Paradise Trinity Day

[Amigoo]

Re: Spiraling Squares of 4(4(4 (new design concept)

Since LR Squares of X( was discarded, it's easy to rationalize a replacement,
but redundant geometry keeps shouting that the portfolio is complete.

So, 4(4(4 has good opportunity to become the final(final(final design
... but "(not) seeing is believing."

Note: "4(4(4" describes the spiraling squares' 4X increase in area
as the squares spiral outward from the center of the spiral.

Rod

[Amigoo]

Re: Spiraling Squares of 4(4(4 design

Geometers will easily notice that the squares actually increase in
increments of sqrt(2) and that another set of spiraling squares overlays
the first set with the areas in both sets increasing by sqrt(2).

However, since my original focus was on every other square in a set,
the 4(4(4 title was my preferred choice (and 444 is a common prompt).

Note: one set of spiraling squares highlights the circles' inscribed squares
and the other set highlights the squares of the same circles. If there are
more sets, I haven't "Been there! Done that!"

Rod ... ...

[Amigoo]

Re: Spiraling Squares of 4(4(4 design

The "wiggly numbers" (geometric dimensions) for these circles and their squares are listed below. The largest circle is not displayed but its side of square equals sqrt(Pi) (the line displayed as an arrow). Does this arrow point to Paradise? It would if the Cartesian drawing is properly aligned in tri-dimensional space:

"Observation of the so-called Milky Way discloses the comparative increase in Orvonton stellar density when the heavens are viewed in one direction, while on either side the density diminishes; the number of stars and other spheres decreases away from the chief plane of our material superuniverse. When the angle of observation is propitious, gazing through the main body of this realm of maximum density, you are looking toward the residential universe and the center of all things."
(UB 15:3.3, http://www.theub.org/part-i.html#P015_3_3 )

Diameters (smallest to largest) and side
of circle's square (both increase by sqrt(2)):

0.125
0.11077836568159475170613546770881..

0.17677669529663688110021109052621..
0.15666426716443753140098533030068..

0.25
0.22155673136318950341227093541762..

0.35355339059327376220042218105242..
0.31332853432887506280197066060135..

0.5
0.44311346272637900682454187083525..

0.70710678118654752440084436210485..
0.62665706865775012560394132120271..

1.0
0.088622692545275801364908374167057..

1.4142135623730950488016887242097..
1.2533141373155002512078826424054..

2.0
1.7724538509055160272981674833411..

Rod

[Amigoo]

Re: Spiraling Squares of 4(4(4 design

This 360 degree geometry shouts that whatever the value of Pi, it must
be evenly divisible by 8: Transcendental / 8 = Return to Terra Firma?

Rod

[Amigoo]

This 360 degree geometry shouts that whatever the value of Pi,
it must be evenly divisible by 8

Oops! The divisor was incorrect - this is the relationship of the
two right triangles, one a subset of the other (after 360 degree spiral),
where hypotenuse = diameter, long side = side of circle's square:

Diameter ~ Side of square:
.125 ~ 0.11077836568159475170613546770882..
1.00 ~ 1.7724538509055160272981674833411..

To restate that bright idea:

This 360 degree geometry shouts that whatever the value of Pi, it must
be evenly divisible by 16: Transcendental / 16 = Return to Terra Firma?

Rod

[Amigoo]

Re: Al Punto (new design)
"Squared Circles Of Unique Persuasion" (SCOUP)

There goes "final(final(final" ... but there's hope!
"(not) seeing is believing" is the clue to a final design.

BTW: This page appeared now when I fat-fingered a key,
but it was neither a programmed key nor an active tab:
http://www.theub.org/ "Who can tell?"

Rod ... ...

[Amigoo]

Re: Al Punto design (redrawn as SCR16*)
"Squared Circles Of Unique Persuasion (SCOUP)"

* Squared Circle Rule of 16 (dimensions of 16 decreasing by /2)
(see: http://aitnaru.org/images/Three_Point_One_Four.pdf )

Rod

[Amigoo]

Re: Al Punto design

From Spiraling Squares of 4(4(4 design:

"This 360 degree geometry shouts that whatever the value of Pi, it must
be evenly divisible by 16: Transcendental / 16 = Return to Terra Firma?"

Al Punto highlights this apparent Squared Circle Rule of 16 (SCR16).
The geometry is displayed as dimensions of 16 decreasing by /2.
Hmmm ... 1,2,4,8,16 comes to mind.

How many circles are squared? all of them, via the
62.4028873643.. degree radius (red right triangle).

How many points are required to square the circle?
Hint: red right triangle (squares concentric circles)

Rod ... ...

[Amigoo]

Re: Al Punto design
"Squared Circles Of Unique Persuasion"

A few more lines were added in the study of a circle
inscribed in a square (green, diameter = 1).

But now, the red right triangle is subdivided and,
after a 360 degree journey in this Cartesian neighborhood,
I've returned to the original question: "What's the point?"

Rod

[Amigoo]

Re: Al Punto design (SCR16)
"Squared Circles Of Unique Persuasion"

Everyone knows that "Al Punto" means "to the point"
... but then they ask "What's the point?"

This would make a great dart board! even better as a projected image
with computer-assisted highlighting as the game progresses.
(get that commercial point?)

Notes: largest circle's diameter = 2, smallest = 2/16 = 0.125;
SCR16 = Squared Circles Rule of 16 (applied to this geometry)

Rod

[Amigoo]

Re: Al Punto design (SCR16)
"Squared Circles Of Unique Persuasion"

BTW (re: circle-squaring right triangles) ...

This geometry of squared circles highlights
the math rule for working with proportions:
[ product of means = product of extremes ]
(diameter : square's side ~ diameter : square's side)

Given the diameters of 2.0 and 1.1283791670955125738961589031215..
(largest circle = 2.0, topmost = 1.1283791670955125738961589031215..)

If 2.0 : sqrt(Pi) ~ 1.1283791670955125738961589031215.. : X,
then, X = (sqrt(Pi) * (2/sqrt(Pi))) / 2 = 1.0

Therefore ...

2.0 : 1.7724538509055160272981674833411..
~ 1.1283791670955125738961589031215.. : 1.0

Consequently, if circle 2.0 is squared,
circle 1.1283791670955125738961589031215.. is squared!

Rod

[Amigoo]

Re: Al Punto design (SCR16)
"Squared Circles Of Unique Persuasion"

largest circle = 2.0, topmost = 1.1283791670955125738961589031215..

"topmost" refers to the green arc - not recently added golden arc.

About that golden arc ...
If an arc with more than half of the circle's circumference can be rounded to "circle",
is an arc with less than half of the circle's circumference just an "arc"? or pre-circle?
And are at least three points required to define an arc?

"What's the point?" Are we there yet?

Rod

[Amigoo]

Re: Secret of St. Gria design
"... but 'What's the secret?'"

Derivation of "St. Gría" (and minimal explanation of the wandering allusion):

While evaluating the Al Punto (SCR16) geometry and thinking that the "golden arc"
looked like a shock top (allusion to an internet moment of creativity), I realized
that the golden arc needed more visual explanation (what's its purpose?).

Well ... its purpose still remains the secret of geometers,
but its name is the result of "wandering allusion":

"St." alludes to "shock top" (ST).
Alegría is the Spanish word for joy.
The missing "ale"? I drank that beer.

Long story very short ....

This St. Gría is a mythical patron saint of joy.
(satisfying progress in the Cartesian neighborhood)

Rod ... ...

[Amigoo]

Re: Secret of St. Gría design (updated)
"... but 'What's the secret?'"

PROCEDURE DIVISION.

MAIN-POINT.
DISPLAY 'BEGINNING OF POST'.

While looking esoteric because of color choices, this design is simply a geometry study of a circle (diameter = 2) and its inscribed square, effectively halved by CSC geometry (circle inscribed in square inscribed in circle ...). And the geometry of the upper squared circle (green) "reflects back" to the center green circle of similar diameter; these two green circles form a vesica piscis, sharing a common radius.

Say what?
Diameters (D) and sides of inscribed square (S), starting from the outer object in this set
(the upper green circle): D = 2, S = sqrt(2), D = sqrt(2), S = 1

Geometric "sleight of hand" for this Cartesian gaming table; a prestidigitation of Pi!

Entertaining! ... but "What's the secret?"

It answers the question: "What's the point?"

Intriguing! ... but "What's the point?"

GO TO MAIN-POINT DEPENDING ON TIME-AVAILABLE.

FINAL-POINT.
DISPLAY 'BEEN THERE! DONE THAT!'.
STOP RUN.

Rod ... ... (back to the gaming table)

[Amigoo]

Re: Secret of St. Gría design (updated again)
"... but 'What's the secret?'"

Now that's esoteric! (not the design, that's basic squared circle geometry -
the way new objects appear during a stroll in this Cartesian neighborhood).

Also esoteric is how confusing it has been to explain the CSC feature.
Another version: large circle's diameter = 2; square of interest has side length of 1.
The CSC relationship is how 2 = 1 (aka "why the food product I purchased is 1 ounce
instead of the 2 suggested by the packaging").

And most esoteric is why the square of interest is not displayed
(but squared circle geometers know where it should be).

Rod

[Sandy]

Hi Rod,
Do you think you could post the URL for your latest creation of geometric wonder... "Secret of St. Gría design" again. I can't find it.
However I must say I have enjoyed perusing through your designs and I absolutely love "Extraordinary Pi Inside Catwalk"! I feel like I could walk into it and feel motion... so lovely! They all are just that, to tell you the truth. I may not be able to understand the mathematics that got you there, but I can sure enjoy the gorgeous designs!
xxSandy

[Amigoo]

Sandy,

The current geometry designs are accumulating here:
http://aitnaru.org/images/Three_Point_One_Four.pdf

The Secret of St. Gría is another geometry study, but I wandered around the
Cartesian neighborhood with coloring "pencils" and whimsical embellishment.
It looks more esoteric than it is, but might contain a secret or two.
I'm working on a companion study that is more geometry than artistry.

Extraordinary Pi Inside Catwalk is a rare study of replicating patterns,
rare because I maintained good focus for many hours of CAD design,
then many hours of its conversion to artistic interpretation.

Rod

[Amigoo]

Re: Outside the Box (new concept)

The title of this design became apparent quickly when the geometry
revealed a 3-dimensional box. So, the foundational geometry seems
certain - conversion to artistic design is next.

Maybe the Secret of St. Gría is now out of the box!

Rod

[Amigoo]

Re: Outside the Box (developing concept)

If not the Secret of St. Gría,
a clue about the path to follow!

Upon discovering a 3-D box within a 3-D box
(nested squared circles with a vesica piscis),
the message of the geometry is obvious:

To think outside the box is grand,
but to be outside the box is sublime!

The real "shock and awe" of the universe.

Rod

[Amigoo]

Re: Outside the Box ("Been there! Done that!")

Ask not "What's outside the box?"
Ask "Whence the walls of the box?"
"To think outside the box is grand,
but to be outside the box is sublime."

The real "shock and awe" of the universe.

Rod ... ...

[Amigoo]

Re: Outside the Box ("new & improved" )
Added the missing vesica piscis (dark blue).

"New & improved" because every line had to be redrawn
to layer the extra circle into the background.

Vesica complained that she has a younger sibling,
but that extra circle clutters the boxes (which is sad
because the extra circle is nestled in Vesica's lap).

Rod

[Sandy]

Thanks Rod,
I do enjoy looking at the complex designs. It is sort of like watching clouds.... when you sit back and with soft focus,allow the design images to come through. Very relaxing.

Have a great week end!
xxSandy

[Amigoo]

Re: Quadrature of Vesica (new design concept)

Speaking of vesica piscis ... Who knew?
Squares of the two vesica circles can also form a vesica piscis!

If the definition of "Vesica Piscis" is:
"a shape that is the intersection of two circles with the same radius, intersecting in such a way that the center of each circle lies on the perimeter of the other",

The definition of "Quadrature of Vesica" (proposed) is:
"a shape that is the intersection of two squared circles with the same radius, intersecting in such a way that the center of each circle lies on the perimeter of the other, and the center of each square lies on the perimeter of the other square"

Rod

[Amigoo]

Re: Quadrature of Vesica design

"Been there! Done that!"

Presenting ... the world's first Quadrature of Vesica.
http://aitnaru.org/images/Three_Point_One_Four.pdf

Rod ... ...

[Amigoo]

Re: Quadrature of Vesica design

Despite the popularity of the "Outside the Box" theme
in this Cartesian neighborhood, I had to be convinced that
connecting the corners of the box was salient geometry.
(lengths of extra lines relate to radii of the circles)

And I'm certain that our Midwayer friends (all 1984+)
consider two dimensions one short of material reality.

Rod