OT: golden ratio

It's the golden ratio formula because at x=1 it is the golden ratio and at x=-1 it is the inverse of the golden ratio 1.618...

and at x equals all other integers (positive and negative) it is golden ratios from the sequence of numbers, ie for positive x:

0.618033988749895 0.414213562373095 0.302775637731995 0.23606797749979 ..

cheers, Jamie

ng-to->the-golden-ratio/

Absolutely. guy had a saying aout cars, never anted to look under the ood. he said "MF be looking for things MF be finding things". He was right. He w as ignorant but he was right. Buddy of mine kept telling me the number 13 c ame up all the time in his life. I told him that was because he was looking for it. I told him to look for the number 12.

I believe tht alot of science is polluted with this. Even though many of th e results are repeatable, I think the results are many time slewed by impro per scietific method. I can't prove this and don't claim to be able to, but those who claim to be able to and claim that it is not true are not being scientific about it. You can't really disprove anything.

Smoking causing lung cancer for examople, totally disputable. Even though t hey think they know the mechanism by which that happens, the fact remains t hat it does not happen to every smoker, even if they live a very long time. The proper conclusion is that their body is not strong enough to withstand the effects of smoking.

Sometimes correlation indicates causation, other times not.

Recently, they said the disappearance os a pulsar as due to space bending. I call bullshit because if it is bending we should be gettingthe pulses of enrgy slightly out of phse with their former timing. There is no reason for it to disappear for that reason. There is obviously some kind of cloud or some shit in the way. No maybe if it was always ALMOST in the way and then space bends, OK. But that is NOT what they said.

Hi,

I came across this:

It shows the formula for the golden ratio: (1+sqrt(5))/2

The formula I made is: (-x+sqrt((x*x+4)))/2

and their formula for the golden ratio is the same as my formula for x=-1, since 1.618.. is the reciprocal to the golden ratio, it is x=-1, instead of x=1 which gives the

0.618.. golden ratio.

(--1+sqrt((-1*-1+4)))/2 (1+sqrt(1+4)/2 (1+sqrt(5)/2 that is 1.618..

So the (1+sqrt(5))/2 is a specific case of my formula for x=-1

cheers, Jamie

You can find a lot of literature on the subject under the heading "Sacred Geometry"- mostly targeted at the arts and historical architecture.

Unfortunately, the series doesn't converge. It diverges slowly like the harmonic series.

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The series N/2(sqrt(1 + 4/N^2) - 1) doesn't converge. Expand sqrt(1 + x) in a Taylor series and substitute 4/N^2 and you can see the series goes like 1/N.

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Hi,

This formula (-x+sqrt((x*x+4)))/2 for the series doesn't converge?

From 1billion integer iterations starting at x=1 it's summed value was the same to 13 decimal places as with 10billion iterations

It seems to be an irrational number that converges.

I don't think it really matters either way, that equation has a lot of interesting properties, for x = -1 that is 1/phi and for x = 1 that is phi

Also there is a neat sequence of outputs that are exact integers:

x y

-9.9 10.00000000000

-4.8 5.00000000000

-2.1 2.50000000000

0 1.00000000000 1.5 0.50000000000 2.1 0.40000000000 4.8 0.20000000000

Also each input x that is an integer outputs a "golden ratio number" ie:

x y

5 0.19258240357

I call it a golden ratio number because 1/y equals 5.19258240357, each of these y numbers contains both the x and the y value in the number (x is the integer part and y is the fractional part of the number)

I think there could be some interesting patterns in this formula still to find, maybe comparing area's of rectangles with sizes defined by some of these golden ratio numbers or the exact integer y numbers..

cheers, Jamie

It fooled me at first also, but no, unfortunately it does seem to diverge. Extremely slowly, but it diverges nonetheless. Wolfram Alpha thinks so as well:

Expand the square root in a Taylor series, so you get a sum of a sum of terms. If there is a 1/N term in the inner sum, you're dead in the water.

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Hi,

Just out of curiosity what is the difference if it converges or not for what it says about the equation (besides the obvious that it is/isn't converging)? Also the wolfram alpha page you linked shows threes tests used to determine convergence and two of the three are inconclusive it says.

"Convergence tests:

The ratio test is inconclusive The root test is inconclusive By the comparison test, the series diverges"

I don't know if that means it diverges or not, I don't know much about it, but I am pretty sure that even if I go to 10^20 iterations it will remain 18.895995... :D The amount added to the sum when the input N goes up 10 also goes down 10 times at least.

cheers, Jamie

I'm not sure exactly what you mean by your first sentence. If it doesn't converge it just means that the limit to infinty of the sequence of partial sums of the sequence you picked doesn't happen to be a finite number.

The ratio and root test being inconclusive means IIRC that the limits involved in the tests evaluate to 1; there are both series that converge and diverge that have this property so those particular convergence tests can't tell you which category your series falls into.

Well, 10^20 is still a long way off from infinity.

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Hi,

I did a bit more testing of sums, and comparing the wolfram alpha to my C# (and switched to some 64bit math in C#)

Here are the results from C# now for sum from 1 to n of (-x+sqrt((x*x+4)))/2

n = 1000000; //13.857175326264198079655 n = 10000000; //16.1597662612313545745845 n = 100000000; //18.47316029644405020512250 n = 1000000000; //19.06388867623006536968105 n = 2000000000; //19.06388867623006536968105 n = 3000000000; //19.06388867623006536968105 n = 10000000000; //19.39583957916890814311855

For wolfram alpha here are the results for sum from 1 to n of (-x+sqrt((x*x+4)))/2

sum (sqrt(N^2 + 4) - N)2 from 1 to 10000000 - WolframAlpha (16.1598) sum (sqrt(N^2 + 4) - N)2 from 1 to 100000000 - WolframAlpha (18.4681) sum (sqrt(N^2 + 4) - N)2 from 1 to 1000000000 - WolframAlpha (19.321) sum (sqrt(N^2 + 4) - N)2 from 1 to 10000000000 - WolframAlpha (19.3556) sum (sqrt(N^2 + 4) - N)2 from 1 to 20000000000 - WolframAlpha (19.3871) sum (sqrt(N^2 + 4) - N)2 from 1 to 30000000000 - WolframAlpha (19.292)

The last one 19.292, is less than the one before so shows where WolframAlpha starts to get buggy.

example wolfram alpha link:

^2+%2B+4%29+-+N%29%2F2+from+1+to+10000000000

For n = 10 billion (10000000000) the C# gave 19.395 and Wolfram gave

19.355 as the result.

In the C# for n = 1billion, n = 2billion, and n = 3billion the output was the same, I guess this is an error but not sure how.

I can see that this might not converge, thanks!

cheers, Jamie

On a sunny day (Thu, 12 Feb 2015 12:56:07 -0800) it happened Jamie M wrote in :

Yes, that is the entry number for the wormgate.

Sorry but it only means that you don't understand numerical analysis :(

Inspection of the low root shows that as N -> infinity it looks asymptotically like

phiN = 1/N ( 1 - 1/(N^2+2) ) = O(1/N) as N-> inf ie. diverges

Sum 1/N is known to diverge to log(N) as N become large.

The reason you are not seeing it is that for sufficiently large N the mantissa is no longer contributing to the sum. You have to sum these series much more carefully - this untested ought to allow you to see how the series is in fact divergent by selecting 2^N terms at a time.

This too will fail when 1/N underflows the mantissa of the partial sum. Then you have to do a more cunning add adjacent terms to form a series half the length until you get your answer. Sweeping backwards from the smallest term will also slightly help with rounding issues.

Adding very small numbers to a large sum is fraught with danger.

for (i = 0; i < two_t; i++) { > x = (-n + Math.Sqrt((n * n + 4))) / 2; > nSum = nSum + x; n++; } textBox1.Text = nSum.ToString();

two_t = 2*two_t; > > }

Basically a numerical version of the proof that taking the 2^N next terms of something that tends to 1/N will sum to a number which is O(1).

Beware of rounding errors!

--
Regards, 
Martin Brown

Thanks! :)

cheers, Jamie

It's just simple arithmetic to show x(N)>1/(N+1), x(N) the Nth term of his ratio sequence. Even he should understand it diverges when it dominates a known divergent series.

Then there this:

"A neat thing about this sequence of golden numbers is the position in the sequence of the number is encoded in the number itself!

ie for the 10th number in the sequence the number is 0.099019513592784

if you take the reciprocal of that number 1/x, it is 10.0990195135928..

this is the same number just 10 larger, and 10 is the position in the sequence of that number too! "

which can be re-phrased as :

Let's define the sequence to be this, so, hey, the individual terms of the sequence turn out to be the way we defined them.

understand it diverges when it dominates a known divergent series.

defined them.

Hi,

I am not so sure that the sequence does diverge, I am still looking into using arbitrary precision floating point numbers, but with 32bit math this is what I get as results of convergence testing:

for the equation: (-x+sqrt((x*x+4)))/2

x y

1 0.618033988749895 10 0.0990195135927845 100 0.00999900019994725 1000 0.00099999899998693 10000 0.000099999999292777 100000 0.00001000000338535755 1000000 0.00000100000761449337 10000000 0.000000099651515483856 100000000 0.00000000745058059692385 [breakdown of code accuracy]

so when x increases at a rate of 10x, if y is decreasing at a rate of less than 10x, then the series will converge I think.

x=1 to x=10:

0.618033988749895/0.0990195135927845 =6.24153x

x=10 to x=100:

0.0990195135927845/0.00999900019994725 =9.9029x

x=100 to x=1000:

0.00999900019994725/0.00099999899998693 =9.99901x

x=1000 to x=10000:

0.00099999899998693/0.000099999999292777 =9.99999007x

x=10000 to x=100000:

0.000099999999292777/0.00001000000338535755 =9.999996543x

x=100000 to x=1000000:

0.00001000000338535755/0.00000100000761449337 =9.99992724x

x=1000000 to x=10000000:

0.00000100000761449337/0.000000099651515483856 =10.0350467x

x=10000000 to x=100000000:

0.000000099651515483856/0.00000000745058059692385 =13.375000x [wrong result] [breakdown of code accuracy]

It is easy to see that the rate that y is decreasing is itself increasing, or approaching 10x, while the rate of x increase remains steady at 10x. To me this is a proof that the series itself is converging, since y is used to increase the sum, it is increasing the sum at a rate lower than required for it to keep growing infinitely.

Even if the series converges or diverges is kind of a stupid question, as obviously this series is "attempting" to be exactly balanced between converging and diverging, as the rates are approximately equal on x and y. This makes sense since these are the Golden number series:

If these numbers sums diverged or converged then it would be unexpected, but really they have to be balanced between converging and diverging, as they are the most irrational numbers!

Wolfram Alpha says the series diverges based on the comparison test, and for the apparently similar harmonic series the comparison test proof can look like this:

from the page:

I believe that proof is flawed though, it is saying the harmonic series diverges since when it's sequence is mapped to the other series (known to diverge) then it must also diverge. However the mapping used means that an infinite number of terms will be required in each comparison term, which as an idea is illogical, just because the series may be infinite doesn't mean you can use an infinite number of terms summed infinitely within it and expect that to be a proof that it diverges, the harmonic series is linear and remapping it so that it is summed up in a non-linear way, so that each new term has more items will of course be able to make it seem like it diverges. The harmonic series is not convergent or divergent either it is balanced so that the rate of growth of x and y are the same, ie x=1 y=10, and x=10 y=1/10.

The real proof is that the idea of infinity used to prove that a balanced series is divergent are flawed is itself a flawed idea of infinity.

cheers, Jamie

From Wolfram Alpha:

^2+%2B+4%29+-+1000000000%29%2F2%29%29+%2F+%28%28sqrt%2810000000000^2+%2B+4%29+-+10000000000%29%2F2%29%29

((sqrt(1000000000^2 + 4) - 1000000000)/2)) / ((sqrt(10000000000^2 + 4)

- 10000000000)/2))

that gives:

9.99999999999990100000000000198989999999995020102000000139498...

so at infinity it will be 10x, and the series will stop growing, up to infinity it will still keep growing, but at an infinitely small (and decreasing rate), therefore the series converges at infinity, but before infinity it is diverging less and less.

cheers, Jamie

Muppet. You have to sum up the terms between the limits to see that in each decade the overall contribution to the infinite sum remains O(1).

Just taking the last term is enough

Compute partial sums in each decade and you might finally understand.

2..10 9x0.099 = 0.891 11..100 90x0.009999 = 0.8999 101..1000 900x0.000999999 = 0.8999999 1001..10000 etc.

See the pattern? In reality they will be even larger!

For large N the closed form analytic formula for the root that I derived earlier is considerably more accurate than numerical computation of sqrt(x*x+4)-x in the limited precision of IEEE FP.

I am a bit surprised that it fails quite so catastrophically for N=10^5 since I'd have expected numerical solutions to hold out to 10^7 ish. But the numerical representation of 1/10^N is rather poor. It will probably behave better numerically for power of two values so that the mantissa is not compromised by its limited representation of 1/10.

Taking the simpler approach of Taylor series expansion

(-x + sqrt(x*x+4)/2 = x ( -1 + sqrt(1+4/x*x) )

~ x/2 ( -1 + 1 + (1/2)4/x*x + (1/2)(-1/2)(4/x*x)^2 + O(x^-6) )

= x/2 ( 2/x*x - 4/x*x*x*x + O(x^-6) )

= 1/x ( 1 - 2/x*x + O(x^-4) )

The correct value is always very slightly less than 1/x but not by enough to allow convergence in the limit x -> infinity

Inspection of the low root shows that as N -> infinity it looks asymptotically like

phiN = 1/N ( 1 - 1/(N^2+2) ) = O(1/N) as N-> inf ie. diverges

Sum 1/N is well known to diverge to log(N) as N becomes large.

--
Regards, 
Martin Brown

Hi,

I found some higher resolution math using cloud.sagemath.com

I still would say the function converges at infinity, and up to that point it is diverging less and less. Is that the same as saying the function diverges?

I think there are two or three main groups of divergent functions:

1. diverge at a decreasing rate towards infinity (converge at infinity as divergence goes to zero)

1. diverge at a fixed steady rate towards infinity (divergence is constant up to infinity)

2. diverge at an increasing rate towards infinity (diverge infinitely at infinity as divergence goes to infinity)

Here is the Sage code showing more resolution:

A = RealField(500) B = RealField(500)

A =

10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000000.0000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000

B = (-A + sqrt((A * A + 4))) / 2

B

9.999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999

999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 999999999999999999999999999999999999999999999999999999999900000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000001570544810264982 87629590395403090125001937115213564048042010155497262370436291678625006469200668186387956648 507965185636659585004639090646962737922459269518769574987656282142079958408589052421556548739 36960387684980027001774650339383168787307309766270001360131673356890354983861484948025682391 209495391735118525540196891737266156976212444617500755285748170952159085288816704068413112936 76858138989256229014152683959870155965628434542754518120215806948093347090075406857433515165 248782551887288544089505136025053166599464039027730910730681998567770138405682656892174698726 79438938810456369320820704503611031236439299913255572919740590735774790884561455628327798215 327618929908751956980499118520961193351008935919737878390798618787812496368814218079993636812 22139634570433128348841536457848682138299105915069441398517737045458517679305538509541997950 445260288655137687678395681515059813781400670999366635852409920070573669100745638759784698167 43732891334162701869450127949851626286719284114168260725284962701275007900119658000094313497 721937107804276241576055324212449658505290169119707163077542459671072638363696617118118824586 96504354129045658933193395783900414480489339436162248407882942065544172511870548949875745781 966738934932466277170933154015831483713498115750938338529769500782540147543546329583657642056 67510923053464955369467546416901547453939438907390913081388823478262173968542494042648022264 614846454607396971957220124936050314768482661524649070675611851841004939178743673442200656472 17377614751689278587823886421764971934822007574743174626961650970999452711355105607959591367 445303555307754092251479154530071045506337036323247202188984428177831904741068750900039747064 21606589406121307434741157938380045234423973877695081283600715957942491228400837853328851365 846586514806969164041476690235630524360668970060204609218692320152480014330685798591962811152 93726577141003684045550324673501144841774406094209353130943619151888993302001825740558715498 501172383721262012915037545018116015260457224650485363632512677200119958150651898806647334816 08685293587117379213599474012831601978552671513133378744115580007001357658538244619476129282 9790206513207251070100439100736934573441e-1093

cheers, Jamie