In mathematics, the silver ratio is a geometrical proportion with exact value 1 + √2, the positive solution of the equation x2 = 2x + 1. The name silver ratio is by analogy with the golden ratio, the positive solution of the equation x2 = x + 1. Although its name is recent, the silver ratio (or silver mean) has been studied since ancient times because of its connections to the square root of 2, almost-isosceles Pythagorean triples, square triangular numbers, Pell numbers, the octagon, and six polyhedra with octahedral symmetry.
Definition If the ratio of two quantities a > b > 0 is proportionate to the sum of two and their reciprocal ratio, they are in the silver ratio: a b = 2 a + b a {\displaystyle {\frac {a}{b}}={\frac {2a+b}{a}}}
The ratio a b {\displaystyle {\frac {a}{b}}} is here denoted σ . {\displaystyle \sigma .} Substituting a = σ b {\displaystyle a=\sigma b\,} in the second fraction,
σ = b ( 2 σ + 1 ) σ b . {\displaystyle \sigma ={\frac {b(2\sigma +1)}{\sigma b}}.} It follows that the silver ratio is the positive solution of quadratic equation σ 2 − 2 σ − 1 = 0. {\displaystyle \sigma ^{2}-2\sigma -1=0.} The quadratic formula gives the two solutions 1 ± 2 , {\displaystyle 1\pm {\sqrt {2}},} the decimal expansion of the positive root begins with 2.414213562373095... (sequence A014176 in the OEIS). Using the tangent function
σ = tan ( 3 π 8 ) = cot ( π 8 ) , {\displaystyle \sigma =\tan \left({\frac {3\pi }{8}}\right)=\cot \left({\frac {\pi }{8}}\right),}
or the hyperbolic sine σ = exp ( arsinh ( 1 ) ) . {\displaystyle \sigma =\exp(\operatorname {arsinh} (1)).}
σ {\displaystyle \sigma } and its algebraic conjugate can be written as sums of eighth roots of unity:
with ω = exp ( 2 π i / 8 ) = i , σ = ω − ω 4 + ω − 1 − σ − 1 = ω 3 − ω 4 + ω − 3 , {\displaystyle {\begin{aligned}{\text{with }}\omega =&\ \exp(2\pi i/8)={\sqrt {i}},\\\sigma &=\omega -\omega ^{4}+\omega ^{-1}\\-\sigma ^{-1}&=\omega ^{3}-\omega ^{4}+\omega ^{-3},\end{aligned}}}
which is guaranteed by the Kronecker–Weber theorem. σ {\displaystyle \sigma } is the superstable fixed point of the Newton iteration x ← 1 2 ( x 2 + 1 ) / ( x − 1 ) , with x 0 ∈ [ 2 , 3 ] {\displaystyle x\gets {\tfrac {1}{2}}(x^{2}+1)/(x-1),{\text{ with }}x_{0}\in [2,3]}
The iteration x ← 1 + 2 x / {\displaystyle x\gets {\sqrt {1+2x{\vphantom {/}}}}} results in the continued radical σ = 1 + 2 1 + 2 1 + ⋯ {\displaystyle \sigma ={\sqrt {1+2{\sqrt {1+2{\sqrt {1+\cdots }}}}}}}
Properties
The defining equation can be written
1 = 1 σ − 1 + 1 σ + 1 = 2 σ + 1 + 1 σ . {\displaystyle {\begin{aligned}1&={\frac {1}{\sigma -1}}+{\frac {1}{\sigma +1}}\\&={\frac {2}{\sigma +1}}+{\frac {1}{\sigma }}.\end{aligned}}}
The silver ratio can be expressed in terms of itself as fractions
σ = 1 σ − 2 σ 2 = σ − 1 σ − 2 + σ + 1 σ − 1 . {\displaystyle {\begin{aligned}\sigma &={\frac {1}{\sigma -2}}\\\sigma ^{2}&={\frac {\sigma -1}{\sigma -2}}+{\frac {\sigma +1}{\sigma -1}}.\end{aligned}}}
Similarly as the infinite geometric series
σ = 2 ∑ n = 0 ∞ σ − 2 n σ 2 = − 1 + 2 ∑ n = 0 ∞ ( σ − 1 ) − n . {\displaystyle {\begin{aligned}\sigma &=2\sum _{n=0}^{\infty }\sigma ^{-2n}\\\sigma ^{2}&=-1+2\sum _{n=0}^{\infty }(\sigma -1)^{-n}.\end{aligned}}}
For every integer n {\displaystyle n} one has
σ n = 2 σ n − 1 + σ n − 2 = σ n − 1 + 3 σ n − 2 + σ n − 3 = 2 σ n − 1 + 2 σ n − 3 + σ n − 4 {\displaystyle {\begin{aligned}\sigma ^{n}&=2\sigma ^{n-1}+\sigma ^{n-2}\\&=\sigma ^{n-1}+3\sigma ^{n-2}+\sigma ^{n-3}\\&=2\sigma ^{n-1}+2\sigma ^{n-3}+\sigma ^{n-4}\end{aligned}}}
from this an infinite number of further relations can be found. Continued fraction pattern of a few low powers
σ − 1 = [ 0 ; 2 , 2 , 2 , 2 , . . . ] ≈ 0.4142 ( 17 / 41 ) σ 0 = [ 1 ] σ 1 = [ 2 ; 2 , 2 , 2 , 2 , . . . ] ≈ 2.4142 ( 70 / 29 ) σ 2 = [ 5 ; 1 , 4 , 1 , 4 , . . . ] ≈ 5.8284 ( 5 + 29 / 35 ) σ 3 = [ 14 ; 14 , 14 , 14 , . . . ] ≈ 14.0711 ( 14 + 1 / 14 ) σ 4 = [ 33 ; 1 , 32 , 1 , 32 , . . . ] ≈ 33.9706 ( 33 + 33 / 34 ) σ 5 = [ 82 ; 82 , 82 , 82 , . . . ] ≈ 82.0122 ( 82 + 1 / 82 ) {\displaystyle {\begin{aligned}\sigma ^{-1}&=[0;2,2,2,2,...]\approx 0.4142\;(17/41)\\\sigma ^{0}&=[1]\\\sigma ^{1}&=[2;2,2,2,2,...]\approx 2.4142\;(70/29)\\\sigma ^{2}&=[5;1,4,1,4,...]\approx 5.8284\;(5+29/35)\\\sigma ^{3}&=[14;14,14,14,...]\approx 14.0711\;(14+1/14)\\\sigma ^{4}&=[33;1,32,1,32,...]\approx 33.9706\;(33+33/34)\\\sigma ^{5}&=[82;82,82,82,...]\approx 82.0122\;(82+1/82)\end{aligned}}}
σ − n ≡ ( − 1 ) n − 1 σ n mod 1 . {\displaystyle \sigma ^{-n}\equiv (-1)^{n-1}\sigma ^{n}{\bmod {1}}.}
The silver ratio is a Pisot number, the next quadratic Pisot number after the golden ratio. By definition of these numbers, the absolute value 2 − 1 {\displaystyle {\sqrt {2}}-1} of the algebraic conjugate is smaller than 1, thus powers of σ {\displaystyle \sigma } generate almost integers and the sequence σ n mod 1 {\displaystyle \sigma ^{n}{\bmod {1}}} is dense at the borders of the unit interval.
Quadratic field Q(√2)
σ {\displaystyle \sigma } is the fundamental unit of real quadratic field K = Q ( 2 ) {\displaystyle K=\mathbb {Q} \left({\sqrt {2}}\right)} with discriminant Δ k = 8. {\displaystyle \Delta _{k}=8.} The integers Z [ σ ] of K {\displaystyle \mathbb {Z} [\sigma ]{\text{ of }}K} are the numbers ξ = a + b σ ( a , b ∈ Z ) , {\displaystyle \xi =a+b\sigma {\text{ }}(a,b\in \mathbb {Z} ),} with conjugate ξ ¯ = ( a + 2 b ) − b σ , {\displaystyle {\overline {\xi }}=(a+2b)-b\sigma ,} norm ξ ξ ¯ = ( a + b ) 2 − 2 b 2 {\displaystyle \xi {\overline {\xi }}=(a+b)^{2}-2b^{2}} and trace ξ + ξ ¯ = 2 ( a + b ) . {\displaystyle \xi +{\overline {\xi }}=2(a+b).}
The first few positive numbers occurring as norm are 1, 2, 4, 7, 8, 9, 14, 16, 17, 18, 23, 25. Arithmetic in the ring O k = Z [ σ ] {\displaystyle O_{k}=\mathbb {Z} [\sigma ]} resembles that of the rational integers, i.e. the elements of Z . {\displaystyle \mathbb {Z} .} Prime factorization is unique up to order and unit factors ± σ ± n ( n = 0 , 1 , 2 , … ) , {\displaystyle \pm \sigma ^{\pm n}(n=0,1,2,\ldots ),} and there is a Euclidean function on the absolute value of the norm. The primes of O k {\displaystyle O_{k}} are of three types:
σ − 1 {\displaystyle \sigma -1} with norm 2 , {\displaystyle 2,} the single rational prime that divides Δk , the factors a + b σ {\displaystyle a+b\sigma } of rational primes p = 8 n ± 1 {\displaystyle p=8n\pm 1} with norm p , {\displaystyle p,} the rational primes p = 8 n ± 3 {\displaystyle p=8n\pm 3} with norm p 2 , {\displaystyle p^{2},} and any one of these numbers multiplied by a unit. The silver ratio can be used as base of a numeral system, here called the sigmary scale. Every real number x in [0,1] can be represented as a convergent series
x = ∑ n = 1 ∞ a n σ n , {\displaystyle x=\sum _{n=1}^{\infty }{\frac {a_{n}}{\sigma ^{n}}},} with weights a n ∈ [ 0 , 1 , 2 ] . {\displaystyle a_{n}\in [0,1,2].}
Sigmary expansions are not unique. Due to the identities
σ n + 1 = 2 σ n + σ n − 1 σ n + 1 + σ n − 1 = 2 σ n + 2 σ n − 1 , {\displaystyle {\begin{aligned}\sigma ^{n+1}&=2\sigma ^{n}+\sigma ^{n-1}\\\sigma ^{n+1}+\sigma ^{n-1}&=2\sigma ^{n}+2\sigma ^{n-1},\end{aligned}}}
digit blocks 21 σ and 22 σ {\displaystyle 21_{\sigma }{\text{ and }}22_{\sigma }} carry to the next power of σ , {\displaystyle \sigma ,} resulting in 100 σ and 101 σ . {\displaystyle 100_{\sigma }{\text{ and }}101_{\sigma }.} The number one has finite and infinite representations 1.0 σ , 0.21 σ {\displaystyle 1.0_{\sigma },0.21_{\sigma }} and 0. 20 ¯ σ , 0.1 2 ¯ σ , {\displaystyle 0.{\overline {20}}_{\sigma },0.1{\overline {2}}_{\sigma },} where the first of each pair is in canonical form. The algebraic number 2 ( 3 σ − 7 ) {\displaystyle 2(3\sigma -7)} can be written 0.101 σ , {\displaystyle 0.101_{\sigma },} or non-canonically as 0.022 σ . {\displaystyle 0.022_{\sigma }.} The decimal number 10 = 111.12 σ , {\displaystyle 10=111.12_{\sigma },} 7 σ + 3 = 1100 σ {\displaystyle 7\sigma +3=1100_{\sigma }\,} and 1 σ − 1 = 0. 1 ¯ σ . {\displaystyle {\tfrac {1}{\sigma -1}}=0.{\overline {1}}_{\sigma }.}
Properties of canonical sigmary expansions, with coefficients a , b , c ∈ Z : {\displaystyle a,b,c\in \mathbb {Z} :}
Every algebraic integer ξ = a + b σ in K {\displaystyle \xi =a+b\sigma {\text{ in }}K} has a finite expansion. Every rational number ρ = a + b σ c in K {\displaystyle \rho ={\tfrac {a+b\sigma }{c}}{\text{ in }}K} has a purely periodic expansion. All numbers that do not lie in K {\displaystyle K} have chaotic expansions.
Remarkably, the same holds mutatis mutandis for all quadratic Pisot numbers that satisfy the general equation x 2 = n x + 1 , {\displaystyle x^{2}=nx+1,} with integer n > 0. It follows by repeated substitution of x = n + 1 x {\displaystyle x=n+{\frac {1}{x}}} that all positive solutions 1 2 ( n + n 2 + 4 / ) {\displaystyle {\tfrac {1}{2}}\left(n+{\sqrt {n^{2}+4{\vphantom {/}}}}\right)} have a purely periodic continued fraction expansion σ n = n + 1 n + 1 n + 1 ⋱ {\displaystyle \sigma _{n}=n+{\cfrac {1}{n+{\cfrac {1}{n+{\cfrac {1}{\ddots }}}}}}}
Vera de Spinadel described the properties of these irrationals and introduced the moniker metallic means. The silver ratio is related to the central Delannoy numbers D n {\displaystyle D_{n}} = 1, 3, 13, 63, 321, 1683, 8989,... that count the number of "king walks" between one pair of opposite corners of a square n × n lattice. The sequence has generating function
1 1 − 6 x + x 2 = ∑ n = 0 ∞ D n x n for | x | < 1 σ 2 , {\displaystyle {\frac {1}{\sqrt {1-6x+x^{2}}}}=\sum _{n=0}^{\infty }D_{n}x^{n}{\text{ for }}\vert x\vert <{\tfrac {1}{\sigma ^{2}}},}
from which are obtained the integral representation
D n = 1 π ∫ σ − 2 σ 2 d t ( t − σ − 2 ) ( σ 2 − t ) t n + 1 {\displaystyle D_{n}={\frac {1}{\pi }}\int _{\sigma ^{-2}}^{\sigma ^{2}}{\frac {\mathrm {d} t}{{\sqrt {(t-\sigma ^{-2})(\sigma ^{2}-t)}}\;t^{n+1}}}}
and asymptotic formula
D n ∼ σ 2 n + 1 2 π ( σ − 1 ) n ( 1 − 11 − 3 σ 32 n + 221 − 36 σ 2 ( 32 n ) 2 + O ( n − 3 ) ) . {\displaystyle D_{n}\sim {\frac {\sigma ^{2n+1}}{2{\sqrt {\pi (\sigma -1)\,n}}}}\left(1-{\frac {11-3\sigma }{32\,n}}+{\frac {221-36\sigma ^{2}}{(32\,n)^{2}}}+{\mathcal {O}}{\bigl (}n^{-3}{\bigr )}\right).}
For an application of the sigmary scale, consider the problem of writing a possible third-order coefficient c in terms of the silver ratio. The decimal value of c is approximately 0.006865233, which can be found with the method of dominant balance using the recurrence relation for the central Delannoy numbers, n D n = ( 6 n − 3 ) D n − 1 − ( n − 1 ) D n − 2 , {\displaystyle n\,D_{n}=(6n-3)D_{n-1}-(n-1)D_{n-2},} with D − 1 = D 0 = 1 , n m a x = 10 5 . {\displaystyle D_{-1}=D_{0}=1,n_{max}=10^{5}.} "The coefficients all lie in Q ( 2 ) {\displaystyle \mathbb {Q} \left({\sqrt {2}}\right)} and h