Paradise Trinity Day

[Amigoo]

New insight (continuing the tradition of knowing so little and claiming so much):

The only reason that Pi is transcendental is that the popular early model (a polygon with an increasing number of sides) was by its nature "transcendental" (there will never be a polygon with an infinite number of sides). Thereafter, developments in math perpetuated the transcendental model, converting it from geometry to algebra and beyond. However, the transcendental model may be the only model we currently envisage.

At least, this insight might explain why the Transcendental Transition geometry hints that Pi (specifically, relationship of a circle's circumference to its diameter) is not really transcendental.
Scheduling for new model auditioning is a future possibility.

Rod

[11light11]

Rod, that is really cool. I am always amazed by these insights of yours . . .and you bring so many new things to my attention that I've never heard of! I can't tell you how much I appreciate that . . . to think that initially, the model of Pi was transcendental, in that it was conceived of as a polygon with an increasing number of sides . . . rendering it infinite (which is impossible, right?! ). . . Again I am just left with the good-kind-of-headache even trying to imagine this . . . Truly awesome!

I thought of you last night -- someone had sent me a link to an amazing occurrence in the Finnish night sky. Perhaps you can share with your friend in Finland! These are some of the most astonishing images I've ever seen. What do you suppose could have caused this?!

http://imgur.com/a/DMfxx

With love to you! Michele
By the way, for what seems like the thousandth time, you posted at a prompt-time!!! . . . AND, I've just noticed: your thread now has 777 posts!!!

[Amigoo]

Michele,

This time, a 3:33 PM prompt was not enough to convince me to post these comments, but then I read your post and decided to risk one more "knowing so little and claiming so much". Claiming that Pi (the real ratio - not the value calculated by popular formulas) is not transcendental suggests ignorance or arrogance. So, I'll include my disclaimer: this is continuing artistic expression of the geometry of squared circles.

Re: http://en.wikipedia.org/wiki/A_picture_ ... sand_words

Well, whoever said it ... this new "Incremental Pi" design (see http://www.aitnaru.org/images/Alright_Triangles.pdf ), drawn to contrast the pink I-Square right triangle, is mostly symbolism of this new concept and, perhaps, an acceptable visual introduction to the math discussed here: http://www.maths.surrey.ac.uk/hosted-si ... fibpi.html

The geometry of this Incremental Pi design helps explain why the "transcendental" model for Pi causes Pi to be transcendental (the math assumes an infinite number of triangles in the calculations). The I-Square triangle is the perfect square of the circle because its Pi has been calculated to many decimal places, but ... (conjecture):

The transcendental model for Pi (the calculations) may need a limit, probably related to a geometric relationship, since the I-Square scalene triangle hints that Pi's circumference/diameter relationship is not really transcendental. Since current science seems to be satisfied with about 100 decimal digits of Pi, the conjectured limit might be as many as thousands of digits ... but not an infinite number.

Rod

[11light11]

Hi Rod!

I'm glad you decided to post anyway -- don't forget the power of 3:33 !!! I saw it a few times yesterday myself. Right and wrong clocks both, respectively.

I enjoyed your discussion. It rang a bell with me, your concluding (conjecturing, if that's a verb!) that the conjectured limit might be as many as thousands of digits -- it is not an infinite number.

I was discussing the other day with a friend, the idea of an 'infinite universe,' or even the supposed 'infinite density' of the singularity at the center of a black hole. We concluded (from our non-physics, non-mathematical framework ) that such calculations could not be right. When you suppose that the universe exploded outward from the Big Bang (assuming it indeed did so), you also suppose that matter flung outward from that central point in all directions -- and has continued to move outward since that point in time. If this is correct, the universe could not possibly be infinite -- instead it is only as large as its point of extension since that time.

However, we came up against a question which has bothered me since early childhood -- whatever the outer limits of all that matter might be -- what exists just on the other side of its edges?? Still more open space, right? Because even if there were 'walls' to the edges of the universe, which there certainly couldn't be -- any wall has another side to it, by definition. So what's on the other side of that wall? Then I start wondering perhaps the universe indeed must be infinite, for limitless space seems the only possibility. Yet I can't conceive of that. Even if we conceive of the universe as spherical, when you envision any sphere, there is the other side to the edges of the sphere, in 360 degrees rotation. So it's a conundrum.

And when you think of the infinite density of the singularity, how indeed could it be infinitely dense? It seemed to us that it would have a finite point of density, however staggering the density might be.

I don't know whether or not these musings in any way relate to your quest -- but I keep sensing spheres and circles in all this, so wondered if it relates back to your conception of Pi to any extent? Do we find Pi in the singularity? In the radius or circumference of the matter that spilled out of the Big Bang???

But something appeals to me about your idea that Pi does not have infinite digits, but instead thousands . . . I have a dream that you will one day prove something about Pi that to this point has merely stumped the world.

As for the Finnish night sky -- I saw your Pi quest there as well. Did you? With your ties to Finland, I wondered if the sky was calling to you! Perhaps it was 'known' that I would find these images and then share them with you . . . these images even host a nucleus of sorts! A central point? Perhaps triangles abound? Even though these images are 2-D, there is something about this 'circular' object in the night sky -- to me it appeared to be more spherical than circular. I wonder if all 2-D abstractions of the circle are indeed 3-D (or beyond) in another dimensional framework? Perhaps it's hard to draw conclusions about Pi when we limit its discussion to a 2-D model? OK enough gibberish from me.

Thanks for sharing!!! With love to you, Michele

[Amigoo]

Michele,

Speaking of stumps ...

Re: new design "Pi in the Sky" (will be added to the PDF file later today)

(literally, "pie in the sky" means within the earth's atmosphere - "the sky is the limit")

Here's an easy way to conceptualize a limit for Pi: [this smile is perfect: the two raised hands suggest that the answer can be found between its left and right hands - the problem is isolated, but is still within a black box].

The combined length of the two red lines in this design is slightly greater than 1/4 of the circumference of this circle. The actual length would be represented by two lines that meet between the circumference of the circle and the point where the red lines meet.

Translating this concept to Pi calculations suggests that, since Pi is an ever increasing (but diminishing) value in popular calculations, another calculation should balance (limit) this value by working backward from a maximum value. How the increasing and decreasing values are perfectly resolved mathematically is the ripe fruit waiting for picking by a mathematician.

This non-mathematician only observes that a certain scalene triangle, inscribed within a circle, hints that the real Pi has a finite limit.

Rod

[Amigoo]

Re: "Pi in the Sky" design (now, more pie)

Admittedly, tighter than Buffalo Gap in west Texas (a natural pass through which bison herds used to travel), these twin peaks of Pi panorama permit good aim (with cameras these days) at the hidden Pi, sandwiched in this range of line lengths (see red lines):

395284708.. ( max.), .392699082.. ( Pi/8 ), .382683432.. ( min.)

A camera aperture of .392699082.. is generally precise enough to capture one still. Later, quench your thirst with a Perini martini while your sizzler sizzles.

Rod

[Amigoo]

Late night reflection on the "wiggly numbers" of mathematics ...

Given: a square with a side length of 1000 units:

Its area is exactly 1,000,000 units; half its area is exactly 500,000 units.
(a diagonal line from one corner of the square to the opposite corner divides the square in half)

So, how can a square with an area of exactly 1,000,000 units and 4 sides whose length is a rational number
be divided in half as isosceles triangles with an area of exactly 500,000 units and two sides whose length is rational
but one side is irrational (hypotenuse has length equal to the square root of 2 x 1000, an irrational number)?

Does math precisely describe the reality of a triangle?
And what about the relationship of a circle's circumference to its diameter?
From whence this almost sacred Pi?

Should we conclude that a circle cannot be squared because the language of math does not permit the square?

All I know this late in the evening is that a triangular slice of Pepper Jack cheese,
with an irrational hypotenuse, on a round multigrain cracker is a very rational bedtime snack.
As for Pi ... no more sweet treats after an overdose of Halloween candy.

Rod

[Amigoo]

Michele,

Re: "the language of math does not permit the square" (my previous comment)

This seems so true when I return to the geometry and see that scalene triangle square the circle ... and then a smaller circle in the same geometry .. and then one more in another part of the geometry!

My friend from Finland will be visiting this week - I'll ask her about the sky photos. Your "sky is calling" could be true - I had a good feeling about the city of Tampere when we visited. It seems to be a modern, comfortable, international city with good transportation options ... perhaps soon becoming an important world center.

Rod

[Amigoo]

Re: http://aitnaru.org/threepoints.html (new design)

Time for a "long winter's nap" ... (holiday break)

A parting reversal of perspective, with a word from our sponsor:
"Sky in the Pi - Like Father, like Son"

Rod ... ...

[Amigoo]

Climbing further out on the limb ...

Intuitively, something seems wrong with language (math) that assigns irrationality* differently to line lengths which, as a set of 3, describe the same geometric objects (isosceles right triangles with same shape, same angles).

Examples (two isosceles right triangles):

1. Hypotenuse = 1000000.0
Side 1 = 707106.781..*
Side 2 = 707106.781..*

2. Hypotenuse = 1414213.562..*
Side 1 = 1000000.0
Side 2 = 1000000.0

So, what's the problem? the language of math? or the symbols (numbers) used by that language?

Somehow, the rational/irrational labeling is allowed to "distort" the language which should first declare these geometric objects to be similar
and only then acknowledge the dissimilar line lengths.

Again, intuitively, there seems to be a slight disconnect between geometric objects (specifically, the circle and the triangle) and the language that describes them. Of course, language (precise or not) is a means of communication and this language (math) has been communicating adequately for centuries. And since the quality of "irrational" exists in both triangles, math has communicated properly ... apparently.

Rod

[Amigoo]

A paradox, "parabox", or common conundrum?

Now, consider this same isosceles triangle:

2. Hypotenuse = 1414213.562..*
Side 1 = 1000000.0
Side 2 = 1000000.0

Create a square with the hypotenuse (an irrational length) as one side.

Since the length of this line did not change, how did the irrationality disappear?
Or did we create a square with the length of all sides equal but irrational?

Rod

[Amigoo]

A paradox, "parabox", common conundrum ... or "paracircle"?

Again, consider this same isosceles triangle:

2. Hypotenuse = 1414213.562..*
Side 1 = 1000000.0
Side 2 = 1000000.0

Draw a circle around this triangle (the hypotenuse becomes a diameter).

Since the length of this line did not change, how did the irrationality disappear?
Or did we create a circle having a diameter with irrational length?

Rod

[Amigoo]

A paradox, "parabox", common conundrum ... or "paracircle"?

And again, consider this same isosceles triangle:

2. Hypotenuse = 1414213.562..*
Side 1 = 1000000.0
Side 2 = 1000000.0

But substitute the scalene triangle, enclosed within the same circle
(one side = side of circle's inscribed square, attached to square root of Pi line;
angle attached to other end of Pi line = 45 degrees).

Is the length of the Pi line still irrational?
(by popular acclaim, Pi is always irrational)

Rod

[Amigoo]

A paradox, "parabox", common conundrum ... or "paracircle"?

Circular reasoning? (consider this same isosceles triangle):

2. Hypotenuse = 1414213.562..*
Side 1 = 1000000.0
Side 2 = 1000000.0

First, draw an enclosing circle having a 2000000 units diameter.
Then, draw two radii (length = 1000000) 90 degrees apart.
The ends of those two radii (where they meet the circle) are fixed points.

How can a straight line connecting those two points have irrational length?
(isosceles right triangle, length of hypotenuse = square root of 2 x 1000000;
hypotenuse can also become a side of an inscribed square)

Intuitive answer: the length cannot be irrational;
math that describes this line is disconnected from geometry
... and is therefore irrational.

Rod

[Amigoo]

A paradox, "parabox", common conundrum ... or "paracircle"?

Infinite length? Draw a straight line with irrational length:

By definition (infinite number of decimal digits), once you start drawing the line you cannot stop ... or the line length will not be irrational! Your forward motion will be decreasing but the drawing of the line will never end (theoretically).

And this will be true even if a computer draws the line!

Rod

[11light11]

Excellent!! It's always so fun jumping on and reading your updates . . .

How is your visit going with your friend? Did you show her those images from the Finnish night sky??

It's 7:33!

Love to you both! Michele

[Amigoo]

A paradox, "parabox", common conundrum ... or "paracircle"?

Michele,

My friend was busy with Texas tourism and didn't want to focus on things "back home" - I'll send her the link after she returns home. Also, her daughter lives in Norway and may be familiar with the phenomena.

Neither have I been busy with things back home (geometry), but as I refreshed my mind with the irrational line observation, a new perspective emerged. For now, I'll call this "apples and oranges Meditation (aoMmm)":

Let apples represent whole numbers and oranges (their segments) represent decimal digits. In squared circle geometry, traditional math compares, contrasts, adds, subtracts, etc., "apples and oranges". Should we be surprised that math cannot define a squared circle without its chacteristics of "irrationality" and "transcendentalism"?

Perhaps, it's possible to separate math operations (isolate the apples and oranges). Intuitively, the oranges seem to be the cause of the irrationality when treated as members of the apples class. Maybe we should develop a new math fruit model: an apple with segments?

"Apporange" came to mind but that term is already in use in business; "orraple" also exists (orange and purple). So, "aoMmm" (as capitalized and suggesting "mentally tasty") has good potential!

Rod ... ...

[Amigoo]

Say what?

Despite the seeming balance of the decimal system (..xxx.xxx.. with zero ("0") defined as less than 1 and greater than decimal digits), all decimal digits are a portion of 1 - not a portion of 2,3,4...

Ooops! Momentarily meandering into monumentally murky mentation: ...

Shouldn't zero ("0") be that value less than all decimal digits - not the value between 1 and all decimal digits?
Maybe irrationality and transcendentalism are the problems caused by inserting 0 between 1 and the decimal digits!

However, a trifecta of absolute but non-existent values may exist: "0.", ".0", and "0 less than .xxx..".
(re: "non-existent values may exist" Say what? )

Rod

[11light11]

Speechless but enjoying it!!!

Thank you!! Michele

[Amigoo]

Mostly meandering into monumentally murky mentation ...

Re: "a trifecta of absolute but non-existent values may exist: '0.', '.0', and '0 less than .xxx..' "
Say what? '0 less than .xxx..' is a bit esoteric, confusing - even redundant!

So, let's correlate with infinity and posit that the decimal system's 0.0 equals:
plus infinity / true zero / minus infinity (the decimal point is zero infinity).
- values greater than zero exist to the left of the decimal point.
- values less than zero exist to the right of the decimal point.

Apparently, we're in the middle of infinity (or somewhere in its stream).

Rod ... aoMmm

[Amigoo]

Re: "all decimal digits are a portion of 1 - not a portion of 2,3,4.."

This mathematical reality helps explain the concept of "proportional value":

Since all decimal digits of a number have value less than 1, the decimal digits correlate with the proportional value of 1 (the number 1 in proportion to the value of the whole number).

For example, the "3.14" digits of Pi have two components: the whole number "3" and the decimal number ".14" (14/100).

".xx" of the decimal number can have no value greater than .99 and this value is less than 1.
Since 1 in the whole number is 1/3 of 3, ".xx" of the decimal number has proportional value less than 1/3 of 3.

Say what?

This concept of proportional value seems to limit the significance of the decimal digits by associating their value directly with 1 (whose value is proportional to the total value of the whole number). Thus, the proportional value of the decimal digits relates to the whole number just as 1 relates to the whole number.

Say what?

Since the 12:22 AM prompt just appeared, I'll retire for the evening and set my alarm to proportional value (translation: some time after I expect to wake up before the alarm goes off).

After checking this discussion for clarity, the 12:34 AM prompt appeared, suggesting that "all is well" ... the discussion can be clarified some other day. Final perspective: "proportional value" might be a clue for limiting the man-made transcendence of Pi.

Rod ... ...

[Amigoo]

Midwayer humor?

A very late night snack to accompany reflection on "proportional value" was followed by a trip to the bathroom to brush teeth, etc. On the way, I turned off lights and checked the door locks, then passed a clock not yet reset for Daylight Savings Time: 3:14 AM.

Rod

[Sandy]

Love it!
xxSandy

[Amigoo]

Carats in the Pi ...

Re: http://aitnaru.org/threepoints.html

With increased precision, a squared circle may display the facets of diamond:

Lengths of the sides of this diamond (half green, half white) = 1,000,000 units
which is equal to the length of the radius of the large circle.

Rod

[11light11]

Rod, that is really beautiful!

Thanks for sharing as always! Love, Michele