Paradise Trinity Day

[Amigoo]

Re: Paleo Pi Pointers* (new design concept)
Contrasting the dimensions of four squared circles:
two sets having one larger and one smaller diameter,
and diameters of one set larger than the other set.

Set A: D = 2, sqrt(Pi)

2 x 1 = 2

1.7724538509055160272981674833411..
x 1.1283791670955125738961589031215..
= 2

Set B: D = 2(sqrt(1/Pi)), 1

1 / 1.7724538509055160272981674833411..
= 0.56418958354775628694807945156077..

1.1283791670955125738961589031215.. / 2
= 0.56418958354775628694807945156077..

* refers to the Greek origin of "Paleo" [palaio-, from palaios]
and alludes to era of Pythagoras, philosopher and mathematician.

Rod

[Amigoo]

Re: Paleo Pi Pointers design
(added to Veritas 27597e PDF)

"Circle-squaring right triangles: primitive but precise."

Same theme - different perspective.
Diameters = 1, 2(sqrt(1/Pi)), sqrt(Pi), 2

Rod ... ...

[Amigoo]

Re: Paleo Pi Pointers design
"Circle-squaring right triangles: primitive but precise."

Intriguing design closure (5 circles) ...
Diameters = 1, 2(sqrt(1/Pi)), Pi/2, sqrt(Pi), 2

1.0
1.1283791670955125738961589031215..
1.5707963267948966192313216916398..
1.7724538509055160272981674833411..
2.0

1.1283791670955125738961589031215..
^2 = 1.2732395447351626861510701069801..

1.5707963267948966192313216916398..
/ 1.2732395447351626861510701069801.. = 1

1.5707963267948966192313216916398..
x 1.2732395447351626861510701069801.. = 2

Who knew?
Pi/2 is the midpoint between 1 and 2.

Rod

[Amigoo]

Re: Paleo Pi Pointers design
"Circle-squaring right triangles: primitive but precise."

My latest grocery store receipt ($111.77) was the herald
of some approaching Paleo "Pi beyond compare!"

We should now be able to calculate the geometric center
between 1 and 2 (exactly .5(Pi)), by calculating forward from 1
and backward from 2. But how? 2(sqrt(1/Pi)) must be a clue!

Rod ... ... (off to a late summer break
to rest before Leftover Pi Day, November 10, 2015).
BTW: 10/10/15 is another plateau ahead.

[Amigoo]

Re: Paleo Pi Pointers design
"Circle-squaring right triangles: primitive but precise."

Geometer's tip:
.5 does not identify the 1-2 midpoint because
the increments relate to squares of circles.

Rod

[Amigoo]

Re: Paleo Pi Pointers design
"Circle-squaring right triangles: primitive but precise."

What = 1? (error corrected)

Consider 5 concentric circles of diameters
1, 2(sqrt(1/Pi)), Pi/2, sqrt(Pi), and 2:

1.0
1.1283791670955125738961589031215..
1.5707963267948966192313216916398..
1.7724538509055160272981674833411..
2.0

1.1283791670955125738961589031215..
^2 = 1.2732395447351626861510701069801..

1.2732395447351626861510701069801..
/ 2 = 0.63661977236758134307553505349006..
x 1.5707963267948966192313216916398.. = 1

1.5707963267948966192313216916398..
x 1.2732395447351626861510701069801.. = 2

So, what does this prove?
Pi/2 is between 1 and 2.
(must be advanced math)

Rod

[Amigoo]

Re: Paleo Pi Pointers design

Another perspective ...
The five squared circles of diameters
1, 2(sqrt(1/Pi)), Pi/2, sqrt(Pi), 2:

1.0
1.1283791670955125738961589031218..
1.5707963267948966192313216916398..
1.7724538509055160272981674833411..
2.0

(1.1283791670955125738961589031218.. ^2)/2
= 0.63661977236758134307553505349036..

1 / 0.63661977236758134307553505349036..
= 1.570796326794896619231321691639..

2 / 2(0.63661977236758134307553505349036..)
= 1.570796326794896619231321691639..

2(1.570796326794896619231321691639..)
= 3.141592653589793238462643383278.. = Pi

Pi/2 is the geometric midpoint of 1 and 2
... if the circle is squared.

Rod

[Amigoo]

Re: Paleo Pi Pointers design

And another perspective ...
The 7 squared circles of diameters:

2.2567583341910251477923178062430.. = 4(sqrt(1/Pi))
2.0000000000000000000000000000000.. = 2
1.7724538509055160272981674833411.. = sqrt(Pi)
1.5707963267948966192313216916398.. = Pi/2
1.1283791670955125738961589031215.. = 2(sqrt(1/Pi))
0.8862269254527580136490837416705.. = sqrt(Pi)/2
1.0000000000000000000000000000000.. = 1

I should have known that 7 squared circles
would be the most accurate perspective!

The revised design will take some time.
(maybe after a week's reflection)

Rod ... ...

[Amigoo]

Re: Paleo Pi Pointers* design (a few numbers)

Integration of 7 squared circles having numerical patterns
suggesting that Pi/2 is the geometric midpoint of 1 and 2
... when the circles are squared precisely.

* "Paleo" - Greek origin: palaio-, from palaios.

Supporting numbers:

2 - .429203673205103380768678308361.. = Pi/2
1 + .570796326794896619231321691639.. = Pi/2
.. 1.000000000000000000000000000000

0.56418958354775628694807945156077.. = sqrt(1/Pi)
x 2 = 1.1283791670955125738961589031215..
^ 2 = 1.2732395447351626861510701069801..
/ 2 = 0.6366197723675813430755350534900..

The seven squared circles,
including their relationship to Pi/2:

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

2.2567583341910251477923178062430.. = 4(sqrt(1/Pi))
2.0000000000000000000000000000000.. = 2
1.7724538509055160272981674833411.. = sqrt(Pi)
1.5707963267948966192313216916398.. = Pi/2
1.1283791670955125738961589031215.. = 2(sqrt(1/Pi))
1.0000000000000000000000000000000.. = 1
0.8862269254527580136490837416705.. = sqrt(Pi)/2


2.2567583341910251477923178062430.. = 4(sqrt(1/Pi))

1.5707963267948966192313216916398..
x (1.1283791670955125738961589031215..)^2 = 2

1.5707963267948966192313216916398..
x 1.1283791670955125738961589031215..
= 1.7724538509055160272981674833411.. = sqrt(Pi)

1.5707963267948966192313216916398.. = Pi/2

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

1.5707963267948966192313216916398..
x 0.6366197723675813430755350534900.. = 1

1.5707963267948966192313216916398..
x 0.5641895835477562869480794515607..
= 0.8862269254527580136490837416705.. = sqrt(Pi)/2

Rod

[Amigoo]

Re: Paleo Pi Pointers design

All the salient numbers with pretty pictures:
http://aitnaru.org/images/Veritas_27597e.pdf

Rod ... ... (off to see no more numbers)

[Amigoo]

Re: Paleo Pi Pointers design ( ... 1 ~ Pi/2 ~ 2 ~ Pi ~ 4 ... )

Is this the intended enlightenment after a wakeup call* during afternoon slumber? Having decided that no human had tapped and that the sound may have been a "return to the geometry sandbox" suggestion, I was soon tap, tap, tapping on a keyboard. * apparently, a Midwayer's loud tap, tap, tap on a wall.

Long story short ...

The Paleo Pi geometry had been insisting on an 8th circle, but I now needed significant persuasion to rework the geometry yet again (this was easy!) and the numbers ("fugetaboutit"!). But I kicked sand long enough to realize that the only enlightenment proffered was that "Pi is the geometric midpoint of 2 and 4". Thus, ... 1 ~ Pi/2 ~ 2 ~ Pi ~ 4 ... describes a continuum of midpoints! ... in both directions ... to infinity!

I will probably add the 8th circle (D = Pi/4) but not change the numbers since the explanation is recorded here
... unless awakened in the future for more keyboard tap, tap, tapping.

Rod ("hunt and peck" typist)

[Sandy]

I'm enjoying the pretty pictures Rod! Every time I refocus my eyes on the scene I seem to see a new pattern. Very good for the mind and concentration I suspect. I'm appreciating your hard work and the Midwayers "a tap tap tapping at your chamber door." (Edgar Allen Poe ) Quite fitting as we have had an enormous and gorgeous black Raven hanging about the yard this week.
hugs,
Sandy

[Amigoo]

Re: Paleo Pi Pointers design (simplified)

A technicality allows the 8th circle to be added without changing numbers:
Since the geometry shows that a certain right triangle squares the circle,
only two sides of the triangle appear in the 8th circle (D = Pi/4).
Therefore, that circle is not squared ... yet.

Why Pi/2 is geometric center of 1 and 2:

0.56418958354775628694807945156077.. = sqrt(1/Pi)
x 2 = 1.1283791670955125738961589031215..
^ 2 = 1.2732395447351626861510701069801..
/ 2 = 0.6366197723675813430755350534900..

Pi/2 = 1.5707963267948966192313216916398..

Let N = 0.6366197723675813430755350534900..
2N = 1.2732395447351626861510701069800..

Therefore, N x Pi/2 = 1, 2N x Pi/2 = 2

Since 1.2732395447351626861510701069800..
+ 0.6366197723675813430755350534900.. = 3N
3N x Pi/2 = 3

Rod (wondering which is the larger half of Pi)

[Amigoo]

Re: Paleo Pi Pointers design (number patterns)

A closing Paleo Pi Pointers vise?
Distance between "bookends" decreases by /2:
(beginning with ... 2 ~ Pi ~ 4 ...)

4
2
1
0.5
0.25
0.125
0.0625
0.03125
0.015625
0.0078125
0.00390625
0.001953125
0.0009765625

... as Pi decreases by /2:

3.1415926535897932384626433832795..
1.5707963267948966192313216916398..
0.78539816339744830961566084581988..
0.39269908169872415480783042290994..
0.19634954084936207740391521145497..
0.098174770424681038701957605727484..
0.049087385212340519350978802863742..
0.024543692606170259675489401431871..
0.012271846303085129837744700715936..
0.0061359231515425649188723503579678..
0.0030679615757712824594361751789839..
0.0015339807878856412297180875894919..

Rod (caught in a Pinary Vise)

[Amigoo]

Re: Paleo Pi Pointers design (number patterns)

Hmmm ...
What does this mean? (the patterns)
Does this relate to "Pi beyond compare!"?

Current observation:
Whereas the number of decimal digits in the "bookends" keep increasing,
the number of decimal digits in Pi keep decreasing* (by definition: Pi/2).
What's the significance when Pi has fewer decimal digits?

* it's time to inaugurate supercomputer races!
(number of decimal digits in Pi vs. Pi/2 vs. bookends/2)
... with occasional breaks for Pinary Plenaries.

Rod ... ... (off to find very slim books)

[Amigoo]

Re: Paleo Pi Pointers design (number patterns)

Whereas the number of decimal digits in the "bookends" keep increasing,
the number of decimal digits in Pi keep decreasing

Hmmm ... a few decimal points shy of real math!
Let's blame the brilliant Paleo Pi Pointers calculator ...
who's a few decimal points shy of a real mathematician.

A restatement to recover in the Veritas tradition ...
Whereas the number of decimal digits in the "bookends" keep increasing,
the number of decimal digits in Pi also keep increasing (but not related
to this current Paleo Pi Pointers enlightening analysis).

Rod

[Amigoo]

Re: Paleo Pi Pointers design

Redrawn to highlight the irregular trapezium (dark blue) and its larger sibling,
just above. If any geometric object is "primitive, penetrating, and precise",
this trapezium is ... and worthy of penetrating analysis!

New aromas make Leftover Pi Day enticing!

Rod

[Amigoo]

Re: Paleo Pi Pointers design (Pinary Vise)

Regarding "bookends" 2/1024 ~ 2/2048 (0.001953125 ~ 0.0009765625)
and the intervening Pi/2048 decrement (a very slim "book"):

0.001953125 / 0.0009765625 = 2 (distance between bookends)

0.001953125000000000000000000000000.. 2/1024
/ 1.2732395447351626861510701069801.. 2(sqrt(1/Pi))^2
= 0.001533980787885641229718087589.. Pi/2048
/ 1.570796326794896619231321691639.. Pi/2
= 0.000976562500000000000000000000.. 2/2048

Say what? Notes for future reference.

Rod

[Amigoo]

Re: Paleo Pi Pointers design (Pinary Vise)

Tonight's 2:56 prompt-inspired focus on 2:
(calculate backward and forward from 2)

= 2.2567583341910251477923178062430.. 4(sqrt(1/Pi))
/ 0.8862269254527580136490837416705.. sqrt(Pi)/2
2.0000000000000000000000000000000..
x 0.8862269254527580136490837416705.. sqrt(Pi)/2
= 1.7724538509055160272981674833411.. sqrt(Pi)

2.2567583341910251477923178062430.... 4(sqrt(1/Pi))
x 1.7724538509055160272981674833411.. sqrt(Pi)
= 4

2:56 = 2 + 5 + 6 = 13 = 1 + 3 = 4

Rod ... ... (off 2 bed at 3:33)

3:33 = 3 x 3 x 3 = 27 = 2 + 7 = 9
3:33 = 3 + 3 + 3 = 9 and 9 + 9 = 18 = 1 + 8 = 9

What does this all mean?
One beer is my bedtime limit.

[Amigoo]

Re: Paleo Pi Pointers design (Pinary Vise)

Explanation of the Pinary Vise was added to this file:
http://aitnaru.org/images/Veritas_27597e.pdf

The mystique of the Pinary Vise is that the significance of the squeeze is debatable.

~ Rod ~ (in the Pinary Vise)

[Amigoo]

Re: Paleo Pi Pointers design (Pinary Vise)

Another page of numbers was added to this file:
http://aitnaru.org/images/Veritas_27597e.pdf

... to highlight the coordinating 2(sqrt(1/Pi))
( 1.128379167095512573896158903121.. )

~ ~ (Rod; on break)

[Amigoo]

Re: Paleo Pi Pointers design (Pinary Vise)

The mystique of the Pinary Vise is that the significance of the Pi values squeeze is debatable

Regardless of such debate, these constant ratios indicate that the relative distances of the binary decrement of Pi to its bookends never change. So, does this not suggest that the number of decimal digits of Pi (especially when stated as hundreds+) is irrelevant in a Pinary Vise?
... and that to be released from a Pinary Vise is a transcendental experience?

Sample bookends (rightmost digits reflect truncation/rounding)
[calculate up from bottom (x) or down from top (/)]

Bookends 4.0 ~ Pi/2 ~ 2.0
= 4.0
x 1.2732395447351626861510701069807.. 2(sqrt(1/Pi))^2
= 3.1415926535897932384626433832795.. Pi
x 1.5707963267948966192313216916398.. Pi/2
= 2.0

Bookends 2.0 ~ Pi/2 ~ 1.0
2.0
x 1.2732395447351626861510701069807.. 2(sqrt(1/Pi))^2
= 1.5707963267948966192313216916398.. Pi/2
x 1.5707963267948966192313216916398.. Pi/2
1.0

Bookends 1.0 ~ Pi/4 ~ 0.5
1.0
x 1.2732395447351626861510701069807.. 2(sqrt(1/Pi))^2
= 0.78539816339744830961566084581988.. Pi/4
x 1.5707963267948966192313216916398.. Pi/2
0.5

~ ~ (Rod, out of the vise)
"Been there! Done that!" ... ...

[Amigoo]

Re: Paleo Pi Pointers design (Pinary Vise)

Coordinating Factor 2(sqrt(1/Pi))
= 1.128379167095512573896158903121..
^2 = 1.2732395447351626861510701069801..
/2 = 0.6366197723675813430755350534903..

Simplified recipe:
Pi/2 or Pi/4 x (CF^2) = 2 or 1
Pi/2 or Pi/4 x (CF^2)/2 = 1 or .5

[calculate up/down from Pi/2]

= 2
x 1.2732395447351626861510701069807..
= 1.570796326794896619231321691641.. Pi/2
x 0.6366197723675813430755350534903..
= 1
x 1.2732395447351626861510701069807..
= 0.7853981633974483096156608458206.. Pi/4
x 0.6366197723675813430755350534903..
= .5

Rod

[Amigoo]

Re: Paleo Pi Pointers design (Pinary Vise)

Multiple reviews of the Pinary Vise numbers were required to put this child to bed (current research - not the researcher who goes to bed whenever). The Answers.com* update today now has good potential for a long rest ... without one more attempt at "improvement".

Interestingly, I expected (and explored many times) to see evidence of project closure in the geometry designs - not in the boring (relative to the colorful designs) numbers and math. However, the numbers are so convincing that the creativity sandbox may indeed be "closed until further notice".

I've used the expression "Move along Hopalong!" in the past, alluding to my occasional, self-directed command to take a break before continuing the research. Today, as if to shout "Move along ...", the universe guided me to a rare viewing of an old TV episode ( https://www.youtube.com/watch?v=-82B7uz5fO0 ). If I understand the symbolism (and timing) correctly, the message is clear: this research has reached a major plateau ... if not closure (I won't believe this until after Leftover Pi Day).

* http://www.answers.com/Q/What_is_a_Pinary_Vise

Rod ... ...

[Amigoo]

Re: Paleo Pi Pointers design (Pinary Vise)

Research notes ...
Derivation of 2/3 to 1/3 proportion (example):

Bookends 4.0 ~ Pi ~ 2.0
= 4.0
x 1.2732395447351626861510701069801..
= 3.1415926535897932384626433832795.. Pi
x 0.6366197723675813430755350534900..
= 2.0

4 / 3.141592653589793238462643383279.. 4/Pi
= 1.2732395447351626861510701069801.. 2(sqrt(1/Pi))^2
2 / 3.141592653589793238462643383279.. 2/Pi
= 0.6366197723675813430755350534900.. (2(sqrt(1/Pi))^2)/2
+ 1.2732395447351626861510701069801..
= 1.90985931710274402922660516047..
x 0.3333333333333333333333333333333..
= 0.6366197723675813430755350534900..

2/3 to 1/3 pattern table
for Bookend 4.0 ~ Pi ~ 2.0

4.0 / 3.30 = 1.212
4.0 / 3.20 = 1.250
4.0 / 3.14 = 1.273 Pi
4.0 / 3.10 = 1.290
4.0 / 3.00 = 1.333

2.0 / 3.00 = 0.666
2.0 / 3.10 = 0.645
2.0 / 3.14 = 0.636 Pi
2.0 / 3.20 = 0.625
2.0 / 3.30 = 0.606

Rod