Silver MeansDate: 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/ |
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