Balancing on the edge, the golden ratio, the Fibonacci sequence and their generalization
BBalancing on the edge, the golden ratio, the Fibonacci sequenceand their generalization
Gautam Dutta , , Mitaxi Mehta , Praveen Pathak Dhirubhai Ambani Institute of Information and Communication Technology,Gandhinagar 382007, India. SEAS, Ahmedabad University, Navrangpura, Ahmedabad 380009, India Homi Bhabha Center for Science Education, Tata Institute of Fundamental Research,Mankhurd, Mumbai, 400088, India E-mail: gautam [email protected]
Abstract
The golden ratio and Fibonacci numbers are found to occur in various aspects ofnature. We discuss the occurrence of this ratio in an interesting physical problemconcerning center of masses in two dimensions. The result is shown to be independentof the particular shape of the object. The approach taken extends naturally to higherdimensions. This leads to ratios similar to the golden ratio and generalization of theFibonacci sequence. The hierarchy of these ratios with dimension and the limit as thedimension tends to infinity is discussed using the physical problem.
In a basic physics course students handle a number of interesting problems on center of massof laminar bodies. In [1] the authors discuss an interesting connection of center of mass(c.m) of uniform laminar bodies to a very interesting irrational number, namely, the goldenratio. When a circular disk of diameter d (cid:48) is removed from a bigger circular disk of diameter d , touching the bigger circle at a single point O , it leaves a crescent like structure as shownin Fig. 1. If we demand the c.m of the remaining portion of the disc to be at P , a pointexactly at the inner edge of the crescent, then the ratio of the diameter of the bigger to thesmaller circle is the golden ratio, i.e., d ø d (cid:48) = 1 + √ a r X i v : . [ m a t h . G M ] M a r d (cid:48) dO Figure 1: A circular cavity of diameter d (cid:48) is cut out from a circular disc of diameter d . Theinner circle is tangential to the outer circle at the point O . The c.m of the crescent likeremaining portion of the disc is at P .The golden ratio [3] is the positive root of the quadratic polynomial P ( x ) = x − x − ϕ , mysteriously crops up in various natural phenomena[4], geometrical constructions [5] and physical problems [6]. One of the most common geo-metrical description of the golden ratio is the ratio of length l to the breadth b of a rectanglesuch that we are left with a rectangle with the same ratio of length to breadth after a squareof side b is removed from it, i.e, b ø l − b = l ø b . This ratio is also the asymptotic ratio of the n th and the ( n − th term of a Fibonaccci sequence. A Fibonacci sequence is a sequenceof numbers such that every term since the third term is a sum of the previous two terms.The first two terms are arbitrary [7]. Usually one considers a Fibonacci sequence of positiveintegers, for e.g. { , , , , , , , , , .... } . The numbers in a Fibonacci sequence arecalled Fibonacci numbers. These numbers are found to occur in several places in naturesuch as the branching of stems in trees [8] and the spiral arrangement of buds in sunfloweror the scales on pineapple [9].We extend this analysis to three and higher dimensions and obtain higher degree analoguesof the polynomial P ( x ). A connection is made with a kind of generalization of the Fibonaccisequence and that of the golden ratio. Let S be the boundary of the given object and S (cid:48) be the boundary of the cavity. Theyare closed curves and they touch each other at O . S and S (cid:48) are similar, with S (cid:48) beingthe scaled down version of S . The common point O is the center of scaling. C is the c.mof the complete object. Every point on the outer curve S has a corresponding point onthe inner curve S (cid:48) . Let OQ be a chord of the closed curve S that passes through the c.m C . P is a point corresponding to Q on S (cid:48) . Let C (cid:48) be the c.m of the region removed by thecavity. Due to similarity and identical orientation of S and S (cid:48) , C (cid:48) lies on the same chord OQ .Let OQ = d and OP = d (cid:48) . We will call d and d (cid:48) as diameter. We make the following2 P QCC' SS'
Figure 2: A laminar object bounded by the curve S . C is the c.m of the object. S (cid:48) is theedge of the cavity. S (cid:48) is similar to the curve S and scaled around the common point O . C (cid:48) isthe c.m of the cavity. P is the c.m of the remaining portion of the object lying on the inneredge S (cid:48) . Q is a point corresponding to P on the outer edge S . The points O, C (cid:48) , C, P and Q are collinear.observations: The area of the given planar object can be expressed as αd where α is ageometrical constant depending upon the shape of the object. For e.g in the case of a circle α = π/ OQ is the diameter. For a square α = 1 when OQ is the length of its sidewhile α = 1 / OQ is the diagonal of a square. The area of the cut portion is αd (cid:48) whilethe area of the shaded region is α ( d − d (cid:48) ). Let OC = βd with 0 < β <
1. Since the shapeof the cavity is similar to that of object, OC (cid:48) = βd (cid:48) . Like α, β is also a purely geometricalfactor. The center of mass of the shaded part will lie on the line joining C and C (cid:48) . Since wedemand this center of mass to be on the edge of the cavity, it has to be the point P . Let σ be the planar mass density. Then the equation relating the three c.m is given as σα ( d − d (cid:48) ) d (cid:48) + σαd (cid:48) βd (cid:48) = σαd βd (1)With d = xd (cid:48) this may be rewritten as( x −
1) = β ( x −
1) (2) x = 1 is an obvious solution of the above polynomial equation. This means the cavity isalmost the size of the object. This can be understood only as x → + . In this limit, thepoint P → Q and most of the mass lies between these points, hence near P . This is not veryinteresting. Hence we look for other solutions. Factoring out x − βx + ( β − x + ( β −
1) = 0 (3)This is the most general form in two dimensional case. A few comments are in order. Theroots of the quadratic polynomial in this equation are always real for 0 ≤ β ≤
1. The tworoots are of opposite signs since the product of the roots is β − β which is negative. Theabsolute value of the positive root is greater than the absolute value of the negative rootsince the sum of the roots is equal to 1 − β ø β , which is positive. These features will beobserved even in higher dimensions.Interestingly Eq. (3) is not dependent on the structure parameter α . It only depends on β .For e.g, for circle, even sided regular polygons and ellipse, the parameter α are different butthe parameter β = 1 /
2. Substituting this in the above equation gives x − x − C is the midpoint of the chord OQ independent ofits orientation. All objects don’t have such a symmetry and this is precisely one of the waywe are generalizing here. If C is not the midpoint of OQ then β = OC ø OQ is either less than1 / /
2, see Fig. 2. If the cavity is excised so that Q is the common pointthen β = QC ø OQ = 1 − β . As we rotate the chord OQ about C , β changes continuouslyfrom β to β , thus taking the value 1 / OQ . So for any object wecan find a chord along which Eq. (4) is satisfied. A case in point is an odd sided polygon(Fig. 3). Here, for the vertex excission, i.e, when the common point between the polygonand the excised cavity is the vertex O in Fig. 3(a), β = OC ø OQ = 1ø1 + cos( π/n ). For thecase of mid point excission, i.e, when the common point is the midpoint Q of a side of thepolygon, β = QC ø OQ = cos( π/n )ø1 + cos( π/n ). These cases are discussed in [1] wherebyit is shown that the ratio d ø d (cid:48) required for balancing at the edge is not the golden ratio. Aswe rotate the chord OQ , C becomes the midpoint of OQ at a particular orientation. Onesuch orientation is shown in Fig. 3(b). The way to make the excission is also shown. P , apoint on the inner edge, is the c.m of the shaded region. The ratio, d ø d (cid:48) = OQ ø OP is thegolden ratio in this case. Note that the excission is not symmetric, unlike the vertex and themidpoint excission, discussed in [1]. OQC rr cos π ø n (a) CO QP (b)Figure 3: An odd sided polygon. (a): For the vertex excision distances are measured fromthe point O . So β = 1ø1 + cos( π/n ). For the midpoint excision distances are measured fromthe point Q . So β = cos( π/n )ø1 + cos( π/n ) . (b): β = 1 /
2. Neither the vertices nor themid points of the common sides of the two polygons coincide. The excission is assymetric. P is the balancing point and OQ ø OP = 1 + √ β = 1 / √ ϕ is the asymptotic limit of the ratio of successive terms of the Fibonacci sequence F n whichis defined as F n = F n − + F n − . So for large n , if F n − = c then F n − = cϕ and F n = cϕ .Since these are the terms of the Fibonacci sequence we have cϕ = cϕ + c i.e. , ϕ − ϕ − − /ϕ is negative and hence not relevantto this problem.The problem, as presented here, makes it easy to generalize beyond two dimensions. Theindependence of the shape and the existence of a chord along which β = 1 / We start with modifying Eq. (1) for a homogeneous object in k dimension. The k dimensionalvolume of the object can be generally written as αd k where α is again a purely geometricfactor depending on the generic shape of the object and d is the length scale in terms of whichthe volume can be determined. Likewise the volume of the cavity which will be removedis given by αd (cid:48) k where d (cid:48) is the corresponding length scale of the cavity. OC = βd and OC (cid:48) = βd (cid:48) where 0 < β < P , a point on the inner edge ofthe cavity then we get the following k dimensional equivalent of Eq. (1) ρα ( d k − d (cid:48) k ) d (cid:48) + ραd (cid:48) k βd (cid:48) = ραd k βd (6)where ρ is the uniform k dimensional volume density of the material. Writing d = xd (cid:48) , Eq.(6) reduces to x k − β = βx k +1 (7)which can be rearranged as β ( x k +1 −
1) = x k − x = 1 is a solution of the above equation which as discussed earlier is not of interest to us.So we divide both sides by x −
1. We note that 1 + x + .... + x k − = x k − x −
1. Usingthis result on both sides we get βx k + ( β − x k − + ( β − x k − + .... + ( β − x + ( β −
1) = 0 (9)This is the k dimensional equivalent of Eq. (3). Again we take the special case β = 1 / x k − x k − − x k − − .... − x − There are various ways to generalize the Fibonacci sequence. We would be interested ina generalization to a sequence of non-negative integers where the n th term is a sum of k F n = F n − + F n − + ..... + F n − k . This is called the k - generalizedFibonacci sequence and the terms of this sequence are called the k -nacci numbers. So k = 2corresponds to Fibonacci numbers. k = 3 gives the tribonacci numbers and likewise wehave the tetranacci and pentanacci numbers. Asymptotically for large n this sequence alsobehaves like a geometric sequence with common ratio say ζ [10, 11]. So for large n if F n − k = c then F n − k +1 = cζ, F n − k +2 = cζ , ...., F n − = cζ k − and F n = cζ k . Substitutingthese in the generalized Fibonacci terms we get ζ k = ζ k − + ζ k − + ...... + ζ + 1 (11) ζ satisfies Eq. (10). It is the equivalent of the golden ratio in k dimension and called the k -nacci constants.Let us denote the polynomial on the l.h.s of Eq. (10) as P k ( x ) which is a polynomial ofdegree k . Let us look at the kind of roots P k ( x ) can have. P k (0) = − , ∀ k . For sufficientlylarge positive value of x , P k ( x ) >
0. So each of these polynomials have a positive root. Foreven k , P k ( x ) > x . So these polynomials willalso have a negative root which is not of relevance to us. Can P k ( x ) have more than onepositive root? In that case which of the positive roots will be the k -nacci constants? In factwe can show that it has exactly one positive root. Let a be a positive root of P k ( x ). Then a k = 1+ a + a + .... + a k − . It can be shown that x k grows faster than 1+ x + x + .... + x k − for x > a . Hence there will be no other positive roots of P k ( x ). Hence for each k we get exactlyone positive root, which are physically relevant to us. They are k dimensional equivalent ofthe golden ratio, called the k -nacci constants. We denote them as ϕ k . We will see that ϕ k is an irrational number between 1 and 2 for all k . The coefficients of P k ( x ) are all integersand the leading coefficient (the coefficient of x k ) is 1. Such polynomials are called integermonic. They can be factored into integer monic [12]. So the real roots of P k ( x ) are eitherintegers or irrational. When k = 1 , P ( x ) = x −
1. So ϕ = 1. It can be shown algebraicallythat the positive roots of P k ( x ) lie between 1 and 2 but in the spirit of this article we give astrightforward justification using the generalized fibonacci sequence. For k > F n > F n − in a k -generalized Fibonacci sequence of positive numbers. Also F n < F n − .This can be proved as follows: F n − = F n − + F n − + .... + F n − k + F n − ( k +1) F n = F n − + ( F n − + .... + F n − k ) < F n − + F n − = 2 F n − (12)So 1 < F n ø F n − < k >
1. This implies ϕ can not be an integer for k >
1. Hencethey have to be an irrational number between 1 and 2. So we conclude that each of the ϕ k for k > ϕ = 1. ϕ = 1 + √ ≈ . ϕ ≈ . ϕ ≈ . ϕ k → k → ∞ . This limit is very interesting and simpleto understand. Let us take an example of such a generalized Fibonacci sequence (cid:104) , , , , , , , , , ......... (cid:105) (13)In this sequence we can start with as many 0 as we like initially. Subsequent to the first non-zero entry, every term is the sum of all the previous terms. Asymptotically, this sequence6ehaves as the geometric sequence 2 n . Hence the asymptotic ratio of successive terms is 2.The generalized golden ratios have a hierarchy, ϕ < ϕ < ϕ < ..... <
2. This hierarchycan be understood algebraically but since these ratios occur in a physical problem, it isinteresting to have a physical understanding for this. The mass of the object is proportionalto d k and that of the cavity goes as d (cid:48) k in k dimension. The removal of the cavity shifts thec.m from C to P (See Fig. (2)). The fraction of the cavity mass increasingly diminishes incomparison to the total mass with rising k since d (cid:48) < d . This would mean that the shift C to P decreases as k increases. As d (cid:48) decreases with k , ϕ k = d ø d (cid:48) increases with k . Thisexplains the hierarchy of ϕ k with k . As k → ∞ , the mass of the cavity becomes negligiblecompared to the bulk. Hence P coincides with C . In this case d ø d (cid:48) ≈ OQ ø OC = 1ø β . If β = 1ø2 , d ø d (cid:48) ≈
2. So ϕ k → k → ∞ . The physical problem of balancing a laminar object on its inner edge after suitably excisinga self similar cavity within it, is found to be related to the golden ratio. We find that thisphenomenon is independent of the shape of the object thus emphasizing the role of thisimportant irrational number in various aspects of nature. The extension of this to a higherdimension k gives a connection of this physical problem with the generalized Fibonaccisequence where every term is a sum of k preceding terms. Due to this connection we give aphysical explanation for the hierarchy of the generalized golden ratios ϕ k with k . ACKNOWLEDGEMENTS
This work is supported by the Physics olympiad programme, HBCSE-TIFR. We thank Prof.Vijay A Singh for discussions and inspiring us with critical questions during this work.
References [1] Praveen Pathak, Vijay A Singh, Yet another encounter with the golden ratio: balancinglaminar bodies on the edge, Eur. J. Phys. (2016) 055001[2] Gautam Dutta, Mitaxi Mehta, Comment on ‘Yet another encounter with the goldenratio: balancing laminar bodies on the edge’, Eur. J. Phys. (2016) 068002[3] R.A. Dunlap 1997 The Golden Ratio and Fibonacci Numbers (World Scientific)[4] Livio Mario 2002
The Golden Ratio The story of Phi, the Worlds Most AstonishingNumber (Broadway Books, New York).[5] P.J. Steinhardt and S. Ostlund,
Physics of Quasicrystals (World Scientific, Singapore,1987)[6] T. P. Srinivasan, “ Fibonacci sequence, golden ratio, and a network of resistors,”
Am. J.Phys. , , 461 (1992). 77] Ball, Keith M 2003, “ Fibonacci Rabbits Revisited ”, Strange curves, Counting Rabbitsand other Mathematical Explorations, Princeton, N.J: Princeton Univ. Press, ISBN 0-691-11321-1.[8] Douady S, Couder. Y (1996) “Phylotaxis as a Dynamical Self Organizing Process”,
Jour-nal of Theoretical Biology , 178 (178): 255-74.[9] Brousseau, A (1969), “Fibonacci Statistics in Coniferi”,
Fibonacci Quaterly (7): 525-32.[10] Jerico B. Bacani and Julius Fergy T. Rabago, “On Generalized Fibonacci Numbers”,arXive:1503.05305v1 [math.NT][11] Noe, Tony; Piezas, Tito III; Weisstein, Eric W., “Fibonacci n-Step Number”, From
MathWorld - A Wolfram Web Resource. http://mathworld.wolfram.com/Fibonaccin-StepNumber.html[12] Any book on algebra, for e.g. I.N. Herstein
Topics in Algebra ndnd