Silver Ratio and Pell Numbers

Home , Cube (algebra), Lucas number, Pell number, Ratio, Silver ratio

JASC: Journal of Applied Science and Computations ISSN NO: 0076-5131

Dr. Vandana N. Purav

P.D. Karkhanis College of Arts and Commerce, Ambarnath

Abstract

Like Fibonacci and Lucas Numbers, Pell and Pell-Lucas numbers are mathematical twins. From the Family of Fibonacci Numbers, Lucas Numbers were invented. This family give rise to Pell and Pell- Lucas Number.

Pell and Pell-Lucas numbers have properties similar to Fibonacci and Lucas Numbers. In this note we try to focus on some of these properties of Pell Numbers related with Silver Ratio

Key words : Silver Ratio , Pell Numbers, Pell-Lucas Numbers

Introduction

Pell numbers arise historically and most notably in the rational approximation of √2 . A sequence of Pell numbers starts with 0 and 1 and then each Pell number is the sum of twice the previous number and the

Pell number before . We denote this Pell Numbers by Pn , recursively it can be written as, Pn = 2Pn-1 + Pn-

2 , n≥ 3 and P1 = 1 and P2 = 2 are the initial conditions. The Pell sequence is 1, 2, 5, 12, 29, 70, 169,…..

If we change the initial conditions as P1 = 1 and P2 = 3 we get Pell-Lucas sequence, we denote it by Qn

and defined recursively as Qn = 2Qn-1 + Qn-2 , n≥ 3 and Q1 = 1 and Q2 = 3, The Pell-Lucas sequence is 1, 3, 7, 17, 41, 99,……

Thus like Fibonacci and Lucas Numbers, Pell and Pell-Lucas Numbers are mathematical twins. We characterize some of the properties.

1) Silver Ratio : The convergence of Pell sequence 1,2,5,12,29,70,169,…..gives rise to Silver Ratio i.e. as n → ∞ , γ = lim which is called Silver ratio. →

2) Binet Like Formulas for Pell Numbers : Let γ = 1+ √2 and δ = 1- √2 , then P = - and Q n n = γn + δn where γ = 1 + √2= 2.4142136... and δ = 1 - √2= - 0.4142136 are solutions of quadratic equations x2 - 2x -1 = 0

γ + δ = 2 , γ - δ = 2 √2 , γδ = -1 or γ = - 1/δ

3) Powers of Silver ratio γ : Silver ratio γ = γ = 1+ √2 = 0+ 1.γ

γ2 = (1+ √2 )2 = 1+2√2 +2 = 1+2(1+√2) = 1+2γ

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γ3= (1+2γ)γ= γ+ 2γ2= γ +2( 1+2γ) = 2+5γ

……

n γ = Pn-1 + Pn γ

4) Powers of δ

δ1 = δ +0 = [ 2; 2,2,2 …..]

δ2 = 2δ +1 = [5; 1, 4,1, 4,1, 4,…..]

δ3 =5δ +2 = [14; 14,14,14,….]

δ4 =12δ+5 = [31;1,32,1,32,….]

n δ = Qnδ + Qn-1

5) Algebraic formulae: As γ = 1+√2 and δ = 1-√2 then we have the following formulae

i) γ + δ = 2 and γ - δ = 2√2 and γδ=-1

ii)γ + 1 = 1+√2 + 1= 2+√2 = √2(1+√2) = √2.γ

iii)δ +1 = 1-√2 +1 = 2-√2 = -√2(1-√2) = -√2.δ

iv) (γ + 1)( δ +1) = -2γδ= -2 (-1) = 2

v)γ2 + δ2 = ( 1+ √2)2 + ( 1- √2)2 = 6

vi) γ2 - δ2 = ( 1+ √2)2 - ( 1- √2)2 = 4√2

vii) (γ2 + δ2)( γ2 - δ2) = 24√2

3 2 2 viii) γ = γ .γ = (1+√2) .(1+√2) = 7+5√2= 5+5√2+2=5(1+√2)+2 =5γ+2 =P3γ + P2

3 ix) δ = 7-5√2 = 5(1-√2)+2 = 5δ+2 = Q3δ +Q2

x) γn.δn=( γ.δ)n = (-1)n = ±1

6) Continued Fraction : Infinite continued fraction 2 + converges to a limit γ ⋯….

i.e.[ 2; 2,2,2, 2, …..] converges to Silver Ratio γ

Pell Number P can also be written as P = [ 1 - ( )n ] n n

Since | δ | < | γ | , ( )n → ∞ as n→∞

Therefore when n→∞ , P ≈ n √

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log P = n log γ - log 8 n

Number of digits in P = 1 + characteristic log P = [ log P ] = [ n log ( 1 + √2 ) - log 8 ] n n n

7) Expressing the Approximation as a Continued Fraction : We may approximate √2 with

- 1 = = = 1 + = 1 + = 1 +

= 1 + = 1 +

8) Generating the Silver ratio by Newton Raphson Method :

Let f(x)= x2 -2x-1, then f'" '(x) = 2x-2

' By Newton's Formula xn+1 = xn - f(xn) / f (xn )

Taking x1 = 2 , we get x2

() x = x - = = 2 1 ()

x = x - () = = 3 2 ()

x = x - () = = 4 3 ()

() x = x - = = 5 4 ()

n n In general , we xn+1 = P2 + 1 / P2

9) Silver ratio and Geometry : The Silver ratio is connected to trigonometric ratios for = = 22.50

( ) Sin = =

( ) Cos =

Tan = √2 - 1

Cot = √2 + 1

So the area of Octagon with length a = 2a2 Cot = 2 ( 1 + √2) a2

Similarly in a Cub octahedron and Tetrahedron are both based on Silver ratio.

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The Cub octahedron is made by connecting the midpoint of each edge of either the cube or octahedron. These solids are both based on the Silver ratio.

In Tetrahedron the dihedral angle is 70031'43" This is same as the acute angle of the Silver Rhombus with a ratio of the short diagonal to the long diagonal being equal to Silver ratio

10) Silver ratio in a triangle

Consider a triangle whose sides are x, y, 2√xy where x > y

Taking x = a2 , y = b2, then the three sides are a2 , b2 and 2ab, and a2 > b2

The by Triangle inequality , we have ( a+b)2 > 0

2ab + b2 > a2 ⇨ (a/ b )2 - 2(a/b) - 1 < 0

⇨ δ < ( a/ b) < γ

⇨ δ < √x/y < γ

Thus we conclude that the square root of the ratio of two sides of such triangle lies between -1/ γ and γ

11) Pythagorean Triples : Pell Numbers implies Pythagorean Triples . If a and b are the two sides of a right triangle and c is the hypotenuse then a2 + b2 = c2

(4,3 , 5) (20, 21, 29) (120, 119, 169) ….

n= 1 , 2 , 3 ….

Applications : a) 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.

Thus Removing a largest square from one of these sheets leaves once again with aspect ratio 1 : √2

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A rectangle whose aspect ratio is the silver ratio is sometimes called a silver rectangle.

b) 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:γ and the 4 sides of the trapezoids are in a ratio of 1:1:1:γ If the edge length of a regular octagon is t, then the radius of the octagon (the distance between opposite sides) is γt, and the area of the octagon is 2γt2

Silver ratio within the octagon Silver rectangle

c) Consider Characteristic equation t2 = 2t +1 i.e. t(t-2) = 1 , i.e. t = 1/ t-2 This gives rise to interesting Geometric property. Consider t × 1 rectangle , now remove two unit squares from it, then by t = 1/t-2 , the remaining rectangle has the same ratio of length to its width as the original rectangle.

d) Ammann-Beeker Tilings: These are discovered recently ,that are closely related to the Silver Ratio (1+ √2 ) and the Pell Numbers.

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Conclusion:. The convergence of Pell sequence gives rise to Silver Ratio. Silver ratio has many interesting algebraic and geometric properties as well as applications. Geometrically Silver Ratio is connected to octagon .

References :

[1] Some algebraic and geometric properties of the Silver number; A. Primo Ramos and , E. Reyes Iglesias, Universidad Autónoma de

[2] Variations on a Theme of the Silver Ratio; Dann Passoja

[3] The infinite sum of the cubes of reciprocal Pell Numbers; Zhefeng Xu and Tingting Wang*

[4] Generating Silver ratio γ using Different Approaches ; Naresh Patel , C.L.Parihar, International e-Journal of Mathematics and Engineering(2010)903-910

[5] ISO-216 Application

[6]Thomas_Koshy: Pell and Pell_Lucas Numbers

[7] https://en.wikipedia.org/wiki/Silver_ratio

[8] https://en.wikipedia.org/wiki/Metallic_mean

[9]http://mathworld.wolfram.com/SilverRatio.html

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