Exploring Metallic Ratios

Mathematics and Statistics 8(4): 388-391, 2020 http://www.hrpub.org DOI: 10.13189/ms.2020.080403 Exploring Metallic Ratios R. Sivaraman Department of Mathematics, Dwaraka Doss Goverdhan Doss Vaishnav College, Chennai – 600 106, Tamil Nadu, India Received April 20, 2020; Revised May 9, 2020; Accepted June 23, 2020 Copyright©2020 by authors, all rights reserved. Authors agree that this article remains permanently open access under the terms of the Creative Commons Attribution License 4.0 International License Abstract Huge amount of literature has been written produces a new pair after one month, assuming that none of and published about Golden Ratio, but not many had heard the rabbit pairs die, how many rabbit pairs would be there about its generalized version called Metallic Ratios, which at the end of one year? are introduced in this paper. The methods of deriving them The answer to this question lies in the sequence 1, 1, 2, 3, were also discussed in detail. This will help to explore 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, . Thus there would further in the search of universe of real numbers. In be 144 pairs of rabbits at the end of one year, as described mathematics, sequences play a vital role in understanding by the sequence. This sequence is called “Fibonacci of the complexities of any given problem which consist of Sequence” named after the proposer though it is known to some patterns. For example, the population growth, earlier mathematicians. This sequence has generated so radioactive decay of a substance, lifetime of an object all much interest in mathematics especially to layman in follow a sequence called “Geometric Progression”. In fact, conveying the beauty of mathematics. In fact, a separate the rate at which the recent novel corona virus (COVID – journal named “Fibonacci Quarterly” is dedicated to study 19) is said to follow a Geometric Progression with common the properties of these fascinating numbers. ratio approximately between 2 and 3. Almost all branches Considering xn to be the number of rabbit pairs th of science use sequences, for instance, genetic engineers generated in n month, the terms of the sequence described use DNA sequence, Electrical Engineers use Morse-Thue above can be written as Sequence and this list goes on and on. Among the vast number of sequences used for scientific investigations, one of the most famous and familiar is the Fibonacci Sequence named after the Italian mathematician Leonard Fibonacci through his book “Liber Abaci” published in 1202. In this We find that, but for the first two terms, beginning from paper, I shall try to introduce sequences resembling the x3, each term is sum of two preceding terms, leading us to Fibonacci sequence and try to generalize it to identify form the recurrence relation xn++21= x nn + xn, ≥→ 1 (1) , general class of numbers called “Metallic Ratios”. where xx12=1, = 1 Keywords Golden Ratio, Silver Ratio, Bronze Ratio, We can solve (1) using the shift operator E defined by r . Metallic Ratios, Shift Operator, Recurrence Relation, E()xk= xk kr+ , ≥≥ 1, r 1 Platinum Ratio, Rhodium Ratio 2 Thus (1) can be written as ()E−− E1xn = 0. The auxiliary equation is: mm2 − −=10. Solving this, we get 15± two real and unequal roots m = . Among these two 2 1. Fibonacci Sequence and Golden roots, the number 15+ is called the “Golden Ratio”. If Ratio 2 we denote this number by φ then the two roots of the Huge amount of work has been devoted to exploring the −φφ properties of Fibonacci sequence and Golden Ratio. So, I auxiliary equation can be expressed as 1 ,. restrict myself to the essential ideas which were useful in Hence the solution to (1) can be expressed as nn proceeding for this paper. We first consider an interesting xn =−+α(1 φ ) βφ for n ≥1. problem posed by Fibonacci in “Liber Abaci” (“The Book From derived above, we can prove that the limit of of Calculations”) xn Starting with one pair of rabbit, assuming that each pair the ratio of successive Fibonacci Numbers tends to the Mathematics and Statistics 8(4): 388-391, 2020 389 15± called the “Bronze Ratio”. Golden Ratio φ = . 2 If we denote this number by µ then the other root will be 3− µ . Hence the solution of (3) can be written as First we note that 15− 1−φφ = =−0.618..., −<− 1 1 < 1. nn 2 xnn =α(3 −+ µ ) βµ , ≥ 1. Here also, it is easy (as proved n →∞ Hence, (1−→φ ) 0 as n . in the case of Golden and Silver Ratios) that the ratio of the If we now consider the ratio of successive terms of the consecutive terms of the sequence described through (3) Fibonacci sequence then we get 3+ 13 tends to the Bronze Ratio µ = . x α(1−+ φ ) nn++11 βφ α(0) + βφ n+1 n+1 = →=φ as 2 nn n We thus have three ratios namely Golden Ratio xn α(1−+ φ ) βφ α(0) + βφ 15± n →∞, proving the claim. φ = , Silver Ratio λ =12 + , Bronze Ratio 2 3+ 13 µ = described through the equations (1), (2) and (3) 2. Generating New Ratios 2 respectively. We also see that the ratio of the successive terms of the corresponding sequences approaches the 2.1. Let us now consider the sequence 1, 1, 3, 7, 17, 41, respective ratios. 99, 239, . We see that 3 = 2(1) + 1, 7 = 2(3) + 1, 17 = 2(7) + 3, 41 = 2(17) + 7, . So, the terms of this sequence form an 3. Generalized Ratio recurrence relation given by x=2 x + xn , ≥→ 1 (2) , n++21 nn We now try to generalize the recurrence relations where xx12=1, = 1 . obtained above to produce more general ratio from which Solving (2) as we did in (1), using shift operator E we get Golden, Silver and Bronze ratios follow as special cases. 2 The more general recurrence relation is defined as (E−− 2E 1) xn = 0 . The auxiliary equation is x= kx + x, n ≥→ 1 (4) where xx=1, = 1 . That is, mm2 −2 −= 10. Solving this, we get two real and unequal n++21 nn 12 after the first two terms which are 1, each term is k times roots given by m =12 ± . Among these two roots, the the previous term and added to last but one term. Forming a number 12+ is called the “Silver Ratio”. If we denote general equation using shift operator we get 2 . The auxiliary equation is 2 . this number by λ then the other root will be 2−λ . Hence (E−−kx E1) n = 0 m− km −=10 the solution of (2) can be written as Solving this we find we get two real and unequal roots nn 2 xnn =α(2 −+ λ ) βλ , ≥ 1. As observed in the case of kk±+4 m = . Among these two roots, we call the Golden Ratio, we can show that the ratio of successive 2 terms of the sequence described through (2) approaches kk++2 4 number ρ = as “Metallic Ratio” of order k. Silver Ratio λ =12 + . k 2 n First we note that −<12 −λ < 1 and so (2−→λ ) 0as The other root is clearly k − ρk . n →∞. x α(2−+ λ )nn++11 β λ α(0) + βλn+1 3.1. Special Cases Thus, n+1 = →=λ as x α(2−+ λ )nn β λ α(0) + βλn n 15+ n →∞. (i) If k =1, then we get ρφ= = , the Golden Ratio. 1 2 + 2.2. We now consider the sequence 1, 1, 4, 13, 43, 142, (ii) If k =2, then we get 28 , the Silver ρλ2 = =+=12 469, 1549, 5116, . 2 Ratio. We see that 4 = 3(1) + 1, 13 = 3(4) + 1, 43 = 3(13) + 4, + (iii) If k =3, then we get 3 13 , the Bronze 142 = 3(43) + 13, . So, the terms of this sequence form an ρµ3 = = recurrence relation given by x=3 x + xn , ≥→ 1 (3) 2 n++21 nn Ratio. where xx12=1, = 1 . 2 Using the shift operator, equation (3) can be written as Thus the Metallic Ratio kk++4 of orders 1, 2 ρk = 2 . The auxiliary equation is 2 − −= . (E−− 3E 1) xn = 0 mm3 10 2 and 3 are precisely the Golden, Silver and Bronze Ratios. Solving this we get two real and unequal roots given by This is in analogue with winning positions of medals in 3± 13 3+ 13 m = . Among these two roots, the number is Olympics and other global sports. 2 2 390 Exploring Metallic Ratios ratios to define more generalized ratios which in turn can We observe that as k is very large, ρk also becomes be used to determine new numbers called Platinum Ratio very large. But we obtain interesting situations, if we and Rhodium Ratio which were not dealt before. In this assume some special values of k. sense, this paper provides a new insight in understanding If = or = then k =1 giving Golden Ratio. k 2 k 3 more general class of “Metallic Ratios”. Moreover these Similarly, if ke= then k =2 giving Silver Ratio and if new numbers will provide an ample opportunity for future research. k = π then k =3 giving Bronze Ratio. Here x is the largest integer ≤ x also called Floor Function of x. 6. Applications 4. Behavior of Metallic Ratio The Golden Ratio finds abundance application in all branches of Engineering and Technology. The number Instead of considering natural number values for k, let us Golden Ratio has been used from Ancient time and it 1 continues to get used in modern times.

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