Silver ratio

This article is about a constant also termed the silver number. For a different constant sometimes called the silver number, see plastic number.

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. 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.

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 silver ratio is here denoted by the $\delta _{S}$. The relation described above can be expressed algebraically:

${\frac {2a+b}{a}}={\frac {a}{b}}=\delta _{S}\,.$

This equation has as its unique positive solution the algebraic irrational number

$\delta _{S}={1+{\sqrt {2}}}=2.41421\,35623\ldots \,$

Definition

The silver ratio ($\delta _{S}$) 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

$(\delta _{S}-1)^{2}=2\,.$

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

Number-theoretic properties

The silver ratio is a Pisot–Vijayaraghavan number (PV number), as its conjugate 1 − √2 $=-{\frac {1}{\delta _{s}}}$ ≈ −0.41 has absolute value less than 1. This means the distance from $\delta _{s}^{n}$ to the nearest integer is ${\frac {1}{\delta _{s}^{n}}}\approx 0.41^{n}$. Thus, the sequence of fractional parts of $\delta _{s}^{n}$, n = 1, 2, 3, ... (taken as elements of the torus) converges. In particular, this sequence is not equidistributed mod 1.

Powers

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

$\!\ K_{n}=2K_{n-1}+K_{n-2}$

For example, using this property:

$\!\ \delta _{S}^{5}=29\delta _{S}+12=[82;82,82,82,\dots ]\approx 82.01219$

Using $\!\ K_{0}=1$ and $\!\ K_{1}=2$ as initial conditions, a Binet-like formula results from solving the recurrence relation

$\!\ K_{n}=2K_{n-1}+K_{n-2}$

which becomes

$\!\ K_{n}={\frac {1}{2{\sqrt {2}}}}{(\delta _{S}^{n+1}-{(2-\delta _{S})}^{n+1})}$

Trigonometric properties

See Exact trigonometric constants

The silver ratio is intimately connected to trigonometric ratios for $\textstyle {\frac {\pi }{8}}=22.5^{\circ }$.

$\textstyle \sin {\frac {\pi }{8}}=\cos {\frac {3\pi }{8}}={\frac {1}{2}}{\sqrt {2-{\sqrt {2}}}}={\sqrt {\frac {1}{2\delta _{s}}}}$

$\textstyle \cos {\frac {\pi }{8}}=\sin {\frac {3\pi }{8}}={\frac {1}{2}}{\sqrt {2+{\sqrt {2}}}}={\sqrt {\frac {\delta _{s}}{2}}}$

$\textstyle \tan {\frac {\pi }{8}}=\cot {\frac {3\pi }{8}}={\sqrt {2}}-1={\frac {1}{\delta _{s}}}$

$\textstyle \cot {\frac {\pi }{8}}=\tan {\frac {3\pi }{8}}={\sqrt {2}}+1=\delta _{s}$

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

$A=\textstyle 2a^{2}\cot {\frac {\pi }{8}}=2(1+{\sqrt {2}})a^{2}\simeq 4.828427a^{2}.$

Silver means

Silver means
0:	½ (0 + √4)	1
1:	½ (1 + √5)	1.618033989
2:	½ (2 + √8)	2.414213562
3:	½ (3 + √13)	3.302775638
4:	½ (4 + √20)	4.236067978
5:	½ (5 + √29)	5.192582404
6:	½ (6 + √40)	6.162277660
7:	½ (7 + √53)	7.140054945
8:	½ (8 + √68)	8.123105626
9:	½ (9 + √85)	9.109772229
...
n:	½ {n + √(n^2 + 4)}

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 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" and other metal names are occasionally coined for subsequent silver means . The values of the first ten silver means are shown at right. 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

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:

$\!\ {S_{m}^{n}-\lfloor S_{m}^{n}\rfloor }=1-S_{m}^{-n}.$

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 S_m = 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

The paper sizes under ISO 216 are rectangles in the proportion 1:√2 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 √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.

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.

However, only the "A4 rectangle", better called the "Lichtenberg Ratio rectangle" has 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 $1:\delta _{S}$, and the 4 sides of the trapezoids are in a ratio of $1:1:1:\delta _{S}$. If the edge length of a regular octagon is $t$, then the inradius of the octagon (the distance between opposite sides) is $\delta _{S}t$, and the area of the octagon is $2\delta _{S}t^{2}$.

External links

• Weisstein, Eric W. "Silver Ratio". MathWorld.
• Explanation of Silver Means