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katastrofa
Posts: 10067
Joined: August 16th, 2007, 5:36 am
Location: Alpha Centauri

### Re: what did the painter do wrong?

The $\sqrt 7$ identity is wrong. The continued fraction should be the golden ratio with the only radical being $\sqrt 5$. So the two side cannot be equal.
The sqrt(7) identity is OK. What do you mean by "The continued fraction should be the golden ratio with the only radical being $\sqrt 5$"?

I don't see any error in the painting, Collector. Where is it?

Posts: 23951
Joined: September 20th, 2002, 8:30 pm

### Re: what did the painter do wrong?

The $\sqrt 7$ identity is wrong. The continued fraction should be the golden ratio with the only radical being $\sqrt 5$. So the two side cannot be equal.
The sqrt(7) identity is OK. What do you mean by "The continued fraction should be the golden ratio with the only radical being $\sqrt 5$"?

I don't see any error in the painting, Collector. Where is it?
But it's not OK.

The continued fraction converges to 0.61803399... or one minus the golden ratio which is (sqrt(5)+1)/2. Sqrt(7) is 2.6457513..., not 2.61803399...

katastrofa
Posts: 10067
Joined: August 16th, 2007, 5:36 am
Location: Alpha Centauri

### Re: what did the painter do wrong?

This is this the 10% when you sound double Dutch to me...

ppauper
Posts: 70239
Joined: November 15th, 2001, 1:29 pm

### Re: what did the painter do wrong?

continued fraction calculator
seems to give the same answer as the painter

collector said the "official" error was the "ln" being added after the event

katastrofa
Posts: 10067
Joined: August 16th, 2007, 5:36 am
Location: Alpha Centauri

### Re: what did the painter do wrong?

Thanks, it indeed looks a bit "added".

And, indeed, I should have also written out p4 = 4 + 1/p5 - and then it's periodic.

Posts: 23951
Joined: September 20th, 2002, 8:30 pm

### Re: what did the painter do wrong?

Yes, a 2, {1,1,1,4} continued fraction does yield sqrt(7). But the painted image does not have the required last term of +1/4+.... which would be large enough to show on the other side of the painted bar. The painting only shows a 3-term series. In fact the position of the final denominator 1 directly under the last numerator 1 (rather than left-shifted to leave room for a longer denominator) suggests that the final term is just 1/1, that it's not a continued fraction, and that the RHS is 2.666666.
Last edited by Traden4Alpha on May 23rd, 2018, 11:43 pm, edited 1 time in total.

lovenatalya
Posts: 287
Joined: December 10th, 2013, 5:54 pm

### Re: what did the painter do wrong?

This is this the 10% when you sound double Dutch to me...
There is no periodicity so far up to P_4. How did you make the leap of faith to ... nowhere? Rather clever...

lovenatalya
Posts: 287
Joined: December 10th, 2013, 5:54 pm

### Re: what did the painter do wrong?

continued fraction calculator
seems to give the same answer as the painter
A bit care reading the instruction of the calculator and its generated numbers will lead to a different answer...

ppauper
Posts: 70239
Joined: November 15th, 2001, 1:29 pm

### Re: what did the painter do wrong?

continued fraction calculator
seems to give the same answer as the painter
A bit care reading the instruction of the calculator and its generated numbers will lead to a different answer...
it's already given us the correct answer which is the one shown in  the painting
The mistake in the painting was that the painter omitted "ln" and had to add it in after the event

katastrofa
Posts: 10067
Joined: August 16th, 2007, 5:36 am
Location: Alpha Centauri

### Re: what did the painter do wrong?

continued fraction calculator
seems to give the same answer as the painter
A bit care reading the instruction of the calculator and its generated numbers will lead to a different answer...
I wasn't talking to you. You can find the derivation in the photo of my notebook and the same result in the calculator.

lovenatalya
Posts: 287
Joined: December 10th, 2013, 5:54 pm

### Re: what did the painter do wrong?

continued fraction calculator
seems to give the same answer as the painter
A bit care reading the instruction of the calculator and its generated numbers will lead to a different answer...
it's already given us the correct answer which is the one shown in  the painting
The mistake in the painting was that the painter omitted "ln" and had to add it in after the event
I am saying you read the result from the calculator wrong. I copied it below. Note the "4" there. It has a period of 4 not 1.
--------------------------------
x = sqrt{7}
expansion in continued fraction of x = 2 + //1, 1, 1, 4//
where the periodic part is marked in bold (the period has 4 coefficients)

lovenatalya
Posts: 287
Joined: December 10th, 2013, 5:54 pm

### Re: what did the painter do wrong?

continued fraction calculator
seems to give the same answer as the painter
A bit care reading the instruction of the calculator and its generated numbers will lead to a different answer...
I wasn't talking to you. You can find the derivation in the photo of my notebook and the same result in the calculator.
My message was not directed at you, but to ppauper. Why do you answer to it? As for your notebook derivation, that was wrong. I have already said as much in my earlier message which was directed at you. I copy it here:
This is this the 10% when you sound double Dutch to me...
There is no periodicity so far up to P_4. How did you make the leap of faith to ... nowhere? Rather clever...

katastrofa
Posts: 10067
Joined: August 16th, 2007, 5:36 am
Location: Alpha Centauri

### Re: what did the painter do wrong?

The derivation is correct. Maybe you will explain why you started the discussion by claiming that the painting was wrong and hallucinating something about golden ratio instead of making fake impressions?

ppauper
Posts: 70239
Joined: November 15th, 2001, 1:29 pm

### Re: what did the painter do wrong?

A bit care reading the instruction of the calculator and its generated numbers will lead to a different answer...
it's already given us the correct answer which is the one shown in  the painting
The mistake in the painting was that the painter omitted "ln" and had to add it in after the event
I am saying you read the result from the calculator wrong. I copied it below. Note the "4" there. It has a period of 4 not 1.
--------------------------------
x = sqrt{7}
expansion in continued fraction of x = 2 + //1, 1, 1, 4//
where the periodic part is marked in bold (the period has 4 coefficients)
lovenatalya, I didn't read anything wrong. You and only you are the one who made a mistake when you insisted that the expansion in the painting was wrong.

lovenatalya
Posts: 287
Joined: December 10th, 2013, 5:54 pm

### Re: what did the painter do wrong?

$\sqrt{7}=2+\overline{1,1,1,4}$ where the number under the bar is the periodic integer sequence in the continued fraction. ppauper's calculator confirms this. ppauper missed the 4. So if the continued fraction is indeed for $\sqrt{7}$, there should have been a $4$ below and to the right of the rightmost 1. There is no 4 to be found.

The continued fraction can only be represented by the whole not part of its periodic part. Otherwise it would be absurd. That would have meant $(1,1,1)$ represents $\overline{1,1,1,2}$ and $\overline{1,1,1,3}$ and $\overline{1,1,1,15,3,7}$ which are all different numbers.

Therefore the continued fraction in the painting can only represent $\overline{1}=\frac{\sqrt5-1}{2}$ the golden ratio as I have said. This is easily checked by $\frac{\sqrt5-1}{2}=\frac{1}{1+\frac{\sqrt5-1}{2}}$ with period $1$. And $\sqrt7\ne 2+\frac{\sqrt5-1}{2}$. The expression in the painting is thus wrong.

As for catastrofa's derivation, the obvious mistake is that just as in the painting, no 4 is to be found there. Also the derivation has no distinct natural numbers $i, j$ shown such that $P_i=P_j$. In other words, there is no periodicity shown. So the derivation is incomplete and wrong.