Paradise Trinity Day

[Amigoo]

Re: Paleo Pi Pointers design (Pinary Vise)

Research notes ...
Bookends pattern persistence table.
(4/2 ratio maintained as Pi decrements)

[ 4.00000 ~ Pi/01 ~ 2.00000 ] 4(1)= 4/2 =1(2)
[ 2.00000 ~ Pi/02 ~ 1.00000 ] 2(2)= 4/2 =2(1)
[ 1.00000 ~ Pi/04 ~ 0.50000 ] 1(4)= 4/2 =4(.5)
[ 0.50000 ~ Pi/08 ~ 0.25000 ] .5(8)= 4/2 =8(.25)
[ 0.25000 ~ Pi/16 ~ 0.12500 ] .25(16)= 4/2 =16(.125)
[ 0.12500 ~ Pi/32 ~ 0.06250 ] .125(32)= 4/2 =32(.0625)
[ 0.06250 ~ Pi/64 ~ 0.03125 ] .0625(64)= 4/2 =64(.03125)

Hmmm ... how does Pi/trillions relate to 4/2?

Rod ... ... (off to find Bluebell)

[Amigoo]

Re: Paleo Pi Pointers design (Pinary Vise)

Research notes ...

Bookends 2.0 ~ Pi/2 ~ 1.0
2.0
x 1.1283791670955125738961589031215..
= 1.7724538509055160272981674833411.. sqrt(Pi)
x 1.1283791670955125738961589031215..
= 1.5707963267948966192313216916398.. Pi/2
x 0.6366197723675813430755350534900..
1.0

Rod

[Amigoo]

Re: Paleo Pi Pointers design (Pinary Vise)

Research notes ...

Since bookends represent enclosure of a decrement of Pi
between two whole numbers, the math creates ...
"The Mystique of the Pi Corral",
starring 2(sqrt(1/Pi)):

Bookends 2.0 ~ Pi/2 ~ 1.0
2.0
x 1.1283791670955125738961589031215..
= 1.7724538509055160272981674833411.. sqrt(Pi)
x 1.1283791670955125738961589031215..
= 1.5707963267948966192313216916398.. Pi/2
x 0.6366197723675813430755350534900..
1.0

Therefore (restatement) ...

(1.1283791670955125738961589031215..)^2
= 1.2732395447351626861510701069801..

2 / 1.2732395447351626861510701069801..
= 1.5707963267948966192313216916399.. Pi/2

Therefore (the mystique) ...
2 / something squared = Pi/2,
a transcendental number?

Rod ... ...

[Amigoo]

Re: Paleo Pi Pointers design (Pinary Vise)

Since bookends represent enclosure of a decrement of Pi between two whole numbers,
the math creates ... "The Mystique of the Pi Corral", starring 2(sqrt(1/Pi)):

Therefore (transcendental math) ...
The final decimal digit of Pi can be determined by
calculating backward from 4 and forward from 2
... until the 2/3 to 1/3 boundary is located.

Say what? Three pieces of Pi.

Rod

[Amigoo]

Re: Paleo Pi Pointers design (Pinary Vise)

The final decimal digit of Pi can be determined by
calculating backward from 4 and forward from 2
... until the 2/3 to 1/3 boundary is located.

"Hmmm ... 2/3 and 1/3 of what?"

Rod

[Amigoo]

Re: Paleo Pi Pointers design (Pinary Vise)

2/3 and 1/3 of what?"

A rhetorical question - answer is 2(sqrt(1/Pi)).
But what is the significance of these values?

Here's the 2/3 to 1/3 math perspective:

2(sqrt(1/Pi))
= 1.1283791670955125738961589031215..
^2 = 1.2732395447351626861510701069801..
/2 = 0.6366197723675813430755350534900..

1.2732395447351626861510701069801..
+ 0.6366197723675813430755350534900..

= 1.90985931710274402922660516047..
/ 1.2732395447351626861510701069801..
= 0.66666666666666666666666666666667..

= 1.90985931710274402922660516047..
/ 0.6366197723675813430755350534900..
= 0.33333333333333333333333333333333..

[ calculate up/down from Pi in this bookend ]

= 4.0
x 1.1283791670955125738961589031215..
= 3.5449077018110320545963349666806.. 2(sqrt(Pi))
x 1.1283791670955125738961589031215..
> 3.1415926535897932384626433832795.. Pi
x 0.6366197723675813430755350534900..
= 2.0

= 4.0
x 1.2732395447351626861510701069801..
> 3.1415926535897932384626433832795.. Pi
x 0.6366197723675813430755350534900..
= 2.0

Rod ... ... (off to find 1/3 of a pie)

[Amigoo]

Re: Paleo Pi Pointers design (Pinary Vise)

There it is! (1/Pi):

1.90985931710274402922660516047..
/ 6 = 0.318309886183790671537767526745.. = 1/Pi

1.90985931710274402922660516047..
/ 1.2732395447351626861510701069801.. = 1.5

1.90985931710274402922660516047..
/ 0.6366197723675813430755350534900.. = 3.0

Rod ... ... (off to find more 1/3 pie)

[Amigoo]

Re: Pinary Vise (new design)

"But that Pinary Vise! Where does it end? (or begin?)"
A simple design, colorfully bold and pointed:
http://aitnaru.org/images/Veritas_27597e.pdf

Rod

[Amigoo]

Re: Pinary Vise design
"But that Pinary Vise! Where does it end? (or begin?)"

A low calorie dessert (small bowl of numbers) after supper tonight:
(defines relationship of diagonals of upward pointing triangles)

2(sqrt(1/Pi)) = sqrt(Pi) / (Pi/2)

1.1283791670955125738961589031215..
= 1.7724538509055160272981674833411..
/ 1.5707963267948966192313216916398..
= 1.1283791670955125738961589031215..

Rod

[Amigoo]

Re: Pinary Vise design
"But that Pinary Vise! Where does it end? (or begin?)"

The rest of the story ...
(design also revised)

2/(sqrt(Pi)) = sqrt(Pi)/(Pi/2) = 2(sqrt(1/Pi))

2 / 1.7724538509055160272981674833411..
= 1.7724538509055160272981674833411..
/ 1.5707963267948966192313216916398..
= 1.1283791670955125738961589031215..

Rod

[Amigoo]

Re: Pinary Vise design
"But that Pinary Vise! Where does it end? (or begin?)"

The rest of the story ...
(design also revised ... again)
with mirror image of Yin and Yang symbolism.

4(sqrt(1/Pi))/2 = 2/(sqrt(Pi)) = sqrt(Pi)/(Pi/2) = 2(sqrt(1/Pi))

2.256758334191025147792317806242..
/ 1.1283791670955125738961589031215..
= 2.0
/ 1.1283791670955125738961589031215..
= 1.7724538509055160272981674833411..
/ 1.1283791670955125738961589031215..
= 1.5707963267948966192313216916398..

Rod ... ... (off to find Yin/Yang balance)

See also: Floor Plan 001 design:
"Six of one – half a dozen of the other."

[Amigoo]

Re: Pinary Vise design
"But that Pinary Vise! Where does it end? (or begin?)"

Yin/Yang balance with hypergeometric essence:

4(sqrt(1/Pi))/2 = 2/(sqrt(Pi)) = sqrt(Pi)/(Pi/2)
= (Pi/2)/(Pi(sqrt(Pi)/4)) = 2(sqrt(1/Pi))

2.256758334191025147792317806242..
/ 1.1283791670955125738961589031215..
= 2.0
/ 1.1283791670955125738961589031215..
= 1.7724538509055160272981674833411..
/ 1.1283791670955125738961589031215..
= 1.5707963267948966192313216916398..
/ 1.1283791670955125738961589031215..
= 1.3920819992079269613212044955304..

Rod ... ... (off to comprehend "hypergeometric")

[Amigoo]

Re: Pinary Vise design
"But that Pinary Vise! Where does it end? (or begin?)"

Top and bottom sides of irregular quadrilaterals (red, magenta;
one right angle; "X" appears inside) have unique relationship:

Top = 0.92650275035220848584275966758914..
/ 0.52272320087706331513679711195272..
= 1.7724538509055160272981674833411.. sqrt(Pi)

Bottom = 0.52272320087706331513679711195272..
/ 0.92650275035220848584275966758914..
= 0.56418958354775628694807945156077.. sqrt(1/Pi)

0.56418958354775628694807945156077..
x 2 = 1.1283791670955125738961589031215.. 2(sqrt(1/Pi))

Rod

[Amigoo]

Re: Pinary Vise design
"But that Pinary Vise! Where does it end? (or begin?)"

Yin/Yang balance with hypergeometric essence

"Hypergeometric" may refer to comparison of half to whole;
circle within circle having diameter equal to radius of larger circle.
In this comparison, a geometric "limit" may be found (calculated).

Rod

[Amigoo]

Re: CosPi 2.0 design
"The subtle cosine wave of squared circles."

Rod ... ...

[Amigoo]

Re: CosPi 2.0 design
"The subtle cosine wave of squared circles."

A week later and the design is still being updated.
But with identification of the "smile of Pythagoras"
(rotate design clockwise 90 degrees), today's update
may be the final. At least, the Texas 'T' portion
seems to contain the salient right triangles.

Rod ... ...

[Amigoo]

Re: Tripartite Soul (new design concept)
"Squared circle harmony of 2, sqrt(2), sqrt(Pi)"
The reason for spirit pleasure?

Having been reduced to a simple design (where its soul was discovered),
this geometric concept now awaits harmony of colors.

Rod

[Amigoo]

Re: Tripartite Soul
"Squared circle harmony of 2, sqrt(2), sqrt(Pi)"

Who knew?
Squared circle geometry even speaks Plato!

But everyone knows: sqrt(2)/2 = .707..

Rod ... ...

[Amigoo]

Re: Tripartite Soul
"Squared circle harmony of sqrt(2), 2, sqrt(Pi);
a correlation of three similar quadrilaterals."

Since the geometric essence is the quadrilateral
(three similar), the largest is now highlighted.

Rod

[Amigoo]

Re: Tripartite Soul
"Squared circle harmony of sqrt(2), 2, sqrt(Pi);
a correlation of three similar quadrilaterals."

The circle-squaring scalene triangle insisted on obvious presence.
Diameters from smallest to largest: 1, sqrt(2), sqrt(Pi), 2

Rod ... ...

[Amigoo]

Re: Tripartite Soul
"Squared circle harmony of 1,sqrt(2),sqrt(Pi),2
in a correlation of 3 similar quadrilaterals."

The inspiration to title this difficult, integrated geometry
preceded my understanding of how the title related.

I now see that 3 circles (green; D = 1, sqrt(2), 2)
are squared by the yellow circle (D = sqrt(Pi)) with its
inscribed square:

Radius squares green circle D = 1
Diameter squares green circle D = 2
Side of inscribed square squares green circle D = sqrt(2)

With this geometry, if there's any doubt that sqrt(Pi)
is the square of a circle whose diameter = 2,
geometers' minds are not participating.

A convincing Cartesian doodle for 10/10/15
(Paradise Trinity Day, 2015)

Rod

[Amigoo]

Re: Tripartite Soul
"Squared circle harmony of 1,sqrt(2),sqrt(Pi),2
in a correlation of 3 similar quadrilaterals."

A convincing Cartesian doodle

Having dared to think - even for a second - that this was not
serious geometry, I was given an assignment: Locate the next
circle in the decreasing progression 2, sqrt(Pi), sqrt(2), 1

Even this Geometer Junior knew that the next diameter
will be sqrt(2)/2 ... but where is this circle? Obviously, near
the center of this Cartesian neighborhood and sharing a point
on the circumference of the next largest circle (D = 1).

Long story short: There it is!
As easy as 1,2,3

Rod ... ...

[Amigoo]

Re: Tripartite Soul
"Squared circle harmony of 1,sqrt(2),sqrt(Pi),2
in a correlation of 3 similar quadrilaterals."

See updated: http://aitnaru.org/images/Veritas_27597e.pdf

For extra credit, I could provide the
Geometry numbers of Tripartite Soul
(to calculate side of circles' square)

What? Eat more Pi? Twist my arm

Rod (now full of Pi - need lotsa cake
... and ice cream)

[Amigoo]

Re: Tripartite Soul (final summary )
"Squared circle harmony of 2,sqrt(2),1 and sqrt(Pi)
in a correlation of four similar quadrilaterals."

(aka, "a circle squared up, down, and sideways")

Geometry numbers of Tripartite Soul
(to calculate side of circle's square)

D = 2.0
/ 1.1283791670955125738961589031215.. 2(sqrt(1/Pi))
= 1.7724538509055160272981674833411.. sqrt(Pi)

D = 1.4142135623730950488016887242097.. sqrt(2)
/ 1.1283791670955125738961589031215.. 2(sqrt(1/Pi))
= 1.2533141373155002512078826424055.. sqrt(Pi/2)

D = 1.0
/ 1.1283791670955125738961589031215..
= 0.88622692545275801364908374167057.. sqrt(Pi)/2

D = 1.7724538509055160272981674833411.. sqrt(Pi)
/ 1.1283791670955125738961589031215..
= 1.5707963267948966192313216916405.. Pi/2

For reference (next circle in progression):

D = 0.70710678118654752440084436210485.. sqrt(2)/2
/ 1.1283791670955125738961589031215..
= 0.62665706865775012560394132120276..

Hmmm ... interesting constant, this 2(sqrt(1/Pi))
(= 1.1283791670955125738961589031215.. )

Rod

[Amigoo]

Re: Tripartite Soul (final summary )
"Squared circle harmony of 2,sqrt(2),1 and sqrt(Pi)
in a correlation of four similar quadrilaterals."

four similar quadrilaterals

Now it's in, now it's out, now it's in ...

I was reluctant to add the fourth green circle because I would have to draw its inscribed quadrilateral. But I noticed an easy solution: the four circles (three green, one yellow) all had similar quadrilaterals and I had claimed "four". So, I could draw the fourth green circle (a similar geometric arc) but not its inscribed quadrilateral.

Say what? It's alluded to in the Numbers (D = sqrt(2)/2).

Rod ... ... (not saving on the Quadrilateral Expressways (QE))