11:11 Angels Message Board

Sandy,

That is a fascinating design. I find I lose myself in its "movement"

I believe that the human brain is wired (evolved from intelligent design) to detect patterns, for this "wiring" seems to acknowledge (at least, sense) complex geometry even without knowing the mathematical rules. For example, each set of two dark blue lines describe a squared circle (only the largest square is drawn): the inner circle when enclosed defines the square of the larger circle. Even without the square visible, our brains seem to be aware of this unique relationship between the two circles. Further, we also sense a geometric relationship amongst the three sets!

And I now suspect that pattern recognition is part of celestial "communication": when permitted by human free will and allowed by our thought adjuster ("spark of God" within), information is stored in our brains as patterns (maybe "neuronal packets"). Then, when the timing is right (pattern recognition/association is active), we're able to access the stored information. But free will still allows us to choose (when given choices) or to include the stored information in our projects and deliberations.

Geometer's tip: The three points of the smallest vertex of the scalene triangles (two yellow; one red with vertical line) define an isosceles right triangle in the lower right of the design (one line not drawn). This small Pythagorean triangle seems to have unique correlating properties for the scalene trio.

Rod

Re: http://www.answers.com/Q/What_is_leftover_pi_day

Fresh Pi may also be served on Leftover Pi Day
... but check with the pie man first:

Simple Simon met a pie man departing from the fair.
Said the pie man to Simple Simon: "My pies are in repair
to lockers cold, thus wrapped - not sold; left over fare is fair
for sweet repose then new expose of pie beyond compare!"

Rod

"Simple Simon met a pie man, departing from the fair.
Said the pie man to Simple Simon: 'My pies are in repair
to lockers cold, thus wrapped - not sold; leftover fare is fair
for sweet repose, then new expose of pie beyond compare!'"

I'm somewhat persuaded that this poem is "geometrically positioned"
to anticipate a new & improved Pi constant (or formula)
... give or take a "few" decimal digits.

Rod

Re: Open Box of Pi design
"Who let the Pi out?"

I'm somewhat persuaded that this poem is "geometrically positioned"
to anticipate a new & improved Pi constant (or formula)

Curiosity about which objects in this geometry would support a "new & improved" Pi constant or formula, resulted in a few technical notes (below). I was focused on the largest scalene triangle (yellow) and the next largest (red) and how these objects relate to the small isosceles right triangle (one red line, two green) in the lower right of the design. "Veddy interesting!" indeed. (hypotenuse of small right triangle = sqrt(Pi/2))

Given: circle's diameter = 2(sqrt(Pi))
= 3.54490770181103205459633496668..
side of circle's square (calculated) = Pi
= 3.1415926535897932384626433832795..

Pi/sqrt(Pi/2) = 2(sqrt(Pi/2))
3.1415926535897932384626433832795..
/ 1.25331413731550025120788264240..
= 2.50662827463100050241576528481..
= side of circle's inscribed square

Rod

I was off the computer yesterday and just finding this today. So I'm going to bring up the design again along with your posts and thoughts on the design and hopefully learn a little something.

It's 2:00 am and I'm heading to the kitchen for serious nibbles
... while the spaghetti sauce is warming!

I'm a world class nibbler while I cook... to the point its down right shameful. I've been craving spaghetti lately. It's not G's favourite, though, so I usually swing towards other things. But am wondering if maybe I should indulge the craving as our body often tells us what we need or am lacking.

Some pie would be nice as well!
xxSandy

EZ visualization exercise.
Imagine two concentric circles, one smaller than the other,
and both circles have an inscribed square, also concentric.

Given: Circle D (largest circle) with diameter = d
= Pi = 3.1415926535897932384626433832795..

Square defined by sqrt(Pi) has side length
= 1.7724538509055160272981674833411..

Diameter of Circle E enclosing this square
= 2(sqrt(Pi/2)) = 2.506628274631000502415765284811..

Side of inscribed square of Circle D
= 2.2214414690791831235079404950303..

Therefore, inner Circle E is squared (re: A = Pi r^2).

Say what?

sqrt(d) defines a square whose enclosing circle (Circle E)
is squared by the inscribed square of Circle D.

Say what?

To square the circle, one must circle the square.

Rod

To square the circle, one must circle the square.

Attempts to confirm this "fuzzy math" suggest another interpretation of this quote:
"You can't get there from here."

Yes, such geometry does exist (two concentric circles with inscribed squares),
but the associated math claimed looks quite fuzzy this morning!

Even worse, the digital antenna in the attic was not performing as well
last night. "Fuzzy math" hints that the house shifted a few degrees during
my afternoon nap ... or maybe it's those tectonic plates ... or maybe
one-too-many plates of impromptu late-night spaghetti ... or ...

For a break, I'll next visit a friend who claims that curious gnomes are
chatting on a regular schedule at night in his house. Maybe it's his antenna!
... and more exercise is prescribed for this fuzzy geometer.

Rod

Re: iSquares Concentricity design

"You can't get there from here."

Perhaps still true, but at least the geometry supports this quote:

"To square the circle, one must circle the square."

The wiggly numbers for
inscribed Squares Concentricity:

Diameter of light blue circle = 2.
Side of light blue square = 1.
Diameter of golden circle = sqrt(2)
= 1.4142135623730950488016887242097..
Side of circle's square = sqrt(Pi/2)
= 1.2533141373155002512078826424055..

Circle-squaring red Pythagorean triangle:
Cosine angle = 27.597112635690604451732204752339..
Diagonal red line (hypotenuse) = 1
Vertical red line = sqrt(Pi)/2

Rod

Re: iSquares Concentricity design
“Perchance, to see ‘there’”

Hmmm ... exploration in alpha mode following a short afternoon nap
resulted in metamorphosis of the design with return of the butterfly theme.
What's this geometry have on ... it's mind?

Incidentally, the alarm of a bedroom clock just rang at 2:55.
(time is 8 min. fast and this alarm has been silent for years)

Maybe I'll learn more if we find the Gnome News Channel
tonight at my friend's house.

This is the same friend that I was driving to visit in late 2009
when Midwayer prompt 7:07 appeared on my van's dashboard clock,
motivating more dedicated geometry research. [sqrt(2)/2 = .707..]

Rod ... ...

Maybe I'll learn more if we find the Gnome News Channel tonight at my friend's house.

Long story short: An assortment of gnomes throughout the house, but all were quiet. Best guess: my aging friend's poor eyesight and hearing is complicated by side effects of medications and short-term memory issues. The only unusual activity was an overweight, female bulldog that kept hunching on my leg, but it's nice to feel wanted.

Rod

Glad to hear your friend was not the victim of a gnome infestation... at least this time around.

Sandy

Re: iSquares Cx4 design

Replication of patterns from iSquares Concentricity, highlighing the
unique scalene triangle that exists in every squared circle, and providing
proof that squared circles must be studied as complex geometry.

Rod

Re: iSquares Cx4 design
"Extraordinary Pi Inside Catwalk"

Who knew? Pi inside an enclosed catwalk!
(maybe the anticipated New Pi)

This design had 14 concentric squares that were
"slightly" modified for better presentation:

Concentric squares of Extraordinary Pi Inside Catwalk
(side length of square from largest to smallest)

01 - S = 4.0
02 - S = 2(sqrt(Pi)) = 3.5449077018110320545963349666823..

03 - S = Pi = 3.1415926535897932384626433832795..

04 - S = 2(sqrt(2)) = 2.8284271247461900976033774484194..
05 - S = 2(sqrt(Pi/2)) = 2.506628274631000502415765284811..

06 - S = Pi/sqrt(2) = 2.2214414690791831235079404950303..

07 - S = 2.0
08 - S = sqrt(Pi) = 1.7724538509055160272981674833411..

09 - S = Pi/2 = 1.5707963267948966192313216916398..

10 - S = sqrt(2) = 1.4142135623730950488016887242097..
11 - S = sqrt(Pi/2) = 1.2533141373155002512078826424055..

12 - S = (Pi/sqrt(2))/2 = 1.1107207345395915617539702475152..

13 - S = 1.0
14 - S = sqrt(Pi)/2 = 0.88622692545275801364908374167057..

Rod ... ...

Re: iSquares Cx4 design (with complete squares)
"Extraordinary Pi Inside Catwalk" (EPIC)

A recap of the adventure ...

What's the meaning of "Pi on a catwalk"? Purr fection.
The "S" in the design? Scalene, of course! (triangle)
"Inside" instead of "on" because catwalk is enclosed.
(What? A square does not enclose?)

Ever been on a Pi catwalk? Watch that distant step!
It's the Sanitas/Morbus Cyclometricus border!

14 concentric squares in this design of 10 squared circles:
(side length of squares from largest to smallest)

01 - S = 4.0
02 - S = 2(sqrt(Pi)) = 3.5449077018110320545963349666823..

03 - S = Pi = 3.1415926535897932384626433832795..

04 - S = 2(sqrt(2)) = 2.8284271247461900976033774484194..
05 - S = 2(sqrt(Pi/2)) = 2.506628274631000502415765284811..

06 - S = Pi/sqrt(2) = 2.2214414690791831235079404950303..

07 - S = 2.0
08 - S = sqrt(Pi) = 1.7724538509055160272981674833411..

09 - S = Pi/2 = 1.5707963267948966192313216916398..

10 - S = sqrt(2) = 1.4142135623730950488016887242097..
11 - S = sqrt(Pi/2) = 1.2533141373155002512078826424055..

12 - S = (Pi/sqrt(2))/2 = 1.1107207345395915617539702475152..

13 - S = 1.0
14 - S = sqrt(Pi)/2 = 0.88622692545275801364908374167057..

Hmmm ... Pi divided by square root of 2 must be special!

So, I viewed: https://www.youtube.com/watch?v=5sKah3pJnHI
and learned to not urinate toward the sun (which seems
less intuitive than not urinating into the wind).

Rod

Re: iSquares Cx4 design (looking for patterns)
"Extraordinary Pi Inside Catwalk" (EPIC)

While "curiosity killed the cat", this catwalk may be eternal!
Trade in your old ride and get the "impossible" iSCx4!

Notice the increment for these squares (side length of square)
and then consider that Pi is "irrational and transcendental".
Also notice that sqrt(2) is the set-to-set increment:

Increment for 3-per-set squares = 2(sqrt(1/Pi))
= 1.1283791670955125738961589031215..

4.0
3.5449077018110320545963349666823..
3.1415926535897932384626433832795.. = Pi

2.8284271247461900976033774484194..
2.506628274631000502415765284811..
2.2214414690791831235079404950303.. = 2(Pi/sqrt(2))

2.0
1.7724538509055160272981674833411..
1.5707963267948966192313216916398.. = Pi/2

1.4142135623730950488016887242097..
1.2533141373155002512078826424055..
1.1107207345395915617539702475152.. = Pi/sqrt(2)

1.0
0.88622692545275801364908374167057..
0.78539816339744830961566084581975.. = Pi/4

A geometric Pi corral!

Rod ... ...

Re: iSquares Cx4 design (looking for patterns)
"Extraordinary Pi Inside Catwalk" (EPIC)

Identification corrections:
2.2214414690791831235079404950303.. = Pi/sqrt(2)
1.1107207345395915617539702475152.. = (Pi/sqrt(2))/2

Rod

Re: iSquares Cx4 design (exploring the limits of TMI)
"Extraordinary Pi Inside Catwalk" (EPIC)

A Squared Circles Table of Dimensions,
including iSCx4 Design (delineated by ---)

Increments/decrements for 3-per-set squares:

1.1107207345395915617539702475152.. = (Pi/sqrt(2)/2
1.1283791670955125738961589031215.. = 2(sqrt(1/Pi))
(= diameter of circle whose square = 1)
1.4142135623730950488016887242097.. = sqrt(2)

Concentric squares (3 per set):

8.0
7.0898154036220641091926699333646.. = 4(sqrt(Pi))
6.283185307179586476925286766559.. = 2(Pi)

5.6568542494923801952067548968388.. = 4(sqrt(2))
5.0132565492620010048315305696221.. = 4(sqrt(Pi/2))
4.4428829381583662470158809900607.. = 2(Pi/sqrt(2))

----------------------------------------------------------

4.0
3.5449077018110320545963349666823.. = 2(sqrt(Pi))
3.1415926535897932384626433832795.. = Pi

2.8284271247461900976033774484194.. = 2(sqrt(2))
2.506628274631000502415765284811.. = 2(sqrt(Pi/2))
2.2214414690791831235079404950303.. = Pi/sqrt(2)

2.0
1.7724538509055160272981674833411.. = sqrt(Pi)
1.5707963267948966192313216916398.. = Pi/2

1.4142135623730950488016887242097.. = sqrt(2)
1.2533141373155002512078826424055.. = sqrt(Pi/2)
1.1107207345395915617539702475152.. = (Pi/sqrt(2)/2

1.0
0.88622692545275801364908374167057.. = (sqrt(Pi))/2
0.78539816339744830961566084581975.. = Pi/4

----------------------------------------------------------

0.70710678118654752440084436210485.. = (sqrt(2))/2
0.62665706865775012560394132120276.. = (sqrt(Pi/2))/2
0.55536036726979578087698512375759.. = (Pi/sqrt(2))/4

0.5
0.44311346272637900682454187083529.. = (sqrt(Pi))/4
0.39269908169872415480783042290994.. = Pi/8

0.35355339059327376220042218105242.. = (sqrt(2))/4
0.31332853432887506280197066060138.. = (sqrt(Pi/2))/4
0.27768018363489789043849256187879.. = (Pi/sqrt(2))/8

0.25
0.22155673136318950341227093541764.. = (sqrt(Pi))/8
0.19634954084936207740391521145497.. = Pi/16

0.17677669529663688110021109052621.. = (sqrt(2))/8
0.1566642671644375314009853303005.. = (sqrt(Pi/2))/8
0.13884009181744894521924628093923.. = (Pi/sqrt(2))/16

0.125
0.11077836568159475170613546770882.. = (sqrt(Pi))/16
0.098174770424681038701957605727484.. = Pi/32

0.088388347648318440550105545263106.. = (sqrt(2))/16
0.078332133582218765700492665150345.. = (sqrt(Pi/2))/16
0.069420045908724472609623140469698.. = (Pi/sqrt(2))/32

0.0625
0.055389182840797375853067733854411.. = (sqrt(Pi))/32
0.049087385212340519350978802863742.. = Pi/64

0.044194173824159220275052772631553.. = (sqrt(2))/32
0.039166066791109382850246332575173.. = (sqrt(Pi/2))/32
0.034710022954362236304811570234849.. = (Pi/sqrt(2))/64

0.03125
0.027694591420398687926533866927205.. = (sqrt(Pi))/64
0.024543692606170259675489401431871.. = Pi/128

0.022097086912079610137526386315777.. = (sqrt(2))/64
0.019583033395554691425123166287586.. = (sqrt(Pi/2))/64
0.017355011477181118152405785117433.. = (Pi/sqrt(2))/128

0.015625
0.013847295710199343963266933463603.. = (sqrt(Pi))/128
0.012271846303085129837744700715936.. = Pi/256

Rod

Re: Two Turns of Tau design (renamed from Pythagorean iSpiral)
(view here: http://aitnaru.org/images/Three_Point_One_Four.pdf)

“What goes around, comes around
... if it goes around twice."

iSCx4 Table of Dimensions is full of numbers and patterns, but is not intuitive
as this Two Turns design. "Two Turns" refers to the double 360-degree rotation
required for the smallest circle-squaring right triangle (two green lines, one yellow)
to reposition exactly like the largest of these triangles ... and hints that Pi
"restarts" every two 360-degree rotations.

Rod

Re: Two Turns of Tau design
“What goes around, comes around
... if it goes around twice."

Diameters of the two right triangles:
small = 0.2215567313631895034122709354176.. = (sqrt(Pi))/8
large = 3.5449077018110320545963349666823.. = 2(sqrt(Pi))

Note: 2(sqrt(Pi)) - (sqrt(Pi))/8 = 16

Rod

Re: Two Turns of Tau design
“What goes around, comes around
... if it goes around twice."

Diameters of the two right triangles:
small = 0.2215567313631895034122709354176.. = (sqrt(Pi))/8
large = 3.5449077018110320545963349666823.. = 2(sqrt(Pi))

Note: 2(sqrt(Pi)) / (sqrt(Pi))/8 = 16

What? Reposted the same information?
Notice the substitution of "/" for "-".

Rod

Re: Two Turns of Tau design
“What goes around, comes around
... if it goes around twice."

2(sqrt(Pi)) / (sqrt(Pi))/8 = 16

This factor of 16 was just too intriguing to ignore - it must exist in the geometry!
Also, the Midwayer prompt at 4:03 AM (I was out of bed for a different reason )
seemed related ... and was found later at a 107.403 degree angle!

The iSquares Cx4 design was revised to reflect this factor and seems to explain
visually why Pi may indeed "restart" after two 360 degree rotations.

Notice the commanding "S" (for "scalene") in this design now, perhaps
because these dual red scalene triangles directly relate to the factor of 16
(one side of each triangle has length equal to sqrt(Pi)/2).

Rod ("impossible" Cartesian workspace)

Re: Two Turns of Tau design
“What goes around, comes around
... if it goes around twice."

After more and more Pi ...

While the Two Turns of Tau geometry doesn't show "how to get there from here" (how to square the circle) it does proffer that Pi must have finite value:

As represented by a unique, circle-squaring triangle that shares an hypotenuse with an inscribed isosceles right triangle (inscribed in a circle), a similar circle-squaring triangle appears in the 9th replication (indicating 8 per set) as the isosceles triangle is replicated inward. The 9th replication has dimensions 1/16 of the originating triangle.

In other words, whatever the triangular shape, that same shape appears in the 9th replication, with similar Cartesian coordinates, and has 1/16 the size of the beginning triangle. Thus (for example), a Pi of a trillion decimal digits can be perfectly divided into 16 slices.

As displayed in the Two Turns of Tau design, the largest circle having a diameter of 3.5449077018110320545963349666823..,
has a square whose side length is between the minimum of 2(sqrt(Pi/2)) = 2.506628274631000502415765284811.. (square inscribed in circle)
and the maximum of 2(sqrt(Pi)) = 3.5449077018110320545963349666823.. (square enclosing the circle).

In summary, the largest circle's circle-squaring right triangle has a hypotenuse with length equal to the circle's diameter and a long side having length between the maximum and minimum lengths, creating that triangle's "whatever" shape ... and located somewhere within this "Pi corral".

On Leftover Pi Day (re: http://www.answers.com/Q/What_is_Leftover_Pi_Day ),
get your serving ASAP, especially if the cutting method is by geometric replication.
The 9th serving will have similar shape, but will be much less caloric.

Rod

Re: Two Turns of Tau design
"Who can tell?" (Posted on another blog about squaring the circle):

Re: Two Turns of Tao design

"Two Turns” refers to the double 360-degree rotation (drawing from largest to smallest) required for the smallest circle-squaring right triangle (two green lines, one yellow) to reposition exactly like the largest of these triangles. This geometry hints that Pi “restarts” after two 360-degree rotations (i.e., Pi may have repeating decimal digits or maybe Pi is not calculated to complement the apparent precision of these "Tau turns".


At first, I considered spelling "Tao" in "Re: Two Turns of Tao design" an amusing Freudian slip, but then awakened to a subtle reality: in the context of squaring the circle, Tao and Tau might indeed be related ... and easily reflects on this research that has been more intuition, pattern recognition, and perseverance than geometry expertise:

Re: http://en.wikipedia.org/wiki/Tao

"Tao is a Chinese concept signifying 'way, path, route', or sometimes more loosely, 'doctrine' or 'principle' ...
The Tao signifies the primordial essence or fundamental nature of the universe."

Fascinating Cartesian dynamics (geometric tension):
If Pi can be divided evenly, infinite division (e.g., 1/2, then 1/2, then 1/2 ...)
suggests that as more Pi digits are discovered, the significance is less
and might even roll Pi back to a "few" signficant decimal digits.

Rod

Re: Leftover Fare (new design concept)

Simple Simon met a pie man parting from the fair.
Said the pie man to Simple Simon: "My pies are in repair
to lockers cold, thus wrapped - not sold; leftover fare is fair
for sweet repose then new expose of pie beyond compare!

A unique isosceles right triangle with two hypotenuses (or hypoteni)
seems to have "leftover fare" potential - geometric guidance for
"new expose of pie beyond compare!"

Rod

Re: Leftover Fare design
"Come one, Come all!"

Looks like a speaker.
Maybe it's for the Leftover Fare!

Rod ... ...