Silver ratio

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Binary 10.0110101000001001111...
Decimal 2.4142135623730950488...
Hexadecimal 2.6A09E667F3BCC908B2F...
Continued fraction 2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{\ddots}}}}
Algebraic form {1+\sqrt{2}}

In mathematics, two quantities are in the silver ratio if the ratio between the sum of the smaller plus twice the larger of those quantities and the larger one is the same as the ratio between the larger one and the smaller (see below). This defines the silver ratio as an irrational mathematical constant, whose value of one plus the square root of 2 is approximately 2.4142135623. 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 silver ratio is denoted by δS.

Mathematicians have studied the silver ratio since the time of the Greeks (although perhaps without giving a special name until recently) because of its connections to the square root of 2, its covergents, square triangular numbers, Pell numbers, octagons and the like.

The relation described above can be expressed algebraically:

 \frac{2a + b}{a} = \frac{a}{b} \equiv \delta_S\,.

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.

Properties[edit]

Number-theoretic properties[edit]

The silver ratio is a Pisot–Vijayaraghavan number (PV number), as its conjugate 1 − 2 = −1/δS ≈ −0.41 has absolute value less than 1. In fact it is the second smallest quadratic PV number after the golden ratio. This means the distance from δn
S
to the nearest integer is 1/δn
S
≈ 0.41n
. Thus, the sequence of fractional parts of δn
S
, n = 1, 2, 3, ... (taken as elements of the torus) converges. In particular, this sequence is not equidistributed mod 1.

Powers[edit]

The lower powers of the silver ratio are

 \!\ \delta_S^0 = [1] = 1
 \delta_S^1 = \delta_S + 0 = [2;2,2,2,2,2,\dots] \approx 2.41421
 \delta_S^2 = 2\delta_S + 1 = [5;1,4,1,4,1,\dots] \approx 5.82842
 \delta_S^3 = 5\delta_S + 2 = [14;14,14,14,\dots] \approx 14.07107
 \delta_S^4 = 12\delta_S + 5 = [33;1,32,1,32,\dots] \approx 33.97056

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 = [82;82,82,82,\dots] \approx 82.01219

Using K0 = 1 and K1 = 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})}

Trigonometric properties[edit]

The silver ratio is intimately connected to trigonometric ratios for π/8 = 22.5°.

\textstyle \sin \tfrac18\pi = \cos \tfrac38\pi = \frac{1}{2} \sqrt{2 - \sqrt 2}=\sqrt{ \tfrac12\delta_s^{-1}}
\textstyle \cos \tfrac18\pi = \sin \tfrac38\pi = \frac{1}{2} \sqrt{2 + \sqrt{2}}=\sqrt{ \tfrac12\delta_s}
\textstyle \tan \tfrac18\pi = \cot \tfrac38\pi = \sqrt{2}-1= \delta_s^{-1}
\textstyle \cot \tfrac18\pi = \tan \tfrac38\pi = \sqrt{2}+1=\delta_s

So the area of a regular octagon with side length a is given by

A = \textstyle 2a^2 \cot \tfrac18\pi = 2(1+\sqrt{2})a^2 \simeq 4.828427 a^2.

Silver means[edit]

Silver means
0: 0 + 4/2 1
1: 1 + 5/2 1.618033989
2: 2 + 8/2 2.414213562
3: 3 + 13/2 3.302775638
4: 4 + 20/2 4.236067978
5: 5 + 29/2 5.192582404
6: 6 + 40/2 6.162277660
7: 7 + 53/2 7.140054945
8: 8 + 68/2 8.123105626
9: 9 + 85/2 9.109772229
  ⋮
n: n + 4 + n2/2

The more general simple continued fraction expressions


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

are known as the silver means or metallic means[1] 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 term "bronze ratio", or terms using other names of metals, are occasionally used to name subsequent silver means. The values of the first ten silver means are shown at right.[2] Notice that each silver mean is a root of the simple quadratic equation

 x^2 - nx = 1,\, where n is any positive natural number.

Properties[edit]

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 K0 = 1 and K1 = 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}  = 1 - S_m^{-n}.

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)}.

In general:

 \!\ S_m^{2n+1} = S_{\sum_{k=0}^n {{2n+1} \over {2k+1}} {{n+k} \choose {2k}} m^{2k+1}}.

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.

The silver mean of m is also given by the integral

 S_m = \int_0^m {\left( {x \over {2\sqrt{x^2+4}}} + {{m+2} \over {2m}} \right)} \, dx.

Paper sizes and silver rectangles[edit]

A size illustration2.svg

The paper sizes under ISO 216 are rectangles in the proportion 1:2 (approximately 1:1.4142135 decimal), sometimes called "A4 rectangles". Removing a largest possible square from a sheet of such paper leaves a rectangle with proportions 1:2−1 which is the same as 1+2:1, the silver ratio. Removing a largest square from one of these sheets leaves one again with aspect ratio 1:2. 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 the paper sizes specified by ISO 216.

Octagon1 svg.png

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 1+2.[3]

However, only the Lichtenberg ratio rectangles (rectangles with the shape of ISO 216 paper) have the property that by cutting the rectangle in half across its long side produces two smaller rectangles of the same aspect ratio.

The silver rectangle is connected to the regular octagon. If a regular octagon is partitioned into two isosceles trapezoids and a rectangle, then the rectangle is a silver rectangle with an aspect ratio of 1:δS, and the 4 sides of the trapezoids are in a ratio of 1:1:1:δS. If the edge length of a regular octagon is t, then the inradius of the octagon (the distance between opposite sides) is δSt, and the area of the octagon is 2δSt2.[3]

References[edit]

External links[edit]