Silver ratio

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The silver ratio is a mathematical constant. Its name is an allusion to the golden ratio; analogously to the way the golden ratio is the limiting ratio of consecutive Fibonacci numbers, the silver ratio is the limiting ratio of consecutive Pell numbers. The name silver number is also sometimes used to refer to the plastic number, the limiting ratio of consecutive Perrin numbers and of the Padovan sequence.

Contents

[edit] Definition

[edit] Definition as 1 plus the square root of 2

The silver ratio (δS) is defined as the irrational number formed from the sum of 1 and the square root of 2. That is:

\delta_S = 1 + \sqrt{2} \approx 2.414\, 213\, 562\, 373\, 095\, 048\, 801\, 688\, 724\, 210

It follows from this definition that

(\delta_S-1)^2=2\, .

[edit] Definition as [2; 2, 2, 2, ...]

The silver ratio can also be defined by the simple continued fraction [2; 2, 2, 2, ...]:

\delta_S = 2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ddots}}}\, .

The convergents of this continued fraction (2/1, 5/2, 12/5, 29/12, 70/29, ...) are ratios of consecutive Pell numbers. These fractions provide accurate rational approximations of the silver ratio, analogous to the approximation of the golden ratio by ratios of consecutive Fibonacci numbers.

[edit] Properties

[edit] Non-equidistribution mod 1

In diophantine approximation, the sequence of fractional parts of

xn, n = 1, 2, 3, ...

is shown to be equidistributed mod 1, for almost all real numbers x > 1. The silver ratio is an exception.

[edit] Powers of the Silver Ratio

The lower powers of the silver ratio are

\!\ \delta_S^0 = 1
\delta_S^1 = \delta_S + 0
\delta_S^2 = 2\delta_S + 1
\delta_S^3 = 5\delta_S + 2
\delta_S^4 = 12\delta_S + 5

The powers continue in the pattern

\!\ \delta_S^n = K_n\delta_S + K_{(n-1)}

where

\!\ K_n = 2 K_{(n-1)} + K_{(n-2)}

For example, using this property:

\!\ \delta_S^5 = 29\delta_S + 12


Using \!\ K_0 = 1 and \!\ K_1 = 2 as initial conditions, a Binet-like formula results from solving the recurrence relation...

\!\ K_n = 2 K_{(n-1)} + K_{(n-2)}

which becomes...

\!\ K_n = \frac{1}{2\sqrt{2}} {(\delta_S^{n+1} - {(2-\delta_S)}^{n+1})}

[edit] Silver means

Silver means
0 0 + √1 1
1 ½ + √1¼ 1.618033989
2 1 + √2 2.414213562
3 1½ + √3¼ 3.302775638
4 2 + √5 4.236067978
5 2½ + √7¼ 5.192582404
6 3 + √10 6.162277660
7 3½ + √13¼ 7.140054945
8 4 + √17 8.123105626
9 4½ + √21¼ 9.109772229

The more general simple continued fraction expressions

n + \cfrac{1}{n + \cfrac{1}{n + \cfrac{1}{n + \cfrac{1}{\ddots\,}}}} = \frac{1}{2}\left(n+\sqrt{n^2+4}\right)\,

are known as the silver means of the successive natural numbers. The golden ratio is the silver mean between 1 and 2, while the silver ratio is the silver mean between 2 and 3. The values of the first ten silver means are shown at right.[1] Notice that each silver mean is a root of the simple quadratic equation

x^2 - nx = 1,\,

where n is any positive natural number.

[edit] Properties of Silver Means

These properties are valid only for integers m, for nonintegers the properties are similar but slightly different

The above property for the powers of the silver ratio is a consequence of a property of the powers of silver means. For the silver mean S of m, the property can be generalized as

\!\ S_{m}^n = K_{n}S_{m} + K_{(n-1)}

where

\!\ K_n = mK_{(n-1)} + K_{(n-2)}

Using the initial conditions \!\ K_0 = 1 and \!\ K_1 = m, this recurrence relation becomes...

\!\ K_n = \frac{1}{\sqrt{m^2 + 4}} {(S_{m}^{n+1} - {(m-S_{m})}^{n+1})}

The powers of silver means have other interesting properties:

If n is a positive even integer:
\!\ {{S_{m}^n - \lfloor S_{m}^n \rfloor} \over S_{m}^{-n}} = S_{m}^n - 1

Additionally,

\!\ {1 \over {S_{m}^4 - \lfloor S_{m}^4 \rfloor}} + \lfloor S_{m}^4 - 1 \rfloor = S_{(m^4 + 4m^2 + 1)}
\!\ {1 \over {S_{m}^6 - \lfloor S_{m}^6 \rfloor }} + \lfloor S_{m}^6 - 1 \rfloor = S_{(m^6 + 6m^4 + 9m^2 +1)}
Also,
\!\ S_{m}^3 = S_{(m^3 + 3m)}
\!\ S_{m}^5 = S_{(m^5 + 5m^3 + 5m)}
\!\ S_{m}^7 = S_{(m^7 + 7m^5 + 14m^3 + 7m)}
\!\ S_{m}^9 = S_{(m^9 + 9m^7 + 27m^5 + 30m^3 + 9m)}
\!\ S_{m}^{11} = S_{(m^{11} + 11m^9 + 44m^7 + 77m^5 + 55m^3 + 11m)}


The silver mean S of m also has the property that

\!\ 1/S_{m} = S_{m}-m

meaning that the inverse of a silver mean has the same decimal part as the corresponding silver mean.

\!\ S_{m} = a + b

where a is the integer part of S and b is the decimal part of S, then the following property is true:

\!\ S_{m}^2 = a^2 + mb + 1.

Because (for all m greater than 0), the integer part of Sm = m, a=m. For m>1, we then have

\!\ S_{m}^2 = ma + mb + 1
\!\ S_{m}^2 = m(a+b) + 1
\!\ S_{m}^2 = m(S_{m}) + 1

Therefore the silver mean of m is a solution of the equation

\!\ x^2 - mx - 1 = 0

It may also be useful to note that the silver mean S of −m is the inverse of the silver mean S of m

\!\ {1/S_m} = S_{(-m)} = S_m - m.

Another interesting result can be obtained by slightly changing the formula of the silver mean. If we consider a number

\!\ \frac{1}{2}\left(n+\sqrt{n^2+4c}\right) = R

then the following properties are true:

\!\ R - \lfloor R \rfloor = c/R if c is real,
\!\ \left({1 \over R}\right)c = R - \lfloor \operatorname{Re}(R) \rfloor if c is a multiple of i.

[edit] Silver rectangles

A rectangle whose aspect ratio is the silver ratio is sometimes called a silver rectangle by analogy with golden rectangles. Confusingly, "silver rectangle" can also refer to a rectangle in the proportion 1:√2, also known as an "A4 rectangle" in reference to the common A4 paper size defined by ISO 216.

Both kinds of silver rectangle have the property that removing two squares from them yields a smaller similar rectangle ([1]). Indeed, removing the largest possible square from either kind yields a silver rectangle of the other kind, and then repeating the process once more gives a rectangle of the original shape but smaller by a linear factor of √2.

However, only the "A4 rectangle", better called the "Lichtenberg rectangle" has the property that by cutting the rectangle in half across its long side produces two smaller rectangles of the same aspect ratio.

[edit] References

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