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Silver Means

Date: 01/26/2007 at 12:30:47
From: Robert
Subject: number similar to the golden section

I've found this number that is similar to the golden section.  The
number is x = 3.3027756377319946465596...  The number is derived by
the formula x = 1 / (x - 3).

To put it differently, 1 / 3.3027756377319946465596... = 
0.3027756377319946465596...  This is a similar concept to the golden
section.

Now, x^3 = 36.027756377319946465596...  Notice the similarity to x
given at the top, but more interestingly, x^3 = 1 / (x^3 - 36).

To put it differently, 1 / 36.027756377319946465596... = 
0.027756377319946465596...

Why does this happen?  Is x a special number like the golden section?
Is there a simple mathematical explanation?



Date: 01/26/2007 at 14:16:58
From: Doctor Douglas
Subject: Re: number similar to the golden section

Hi, Robert.

Yes, x is a special number defined in an analogous way to the golden
mean.  One way of defining these types of numbers is through their
continued fraction expansion.  The "golden section" (or "golden mean"
or "golden ratio") is

  phi = 1 + 1/(1 + 1/[1 + 1/{1 + ...}])     

It satisfies

  phi = 1 + 1/phi,

an equation similar to the one you are using for x.  The number phi 
can be found by solving this equation (e.g. by using the quadratic 
formula) to get

  phi = [1 + sqrt(5)]/2 = 1.61803398874989... 

In the continued fraction expansion of phi, the sequence of 
"approximating digits" is infinite and consists entirely of 1's. 
We can express this as phi = [1,1,1,...].  

There is another number whose continued fraction expansion consists
of all 2's:

  d = [2,2,2,...]
    = 2 + 1/(2 + 1/[2 + 1/{2 + ...}])

where it is now easier to see what I mean by "approximating digits"--
the ones that you get if you stop at some point:  [2,2,0,0,0,...] =
2 + 1/(2 + 0) = 2+1/2 = 2.5.  The number d has the numerical value

  d = [2 + sqrt(8)]/2 = 1 + sqrt(2) = 2.41421356237309...

This number, d, has a special name too, the "silver mean" or "silver
ratio", by a connection with the next noble metal.

Your number, x, has the continued fraction expansion x = [3,3,3,...]:

   [3,0,...] = 3
   [3,3,0,...] = 3 + 1/3 = 3.33333333...
   [3,3,3,0,...] = 3 + 1/(3 + 1/3) = 3.3
   [3,3,3,3,0,...] = 3.303030...

   x = [3,3,3,...] = [3 + sqrt(13)]/2
     = 3.30277563773199...

and you can verify that the x in the form [3 + sqrt(13)]/2 satisfies
the conditions for your original equation:  x = 1/(x - 3).

So I suppose that you could refer to x as a "bronze mean".  Usually,
however, the numbers whose continued fractional expansions are

  x[j] = [j,j,j,...]

i.e., 

  x[1] = phi = 1.61803398874989... 
  x[2] = d = 2.41421356237309...
  x[3] = x = 3.30277563773199...
  x[4] = 4.23606797749978...
  ...

are referred to as the "silver means":

  Ron Knott "The Silver Means"
    http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/cfINTRO.html#silver 

  Eric W. Weisstein's MathWorld "Silver Ratio"
    http://mathworld.wolfram.com/SilverRatio.html 

I hope that this helps.

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Fibonacci Sequence/Golden Ratio

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