CSOP*) benefits from careful selection
of contrasting dimensions for four circles and their squares.

* also related to another CSOP acronym:
"Command Standard Operating Procedure"
(for precise squared circle geometry)

Dimensions from smallest diameter to largest:
(D = diameter, SoCS = Side of Circle's Square)

If D = 2.0000000000000000000000000000000.. = 2,
SoCS = 1.7724538509055160272981674833411.. = sqrt(Pi)

If D = 2.256758334191025147792317806243.. = 2(2(sqrt(1/Pi))),
SoCS = 2.

If D = 2.828427124746190097603377448419.. = 2(sqrt(2)),
SoCS = 2.506628274631000502415765284811.. = 2(sqrt(Pi/2))

If D = 3.544907701811032054596334966682.. = 2(sqrt(Pi)),
SoCS = 3.141592653589793238462643383279.. = Pi

Rod ... ...

[Amigoo]

Re: Concise Summary of Pi design
( http://aitnaru.org/threepoints.html )

BTW: The fourth circle completed the visual
geometric juxtaposition of sqrt(Pi) and sqrt(2).

Rod

[Amigoo]

Re: Concise Summary of Pi design
( http://aitnaru.org/threepoints.html )

"Sqrt(Pi) defines the square of a circle whose diameter equals 2. Conversely, a circle
whose diameter equals 4(sqrt(1/Pi)) is squared when that side length equals 2."

Something was wrong in the comparison of a squared circle and circle squared;
further analysis explained the problem: geometric perspective must include the similar
Pythagorean triangles (red) that prove correspondence between these circles.

Rod

[Amigoo]

Re: Concise Summary of Pi design
Final, final compromise (for best presentation).

This will allow you to go to the front of the line at the coffee shop
(where you might be invited to pay for everyone else in line).

Customer: I'll have a cappuccino with a sqrt(Pi).
Barista: I'm sorry, we don't serve squirts of pie in our coffee.

Rod

[Amigoo]

Re: Concise Summary of Pi design

The design in the PDF file has one more line (upper Pythagorean triangle, horizontal top line)
to illustrate the foundational geometry of a squared circle:

Vertical line = Pi/2, hypotenuse = sqrt(Pi), circle's diameter = 2,
lower vertex = 27.597112635690604451732204752339.. degrees.

Area of circle whose diameter = 2
= ((cos 27.597112635690604451732204752339..) x diameter) squared
= ((0.88622692545275801364908374167057..) x 2) squared
= (1.7724538509055160272981674833411..) squared
= 3.1415926535897932384626433832795..
= Pi

Rod

[Amigoo]

Re: Concise Summary of Pi design

Sufficient closure for a long coffee break ...

"Coffee table" version: http://aitnaru.org/images/CSoP.pdf
Updated online page: http://aitnaru.org/threepoints.html
39-page PDF: http://aitnaru.org/images/cPoP_5253.pdf

Rod ... ...

[Amigoo]

Re: Concise Summary of Pi design

What is the Pi-complement* of the vertex angle
of the Pythagorean triangle? arccos (sqrt(Pi)/2)

arccos(.88622692545275801364908374167057..)
= 27.597112635690604451732204752339.. degrees.

* the vertex angle that is as accurate as Pi
(scientific applications often require less than 40 digits)

Rod

[Amigoo]

Re: Concise Summary of Pi design
A related perspective on this unique vertex angle:

What is arccos(sqrt(Pi)/2)?

The trigonometric function that defines the vertex angle of a Pythagorean triangle that squares the circle:
arccos(.88622692545275801364908374167057..) = 27.597112635690604451732204752339.. degrees.

For a circle having a diameter equal to 2, the triangle's long side (circle's chord; side of its square)
= sqrt(Pi) and its hypotenuse = 2 (circle's diameter), with the vertex point on the circumference.

Postscript: An expert says "arccos(pi^(1/2)) is undefined because the square root of pi is greater than 1."
So, the geometry in the design only looks like squared circles ... apparently.
... or is there a disconnect between the math and the geometry?

Rod

[Amigoo]

Re: Concise Summary of Pi design
A related perspective on this unique vertex angle:

The expert's opinion was removed from that blog ??
Apparently, arccos(sqrt(Pi)/2) is still valid conjecture.

Not "how to get there" but "what there looks like".

Rod " "

[Amigoo]

Re: Aye Captain (new design concept)

I've been lamenting how no geometry exists where objects (alone or in a group) can be divided in half to prove that the circle is squared. While this new design concept seems to emphasize (again) that the creativty toy box is bottomless, it also teases that proof-by-halves may indeed exist.

Building on the foundational geometry of Concise Summary of Pi, this design incorporates an "eye" in the middle of an isosceles trapezoid. And this visual hint suggested that "Aye Captain", as a design title, would offer acknowledgment of the possibility as well as agreement with the invitation to keep exploring.

... but no galloping across the Cartesian plane, this time.
"Aye Captain!"

Rod

[Amigoo]

Re: Aye Captain design
"Salut! these uncharted waters."

Having drifted intentionally far off course, the captain ordered a new flag hoisted:
"Let they who salute our wayward voyage reveal themselves!" he bellowed.

Rod