NationMaster - Encyclopedia: Silver ratio

The silver ratio is a mathematical constant. Its name is an allusion to the golden ratio. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. ... The golden section is a line segment sectioned into two according to the golden ratio. ...

1 Definition
- 1.1 Definition as 1 plus the square root of 2
- 1.2 Definition as 2, 2, 2...
2 Properties
3 Silver means
4 Silver rectangles
5 References

Definition

Definition as 1 plus the square root of 2

The silver ratio ( $δ S$ ) is defined as the irrational number formed from the sum of 1 and the square root of 2. That is: In mathematics, an irrational number is any real number that is not a rational number, i. ... A number is an abstract entity that represents a count or measurement. ... Look up one in Wiktionary, the free dictionary. ... The square root of 2 is equal to the length of the hypotenuse of a right triangle with legs of length 1. ...

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

It follows from this definition that

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

Definition as 2, 2, 2...

The silver ratio can also be defined by the simple continued fraction [2, 2, 2, 2, ...]: In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ...

$delta_S = 2 + frac{1}{2 + frac{1}{2 + frac{1}{2 + cdots}}}, .$

Properties

In diophantine approximation, the sequence of fractional parts of In number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of real numbers by rational numbers. ... In mathematics, the floor function is the function defined as follows: for a real number x, floor(x) is the largest integer less than or equal to x. ...

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. In mathematics, a sequence { an : n = 1, 2, 3, ... } is equidistributed modulo 1 precisely if for every interval (a, b) within the larger interval [0, 1), In other words, the long-run proportion of fractional parts of an that fall within any subinterval is just the length of the subinterval. ... In mathematics, the phrase almost all has a number of specialised uses. ...

The lower powers of the silver ratio are

$! delta_S^0 = 1$

$delta_S^1 = delta_S + 0$

$delta_S^2 = 2delta_S + 1$

$delta_S^3 = 5delta_S + 2$

$delta_S^4 = 12delta_S + 5$

The powers continue in the pattern

$! delta_S^n = K_ndelta_S + K_{(n-1)}$

where

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

For example, using this property:

$! delta_S^5 = 29delta_S + 12$

Using $! K_0 = 1$ and $! K_1 = 2$ as initial conditions, a Binet-like formula results from solving the recurrence relation... Alfred Binet (July 11, 1857 – October 18, 1911), French psychologist and inventor of the first usable intelligence test, the basis of todays IQ test. ...

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

which becomes...

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

Silver means

The more general expressions $[n,n,n,dots]=frac{1}{2}left(n+sqrt{n^2+4}right)$ are known as the silver means. The golden ratio is the silver mean for $n = 1$ , while the silver ratio is the silver mean for $n = 2$ . The values of the first ten silver means are^[1]:

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 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:

$! {{S_{m}^n - int(S_{m}^n)} over S_{m}^{-n}} = S_{m}^n - 1$

Also,

$! S_{m}^3 = S_{(m^3 + 3m)}$

$! S_{m}^5 = S_{(m^5 + 5m^3 + 5m)}$

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. Using this property, silver means can be defined as all numbers a that satisfy

$! x equiv x^{-1} pmod 1$

If we break down the silver mean S of m so that

$! S_{m} = a + b$

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:

$! R - int(R) = c/R$

if c is real

and

$! left({1 over R}right)c =$ R's complex conjugate

if c is a multiple of i

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 A4 paper size defined by ISO 216. In geometry, a rectangle is defined as a quadrilateral polygon in which all four angles are right angles. ... The large rectangle BA is a golden rectangle. ... A comparison of different paper sizes A4 is a standard paper size, defined by the international standard ISO 216 as 210Ã—297 mm (roughly 8. ... ISO 216 specifies international standard (ISO) paper sizes, used in most countries in the world today. ...

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.

References

Weisstein, Eric W., Silver Ratio at MathWorld.
Explanation of Silver Means

Categories: Algebraic numbers | Irrational numbers | Mathematical constants MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ...

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