Deeper than primes

[doronshadmi]

zooterkin

 

[doronshadmi]

What? That “(1/2+1/4+1/8+…) - (1/2+1/4+1/8+…) = 0” is a key element of the proof that such a convergent series has a sum.

Not at all.

Two sizes of the same incompleteness have result 0 if they are subtracted from each other.

The real problem here is 1 – (1/2+1/4+1/8+…) and Archimedes did prove that it = 0.


The self similarity over scales clearly shown also by the following diagram, where the values 1,2,4,8,16,32,… etc. are not reached, ad infinituum:

In other words:

1 – (1/2+1/4+1/8+…) > 0

2 - (1/1+1/2+1/4+...) > 0

4 - (2/1+1/1+1/2+...) > 0

...

etc. ad infinituum/

 

[The Man]

Not at all.

Two sizes of the same incompleteness have result 0 if they are subtracted from each other.

Since it is the 2 times series (the series added to itself) minus the series, that just leaves the series as the difference between the 2 times series and the series, which in this case is 1.

The real problem here is 1 – (1/2+1/4+1/8+…) and Archimedes did prove that it = 0.

What so now 1 has “the same incompleteness” as “(1/2+1/4+1/8+…)”? Again what Archimedes proved is that an infinite convergent series has a finite sum and in this case it is 1

The self similarity over scales clearly shown also by the following diagram, where the values 1,2,4,8,16,32,… etc. are not reached, ad infinituum:

[qimg]http://farm5.static.flickr.com/4062/4405947817_0146693fb4_o.jpg[/qimg]

In other words:

1 – (1/2+1/4+1/8+…) > 0

Well that goes against your statement above.

The real problem here is 1 – (1/2+1/4+1/8+…) and Archimedes did prove that it = 0.

Seems you do have a “real problem here”, but it is just with yourself.

2 - (1/1+1/2+1/4+...) > 0

4 - (2/1+1/1+1/2+...) > 0

...

etc. ad infinituum/

Further assumptions based on your simple assumption that the “(1/2+1/4+1/8+…)” has no sum are just as invalid as your initial assumption, which again was proven wrong some 2,300 years ago.

 

[The Man]

Oh, and happy birthday zooterkin.

 

[jsfisher]

All you did is to show that an infinite interpolation has no accurate sum (in your language it is about 1.814…), which is exactly the same result of my infinite converges series, which has an invariant length of Koch's fractal > 0 that can't reach the limit value, which is exactly 0.

And I also showed it without using any circle, but simply show how the convergent length between the opposite edges of the Koch's fractal can't be 0, as long as these edges are in common with the invariant length of the Koch's fractal, upon infinitely many bended levels

No, you got a big fail on this for multiple reasons.

First, you still continue to refer to the generations of Koch's curve as a fractal. This is pure ignorance on your part; only the generation limit is a fractal.

Second, you have not actually proven the curve generations under your length invariant constraint fit your "lute of doron" shape. It is trivial to do, but you seem unable to actually do it.

Third, you claim (in effect) the distance between the end points of any Koch curve generation cannot be zero in your construction. While the statement is true (but completely beyond your ability to prove, I suspect), it does not let you draw your bogus conclusion about the limit.

Fourth, you make a baseless assumption relating end-point distance and overall curve length for the limiting case in your Koch curve construction.

Fifth, you still insist on using the bogus word "bended".

For your original construction of Koch's snowflake, your argument was equally deficient, but the focus then was the relationships of area and perimeter.

 

[zooterkin]

zooterkin

Oh, and happy birthday zooterkin.

Thanks!

 

[jsfisher]

Thanks!

Uh-oh. Not just any birthday, either. Happy Mid-century Day.

Be very careful, though. Body parts no longer just atrophy; some actually fall off.

 

[doronshadmi]

What so now 1 has “the same incompleteness” as “(1/2+1/4+1/8+…)”? Again what Archimedes proved is that an infinite convergent series has a finite sum and in this case it is 1

Further assumptions based on your simple assumption that the “(1/2+1/4+1/8+…)” has no sum are just as invalid as your initial assumption, which again was proven wrong some 2,300 years ago.

The Man, 1 is an accurate value where (1/2+1/4+1/8+…) does not have an accurate value.

As a result 1 – (1/2+1/4+1/8+…) = 0.000…1[base 2]

X = (1/2+1/4+1/8+…)

If you think that X+X = accurate value, you simply wrong, because 2X, 4X, 8X etc … do not have accurate values, and Archimedes did not prove that any of 1X, 2X, 4X, 8X ,16X , 32X … etc. have accurate values, and the following diagrams are a proof without words that Archimedes initial assumption was wrong:

 

[doronshadmi]

only the generation limit is a fractal.

Exeactly the opposite, only the incomplete generation is an infinite fractal.

Again, 1 is not a fractal and (1/2+1/4+1/8+1/16...) is a single path along a fractal exactly because 1-(1/2+1/4+1/8+1/16...)=0.000...1[base 2], exactly as shown in:

 

[jsfisher]

Exeactly the opposite, only the incomplete generation is an infinite fractal.

It continues to be a great disappointment that you were unable to receive a dictionary for Hanukkah. Have you considered just buying one on your own?

Again, 1 is not a fractal....

What are you on about now? Nobody has even suggested that 1 be a fractal. That would be idiotic.

 

[The Man]

The Man, 1 is an accurate value where (1/2+1/4+1/8+…) does not have an accurate value.

As a result 1 – (1/2+1/4+1/8+…) = 0.000…1[base 2]

Again just your baseless assumption that has already been proven wrong some 2,300 years ago.

X = (1/2+1/4+1/8+…)

If you think that X+X = accurate value, you simply wrong, because 2X, 4X, 8X etc … do not have accurate values, and Archimedes did not prove that any of 1X, 2X, 4X, 8X ,16X , 32X … etc. have accurate values, and the following diagrams are a proof without words that Archimedes initial assumption was wrong:
[qimg]http://farm5.static.flickr.com/4062/4405947817_0146693fb4_o.jpg[/qimg]

Just what do you think “Archimedes initial assumption was”? Unlike you he did not simply assume something then draw some images representing that assumption and claim it’s “a proof without words”. Again the basis of Archimedes proof are self similarity, that X=X and X+X-X=X. To refute the proof your have to refute one or all of those aspects the proof is actually based on not simply repeating your assumed and imaginary “accurate value” nonsense. If think you can refute one or all of those aspects to refute that proof you are going to end up just digging a hole for yourself as even your OM depends upon those aspects being valid.

 

[doronshadmi]

No, you got a big fail on this for multiple reasons.

First, you still continue to refer to the generations of Koch's curve as a fractal. This is pure ignorance on your part; only the generation limit is a fractal.

Second, you have not actually proven the curve generations under your length invariant constraint fit your "lute of doron" shape. It is trivial to do, but you seem unable to actually do it.

Third, you claim (in effect) the distance between the end points of any Koch curve generation cannot be zero in your construction. While the statement is true (but completely beyond your ability to prove, I suspect), it does not let you draw your bogus conclusion about the limit.

Fourth, you make a baseless assumption relating end-point distance and overall curve length for the limiting case in your Koch curve construction.

Fifth, you still insist on using the bogus word "bended".

For your original construction of Koch's snowflake, your argument was equally deficient, but the focus then was the relationships of area and perimeter.

You still do not get it do you?

So let us improve the proof without words, by using Koch's fractal.

1) Take a straight 1-dim with length X.

2) Bend it and get 4 equal sides along it.

3) Since the length between the opposite edges is changed to the sum of only 3 sides, and since the number of the sides after the first bending is 4 sides, we have to multiply the bended 1-dim element by 1/(the number of the bended sides), in order to get back length X.

As a result the bended 1-dim element has length X, but the length between its opposite edges becomes smaller (it converges).

In general, this convergent series of 1/(the number of the bended sides) is resulted by 1/1+1/4+1/16+1/64+1/256+... , where X is subtracted by (2a+2b+2c+2d+…)

Here is the result:

4) By Standard Math X – (2a+2b+2c+2d+…) = 0


5) (4) is false because (2a+2b+2c+2d+…) can be found as long as X is found.

6) Since X is found upon infinitely many scale levels then (2a+2b+2c+2d+…) must be < X , and as a result X - (2a+2b+2c+2d+…) > 0.

7) Conclusion: (2a+2b+2c+2d+…) does not have sum X.

 

[doronshadmi]

Again the basis of Archimedes proof are self similarity

So what?

Also a non-accurate value has self similarity.

 

[The Man]

No, it has an incomplete area based on infinite interpolation that has no accurate sum.

Why does the area encompassed by the Koch snowflake need to be a sum? Or based on some imaginary reference of your “interpolation that has no accurate sum”?

The area is complete and finite.

You still do not get that any infinite collection is inherently incomplete, whether it is expressed as infinite interpolation or infinite extrapolation.

You still don’t get assumptions as well as imaginary reference like “infinite interpolation or infinite extrapolation” do not constitute facts or proofs, Even "proof without words" which none of your so called "proofs" are actualy without (the words that is, they are certianly without and proof)

All you did is to show that an infinite interpolation has no accurate sum (in your language it is about 1.814…), which is exactly the same result of my infinite converges series, which has an invariant length of Koch's fractal > 0 that can't reach the limit value, which is exactly 0.

Again the exact value of that sum of your infinite convergent series was give as “3*p*{[Tan(60)*1/6]-[Tan(30)*1/6]}” and “3/2 times the circumference of your first circle”, 1.814 is just an approximate decimal rounded off representation of that value, which again is why I specifically said “or about 1.814”.

And I also showed it without using any circle, but simply show how the convergent length between the opposite edges of the Koch's fractal can't be 0, as long as these edges are in common with the invariant length of the Koch's fractal, upon infinitely many bended levels, here it is again (no circles are used):

[qimg]http://farm5.static.flickr.com/4034/4423020214_87676ef96b_o.jpg[/qimg]

​

Who has claimed it would be 0 other then you and your deliberately erroneous “reasoning” that you just like to ascribe to others? Again the Koch curve increase in length by 1/3 with each iteration. So you still have to scale down each iteration to keep your constant length. Thus you still end up with a common ratio for the series of 1/3. Making the difference between the original series and the 3 times series still equal to twice the sum of the original series and 3 times your original series starting value. So the sum of your original series is still 3/2 times your starting value. If that starting value is 1 then the sum of the series is just 1.5.

You simply have no real notion of the inherent incompleteness of infinite collection (and in this case, infinite collection of bended levels).

You simply have no real notion of the easily demonstrable invalidity of you assumptions, even some 2,300 years ago.

A finite value has an accurate sum, an infinite value does not have an accurate sum, but you simply can't get it because you do not understand the real nature of infinite interpolation\extrapolation.

Again simply your assumption and the fact that an infinite convergent series has a finite sum was proven some 2,300 years ago.

Again, the convergent length between the opposite edges of the Koch's fractal can't be 0 (where 0 is the value of the limit), as long as these edges are in common with the invariant length of the Koch's fractal, upon infinitely many bended levels, as shown above, but you prefer to ignore this proof without words, because it does not fit your Limit-oriented notion, which is obsolete.

Again

Who has claimed it would be 0 other then you and your deliberately erroneous “reasoning” that you just like to ascribe to others? Again since the Koch curve increase in length by 1/3 with each iteration. So you still have to scale down each iteration to keep your constant length. Thus you still end up with a common ratio for the series of 1/3. Making the difference between the original series and the 3 times series still equal to twice the sum of the original series and 3 times your original series starting value. So the sum of your original series is still 3/2 times your starting value. If that starting value is 1 then the sum of the series is just 1.5.

Exactly because the Koch's fractal has an invariant length > 0 , there are infinitely many bended levels that can't reach the value of the limit, which is exactly 0.

Oh so now it has a limit again?

The limitation is entirely the result of your Limit-oriented approach, that prevents the notion of infinite interpolation\extrapolation, which is inherently non-limited.

No Doron the limitation is simply yours since you can’t seem to make up you mind what you think you what to claim you prove “without words”. The series has no sum, the series has no limit or the sum simply does not equal the limit. The real reason you can’t make up you mind it that you simply do not know what you are talking about and you do prove that with words.

You do not get that in my model there is no dichotomy between invariant AND convergent length, as clearly shown and explained above, but you can't get this anomaly because you are using a model that is based on dichotomy.

A significant problem for your “model”, but hey it’s your model and your problem.

Both Wiki sources are not based on the qualitative difference between the local and non-local atomic aspects that send at the basis of any complex like Set (for example).

So they are not based on your unsupported and often proven wrong assumptions, I’m sure everyone will be so dismayed.

Again an obsolete knowledge is used by you The Man, which is irrelevant to OM's novel notions, of this subject.

Again Doron your simple reliance on your fantasies, misunderstanding and misrepresentation are not “novel notions”, you just happen to think they are because they are yours.

No, the incompleteness of a collection is not closed under Class, where a set is a particular case of a collection that is closed under Class.

Unbounded sets are still a class of sets just as bounded sets are.

Since any infinite collection (where a set is a particular case of a collection that is closed under Class) is inherently incomplete (it has a “Trojan horse”), it does not have "closed gates".

The inclusion in the set of the successor of any element in that set is still specifically what closes the set under an operation of succession on any element of that set. If you think it is a “Trojan horse” that attacks your notions that’s just because you brought it in yourself.

You simply ignore the inherent incompleteness of any collection, whether it is closed under Classes or not.

You simply assume that “inherent incompleteness of any collection” and think since it is your assumption it should be significant.

Again, the "“sudden effect” is the result of your obsolete Limit-oriented notion of this subject.

Nope just your imagination since you’re the only one who thinks they experience your “sudden effect”.

No, it is about an anomaly of the standard model, which shows how a length can be both invariant AND variant (converges, in this case) in a one model.

Again the anomaly is just in your brain, you required the “length” of your pseudo-Koch snowflake to be invariant (1 in this case). So your imaginary “anomaly of the standard model” of it being both “invariant AND variant” is just your simple or deliberate confusion of the “length” you required to be invariant with an actual, well, variable.

This is another example of your inability to understand the generalization of this subject.

This is another example of you thinking “generalization of this subject” means just claiming whatever nonsense suits you.

Any infinite collection is inherently incomplete, or in other words, any given element of that collection is not its final element, and order has no impact on that fact.

Again, If it is a member (and least upper bound) then it is the finail element of that collection.

Your simple assumptions, misinterpretations and misrepresentations have “no impact on that fact”

On the contrary, your Limit-oriented notation, can't get the model where both invariant AND variant (converges, in this case) length are found.

Your built-in dichotomy of your notions simply can't get this anomaly of Standard Math model of this subject.

Again your anomaly is only in your brain and results from you simply or deliberately confusing some invariant value (one that you required in this case) with an actual, well, variable.

A typical reply of a person that uses built-in dichotomy and Limit-oriented notions of this subject.

How boring.

A typical reply of some one who can not see that they have deliberaly constructed some imaginary “anomaly” to attribute to others since they can not acctualy address what they to call “Standard Math”

How sad.​

 

[doronshadmi]

How sad.

How sad that you can't get http://www.internationalskeptics.com/forums/showpost.php?p=5715516&postcount=9032 or http://www.internationalskeptics.com/forums/showpost.php?p=5715530&postcount=9033.

 

[jsfisher]

You still do not get it do you?

Project much?

So let us improve the proof without words, but using Koch's fractal.

1) Take a straight 1-dim with length X.

You mean a line segment of length X? You really, really need that dictionary.

2) Bend it and get 4 equal sides along it.

Sloppy, sloppy, sloppy. Koch was never so sloppy as this in describing his procedure for generating each next iteration. You need to accurately describe what the result of this bending needs to be, yet you didn't.

By the way, line segments, even bent ones, don't have sides.

3) Since the length between the opposite edges is changed to the sum of only 3 sides

That presupposes many things not stated in Step 2. So, this is another fail for doron.

...and since the number of the sides after the first bending is 4 sides, we have to multiply the bended 1-dim element by 1/(the number of the bended sides), in order to get back length X.

How about we just set aside your gibberish and assume you meant for Steps 2 and 3 that we applied Koch's procedure and then shrunk it by 3/4 to hold the overall length of the result constant.

As a result the bended 1-dim element has length X, but the length between its opposite edges becomes smaller

Under my description of the steps, sure; under yours, not so much. In other news, "bended" is still not the right word to use here.

...(it converges).

Another doron failure. So far, your steps take you from a line segment to line segment with a triangular bump in the middle. Perhaps you meant to reapply Steps 2 and 3 to the individual line segments in the current result to get the next result?

No wonder you get so much wrong. You just keep jumping from assumption to wrong conclusion without the least consideration for any gap between the two.

So, why not go back to my post and actually address my points rather than wasting our time with this failed tangent.

 

[zooterkin]

You still do not get it do you?

So let us improve the proof without words, by using Koch's fractal.

1) Take a straight 1-dim with length X.

2) Bend it and get 4 equal sides along it.

Congratulations, you now have a square.

 

[doronshadmi]

Project much?



You mean a line segment of length X? You really, really need that dictionary.



Sloppy, sloppy, sloppy. Koch was never so sloppy as this in describing his procedure for generating each next iteration. You need to accurately describe what the result of this bending needs to be, yet you didn't.

By the way, line segments, even bent ones, don't have sides.



That presupposes many things not stated in Step 2. So, this is another fail for doron.



How about we just set aside your gibberish and assume you meant for Steps 2 and 3 that we applied Koch's procedure and then shrunk it by 3/4 to hold the overall length of the result constant.



Under my description of the steps, sure; under yours, not so much. In other news, "bended" is still not the right word to use here.



Another doron failure. So far, your steps take you from a line segment to line segment with a triangular bump in the middle. Perhaps you meant to reapply Steps 2 and 3 to the individual line segments in the current result to get the next result?

No wonder you get so much wrong. You just keep jumping from assumption to wrong conclusion without the least consideration for any gap between the two.

So, why not go back to my post and actually address my points rather than wasting our time with this failed tangent.

As I said, you simply can't get that X-(2a+2b+2c+2d+...) > 0 as proved in http://www.internationalskeptics.com/forums/showpost.php?p=5715516&postcount=9032 by using a proof without words (http://en.wikipedia.org/wiki/Proof_without_words).

Your rambling will not help you here, jsfisher.

 

[doronshadmi]

Congratulations, you now have a square.

EDIT:

Where is the square in _/\_ ?

 

[sympathic]

Where is the squere in _/\_ ?

1. should be "square"

2. is this the AutoCAD program? if so - your AutoCAD probably uses "direct perception" as well....