Paradise Trinity Day

[Amigoo]

Re: Scalene Correlating Redundancy (SCoRe; new design concept)

After seeing the first draft on a clean Cartesian whiteboard,
I get the point! This is Pi for the rest of the universe!
... not Simple Simon's country fair recipe.

A convincing composition* of 7 circles, all squared, in increasing sizes:
sqrt(Pi)/4, sqrt(Pi/2)/2, sqrt(Pi)/2, sqrt(Pi/2), sqrt(Pi), 2(sqrt(Pi/2)), 2(sqrt(Pi))

Pythagorean promotion of the heavenly state of Sanitas Cyclometricus.

* with scalene and isosceles right triangles upon a foundation of
two overlapping inscribed squares within the largest circle.

Answer quickly! How many 2s were typed above? Good guess! (neves)

Rod

[Amigoo]

Re: Scalene Correlation Replication design (SCoRe; better name)

The smallest replicants, a Circle and its Square,
are discussing this Cartesian progression:

C: What's the SCoRe?
S: What's the point?"

C: Are we ahead?
S: Just behind.

C: What's the point?
S: We're behind
the point ahead.

C: Oh! The SCoRe!
S: Oh! The point!

(funnier when you're on the bus,
wondering "Are we there yet?")

Rod " "

[Amigoo]

Re: Scalene Correlation Replication design (SCoRe)

Ooops! Missed a few line colors! (then updated the humor below)
What? Geometry this complex would have simple humor?

Scalene Correlation Replication (SCoRe)
Pythagorean promotion of the heavenly state of Sanitas Cyclometricus.

Upon seeing the first draft on a clean Cartesian whiteboard, I get the point!
This is Pi for the rest of the universe - not Simple Simon's county fair recipe.

The smallest replicants, a Circle and its Square, are discussing
this Cartesian progression of seven circles*, all squared:

C: What's the SCoRe?
S: What's the point?

C: Are we ahead?
S: Just behind.

C: What's the point?
S: We're just behind
the point ahead.

C: Oh! The SCoRe!
S: Oh! The point!

C: Surely, you jest.
S: My name's not ...
Oh! What's the point?
Oh! What's the SCoRe?
Oh! Where's the finity?
(Oh! WTF?)

(funnier when you're on the bus,
wondering "Are we there yet?")

* Diameters from smallest to largest:
sqrt(Pi)/4, sqrt(Pi/2)/2, sqrt(Pi)/2, sqrt(Pi/2),
sqrt(Pi), 2(sqrt(Pi/2)), 2(sqrt(Pi))

Rod

[Amigoo]

Re: PS5253 (new design concept)

"Postscript" for the mysterious x:52:53 Midwayer time prompts.
(x:52 immediately followed by x:53)

Geometry in SCoRe convinces me that x:52:53 refers to
sqrt(Pi)/2 (prompt x:52 = 0.8862 26 9254.. twice) and
sqrt(Pi) (prompt x:53 = 1.7724 53 8509..).

Geometers will notice the influence of Pythagorean right triangles
in this display of three nested circles, all squared.

This Cartesian progression?
D = sqrt(Pi)/2, sqrt(Pi), 2(sqrt(Pi))

Rod ... ...

[Amigoo]

Re: PS5253 design

Who knew? Even a postscript design can evolve!

I was determined to draw a lower-left to upper-right diagonal
to highlight the large trapezoid (includes the red scalene triangle),
but the design refused this intrusion in its geometric balance.

While contemplating the impasse, I was persuaded that
"The geometry is accurate - just make it interesting."

Been there! - Done that!

Rod

[Amigoo]

Re: iSpiral (new design concept)

If squaring the circle is impossible, this Pythagorean Cartesian spiral
will not be geometrically real - it just looks that way.

Nonetheless, it's an impressive spiral of seven isosceles right triangles,
reminiscent of the tumble down the rabbit hole (I forgot which way
I was tumbling upon closure of that adventure).

It may not matter ... we would comprehend neither the beginning
nor the end of infinity; such is the essence of squared circles en spiral.

But I did learn a valuable lesson in the rabbit hole:
select the recipe that allows you to have your Pi
and nibble its transcendental bits.

Rod

[Amigoo]

Re: iSpiral (new design concept)
From the Notebook of Wiggly Numbers:

The vertex point of the smallest angle of a right triangle,
when positioned in a corner of the square of the largest circle,
identifies the exact center of this Pythagorean iSpiral.

long side = 2.221441469079183123507940495030..
hypotenuse = 2.50662827463100050241576528481..
vertex angle = 27.59711263569060445173220475..

2.22144.. = length of one side of the square
of a circle whose diameter = 2(sqrt(Pi/2))
= 2.50662827463100050241576528481..

diameter of largest circle = 2(sqrt(Pi))
= 3.5449077018110320545963349666823..
side of this circle's square = Pi
= 3.1415926535897932384626433832795..

Of course, this is theory and the numbers have been
rounded to minimize sharp objects in the bus whose
occupant is wondering "Are we there yet?".

Rod

[Amigoo]

Re: iSpiral (new design concept)

The vertex point of the smallest angle of a right triangle,
when positioned in a corner of the square of the largest circle,
identifies the exact center of this Pythagorean iSpiral.

It didn't take long to find the geometric flaw in this logic:
A corner-positioned right triangle is not relative to all circle+square sets
along the spiral. However, tracing these sets inward showed that the
trapezoidal pattern (its Cartesian placement) repeats every 9th circle.

Not only does every circle in the spiral contain this similar trapezoid,
but each set of 8 circles begins with the similar trapezoid aligned
the same in this Cartesian neighborhood.

What's so special about this repeating pattern?

Pi is perfectly tracked in 360 degrees of this spiral,
along with its irrational, transcendental mathematical essence.
Which seems to imply that at one finite point on the 360 circle,
Pi is neither irrational nor transcendental.

"Who can tell?" just from the window of a bouncing bus?

Rod

[Amigoo]

Re: iSpiral (new design concept)

If focus is on each right triangle's 90-degree vertex,
three 360-degree turns around the spiral are required to reach
the 9th circle, advancing clockwise and inward 135 degrees
from focus point to focus point.

The trapezoid consists of an isoceles right triangle
and a non-isosceles right triangle - the 90-degree vertex
of focus is in the non-isosceles triangle.

And this requires slight correction of the claim:
"seems to imply that at one finite point on every third 360 circle,
Pi is neither irrational nor transcendental."

Say what? You can't get there from here unless you
travel in circles ... all squared.

Rod

[Amigoo]

Re: iSpiral (new design concept)
From the Notebook of Wiggly Numbers:

iSpiral Landmarks Down the Rabbit Hole
(aka, proof of circular transit DTRH)

C. = Circle, Incre = Increment in degrees,
Degrees = finite points on the circles

C. Incre - Degrees
1. 0000 - 332.403
2. 0135 - 197.403
3. 0270 - 062.403
4. 0405 - 287.403
5. 0540 - 152.403
6. 0675 - 017.403
7. 0810 - 242.403
8. 0945 - 107.403

9. 1080 - 332.403 (1080/360 = 3)

Note: Degrees are rounded;
90-degree vertex of right triangle

BTW: Would not such geometry identify
a repeating pattern in the decimal digits of Pi?

Rod

[Amigoo]

Re: iSpiral design

Replication inward was easy ...
once I learned to follow the fat angle (90-degree vertex)

Would not such geometry identify
a repeating pattern in the decimal digits of Pi?

iSpiral is so visually convincing that, if the repeating pattern of Pi's decimal digits
is not found, the model for calculating Pi must be incorrect.

Rod

[Amigoo]

Re: iSpiral design

Geometer's TIP: the precise center of the spiral is identified by triangulating
the 90-degree to 90-degree vertices across three sets of eight squared circles
(continue inward, drawing two more circles).

Say what? This is the geographic center of infinity
(for this Cartesian neighborhood).

Rod

[Amigoo]

Re: iSpiral design

iSpiral is so visually convincing that, if the repeating pattern of Pi's decimal digits
is not found, the model for calculating Pi must be incorrect.

I was just reminded by a 4:03 prompt that while a repeating pattern might be found,
it would not necessarily be THE repeating pattern - just an interim pattern.

This perspective, of course, acknowledges that 4:03 relates to a rounded number*
and that many more decimal digits follow the rounding (aka, neuronal bashing).

* 90 - 27.597112635690604451732204752339..
= 62 .402 88736430939554826779524767..

So little time and so many recipes of Pi to savor!

Rod

[Amigoo]

Re: iSpiral design (an observation)

The center of every isoceles right triangle (center of the spiral)
is located such that a line drawn from the center to each vertex
creates three angles: 135, 135, 90 degrees.

Apparently, traveling DTRH is quite predictable, geometrically speaking
... when the spiral consists of perfectly squared circles.

Rod

[Amigoo]

Re: Pythagorean iSpiral design

The center of every isoceles right triangle (center of the spiral)
is located such that a line drawn from the center to each vertex
creates three angles: 135, 135, 90 degrees

Now the design complements this quote, displaying tight integration
then replication of the trapezoidal geometry ... in 9 circles squared.
(longest green line of right triangle squares its enclosing circle).

Instead of corner-to-corner linkage, the end of a short green line
links to the end of a similar short green line in the 4th circle inward.
Tumbling DTRH with Cartesian precision!

Rod

[Amigoo]

Re: Pythagorean iSpiral design

Having experiential knowledge about tumbling in squared circles
(in Cartesian neighborhoods), I contemplated the logic of the lessons:

If P then Q. Q. Therefore P.

1. Pythagoras first proved the Pythagorean Theorem to a querulous Quorum of competent Cartesian mathematicians.

2. The Pythagorean Theorem describes a Quorum of squared circles in Cartesian rabbit holes of trapezoidal construction.

3. Therefore, Pythagoras proved that rabbit holes exist in querulous Quorums of competent Cartesian mathematicians.

Rod

[Amigoo]

Re: Pythagorean iSpiral design

Falling both ways in a squared circles rabbit hole was unnecessary
since the large golden isosceles right triangle contains within its points
clear direction to the spiral center, inspiring poetic tribute to:

The Rabbit Hole of Pi

135, 135, 90 degrees,
isosceles segments - triangular tease,
since "X" reveals center
of spiraling threes in
trapezoidal construction
(rabbit hole g'ometry).

Rod

[Amigoo]

Re: Pythagorean iSpiral design

since "X" reveals center of spiraling threes

Now the "X" is colored red and includes a red diagonal pointer.

It's easier to "trap a zoid" in this improved "rabbit hole g'ometry",
but stay close to the red diagonal of length 2(sqrt(Pi/2)) if you
plan to return (aka, "falling up the rabbit hole").

sqrt(Pi/2) is the length of a midpoint-to-midpoint diagonal
(not displayed) that can be created in the largest circle-squaring
scalene triangle (not completely displayed).

What's the point? "triangular tease"

Rod ... ...

[Amigoo]

Re: Pythagorean iSpiral design

Having just read that
"Once a Zoid gets inside
There is nothing you can do",
I quandered then queried
srqt(Pi slash 2):

"To trap a zoid, probe
rabbit hole g'ometry
where up falling down
and down falling up
is likely to be
"impossible". See?
where "X" reveals center
of spiraling threes
and trap-e nets trap-a
... esoterically."

Rod

[Amigoo]

Re: Three Point One Four (new design concept)

This seems to be a refresher lesson in squared circle geometry:
"The toy box is bottomless - rummage around as much as you like and
enjoy the pattern replication, but quit expecting a simple solution."

So, I'll complete the exploration, then the design, once I get a round toit
after finding my exploration Pi shoe* (size 8(sqrt(Pi/2)), lost in the toy box.

* nicknamed Macho Picchu for brazen "out of the box" conjecture and
Machu Picchu's disputed reference: "Lost City of the Incas".
(but this "Picchu" is pronounced "Pi shoe")

Rod

[Amigoo]

Re: Three Point One Four design
"The replicating points of Pi"

Six circles, all squared, promoting transcendental appreciation
of the beloved mathematical constant known as "Pi".

Rod

[Amigoo]

Re: Three Point One Four design
"The replicating points of Pi"

Numbers are included on the page following the design, making this
a good visual reference for Pi's relationship to squared circles
and for the circle-squaring scalene triangle.

Most squares of circles are not drawn (for better presentation),
but the geometry identifies how to draw them.

Rod ... ...

[Amigoo]

Re: Three Point One Four design
"The replicating points of Pi"

Even before finding my Pi shoe ("Macho Picchu", size (8( sqrt(Pi/2))),
I was able to rummage sufficiently to find a most intriguing collection
of geometric objects; "Points of Order" will be its design name.

This geometry contrasts similar points in two squared circles
in a way that brings focus back to the right triangle that is
part of the circle-squaring scalene triangle, together
creating the circle-squaring trapezoid.

Rod

[Amigoo]

Re: Points of Order design
“What’s the point?” indeed!”

135, 135, 90 degrees,
isosceles segments - triangular tease,
since "X" reveals center

Since the red "X" in this geometry supports a smaller circle,
I can only say "Been there! Done that!"

As usual, some lines are not drawn for better presentation,
but the small blue circle well-complements the sub-title":
“What’s the point?” indeed!”

... for if a certain point on that circle is identified,
the line that squares that circle can be drawn!

This file contains only "Three Point One Four" and "Points of Order":
http://aitnaru.org/images/Three_Point_One_Four.pdf

This file contains 47 pages of recent designs:
http://aitnaru.org/images/cPoP_5253.pdf

Rod

[Amigoo]

Re: Points of Order design
“What’s the point?” indeed!”

135, 135, 90 degrees,
isosceles segments - triangular tease,
since "X" reveals center

I sensed something unusally important about the golden yellow "T"s,
finally discovered that significance, and recolored the scalene triangle
to show that the red "X' and the "T"s correlate geometrically.
"triangular tease" indeed!

Rod ... ...