BRIDGES Mathematical Connections in Art, Music, and Science A Fresh Look at Number Jay Kappraff Gary W. Adamson Department of Mathematics P.O. Box 124571 New Jersey Institute of Technology San Diego, CA 92112 - 4571 Newark, NJ 07102 [email protected] Abstract A hierarchy of rational numbers is derived from the integers and shown to be related to naturally occurring resonances. The integers are also related to the Towers of Hanoi puzzle. Gray code is introduced as a tool to aid in understanding Towers of Hanoi and also used to predict the symbolic dynamics of the logistic equation of dynamical systems theory. The Towers of Hanoi and Gray code are both generalized to number systems base n and used to derive a probability density function for the divisibility of integers. The number system based 4 expressed in generalized Gray code is shown to be a natural framework for the representation of the 64 codons of DNA. 1. Introduction What do the divisibility properties of the positive integers have to do with the Towers of Hanoi puzzle, Gray code, dynamical systems theory, and the structure of DNA? This paper explores these relationships. 2. A Natural Hierarchy of Numbers It is not commonly knoWn that when one counts the positive integers one also counts a hierarchy of rational numbers in lowest terms in which the "most important" rational numbers appear higher on the list. Our meaning of "most important" will be defined below, but first we describe a procedure for determining location of a number in the hierarchy by giving an example. What is the 19th rational number in this hierarchy? To answer this, first write 19 in binary, i.e., 19 = (10011)2' Next duplicate the last digit and separate the contiguous l's and O's as follows: 100111 corresponds to [1,2,3] where the numbers in brackets are the number of l's and O's in each contiguous group, i.e., 1 one, followed by 2 zeros, followed by 3 ones. The numbers in brackets are the indices of the continued fraction expansion [1],[2] of 19th rational number in the hierarchy, i.e., 19 corresponds to [1,2,3] = _1 _ = 1 1 1 = 7/10 1+_1_ 1+2+3 2+1 3 256 Jay Kappraff and Gary W. Adamson Note that the boldface indices appear as elements of the continued fraction. This leads to the following algorithm for determining the rational number corresponding to any integer of the hierarchy. Algorithm 1: a) Write the number in binary. b) Duplicate the last digit and write the numbers of D's and l's in each contiguous group, referred to as the indices, beginning from left to right. c) These are the indices of the continued fraction expansion of the rational number in lowest terms, Le., p/q = [at,a2,· •• ,an] = 1. + 1 + ... + 1 , at a2 an Note that [at,a2' ...,a n] = [at'~/... ,an-1,l]. By duplicating the last digit of the binary notation we have chosen the first rather than the second representation. This procedure can also be carried out in reverse to determine the hierarchy number of a given rational number. Using this algorithm 1/2 = [2] corresponds to the integer 1 after eliminating the duplicated last digit. Therefore 1/2 sits atop the hierarchy. Table 1 lists the first 15 numbers in the hierarchy. In this Table, the 2n-t integers with n digits are grouped in blocks and their corresponding rational numbers have continued fraction representations with indices ,that sum to n+1 ,e.g, there are 8 integers with 4 digits whose corresponding rationals have indices that sum to 5. Table 1. A Hierarchy of Rational Numbers and their Representations as Binary, Gray Code, and Tower of Hanoi Positions N Binary Gray Moduli Indices Fraction Pegs A B C TOH 3210 0 0 0 [0] 0/1 (Start) 1 1 1 0 [2] 1/2 1 1 2 10 11 1 [1,2] 2/3 1 2 2 3 11 10 0 3 [3] 1/3 1/2 1 4 100 110 2 [1,31 3/4 3 1/2 3 5 101 111 0 3 [1,1,2] 3/5 1 3 2 1 6 110 101 1 5 [2,2] 2/5 1 2/3 2 7 111 100 0 3 [4] 1/4 1/2/3 1 8 1000 1100 3 [1,4] 4/5 1/2/3 4 4 9 1001 1101 0 3 [1,2,2] 5/7 2/3 1/4 1 10 1010 1111 1 5 [1,1,1,2] 5/8 2 3 1/4 2 11 1011 1110 0 3 [1,1,3] 4/7 1/2 3 4 1 12 1100 1010 2 7 [2,3] 3/7 1/2 3/4 3 13 1101 1011 0 3 [2,1,2] 3/8 2 1 3/4 1 14 1110 1001 1 5 [3,2] 2/7 1 2/3/4 2 15 1111 1000 0 3 [5] 1/5 1/2/3/4 1 It should be noted that, strictly speaking, it is the blocks in Table 1 that are ordered in the hierarchy. Within each block there is no strict ordering of "importance," e.g., in block 4, 3/4 is no more "important" than 1/4, 3/5, or 2/5. The numbering of rational numbers within each block of Table 1 follows the well-known Farey sequence shown in Table 2 [2], [3],[4]. A Fresh Look at Number 257 Table 2. Farey Sequence 0/1 1/0 Row 0 1/2 Row 1 1/3 2/3 Row 2 1/4 2/5 3/5 3/4 Row 3 1/5 2/7 3/8 3/7 4/7 5/8 5/7 4/5 In this table each rational number is generated from the two that brace it from above by adding numerators and adding denominators of this pair, e.g., 5/8 is braced by 3/5 and 2/3 so that 5/8 = (2 +3)/(5+3) and 5/7 is braced by 2/3 and 3/4. Beginning in row 0 and counting right to left, we find that 5/8 is the 10th rational in the hierarchy. Applying the above procedure, 10 corresponds to 1 0 1 00 (with last digit duplicated) which corresponds to the continued fraction [1,1,1,2] = 5/8. Continuing to the next row, you can check that 7/10 is indeed the 19th fraction in the hierarchy. It should also be noted that any term x in a row of the Farey sequence gives rise to two terms l/(l+x) and x/(l+x) in the next row, e.g., x= 2/5 in row 2 gives rise to 5/7 and 2/7 in row 3. Figure 1 illustrates the so-called devil's or satanic staircase generic to almost all dynamic systems [3]. This figure graphs the winding number m vs a , , , l 1 ~k~. I 1 t .\.~. I 4ll- I t 7- Z .. 3 tlt.. - Q24 g.- . I I .J..1 - ! t li·· I.k:'!·· Q22 ~ .. •1 - .t., 0 0 0.2 0.4 0.6 0.8 1 n Figurel. Devil's stairca~e with plateaus at every rational number. From Fractals, Chaos and Power Laws by M. Schroeder. By permission of W.H. Freeman and Company. frequency ratio Q that represents the ratio of a driving force frequency and the resonance frequency of an oscillator fora system known as the circle map [3]. In this map a sequence of points zo' Zl' Z2 ••• are generated by the application of the function, Zj+l = R(zj) where R(z) = Z + Q - k/21t sin 21tz. The j indices can be thought of as time intervals. The winding number m of the map is defined as the limit of (zn-zo)/n as n~ 00. Map R depends on a parameter k related to the energy of the system. As k~l, a critical value, the system approaches a chaotic state in which every winding number from the unit interval [0,1] is obtained depending on the value of Q with rational values of co phase locked to finite intervals of Q. By phase locking we mean that the same winding numbers are manifested for a finite interval of Q values. The phase locked intervals for the irrationals have zero width and so the irrationals form a kind of 258 Jay Kappraff and Gary W. Adamson "dust" between the rationals. We see that the higher a rational number is in the hierarchy the wider is the phase locked interval corresponding to it. Winding numbers represented by larger intervals correspond to resonances of dynamical systems with greater stability justifying our reference to the rationals corresponding to these intervals as being "more· important." For example the three widest plateaus occur for 1/2, 2/3, and 1/3 corresponding to the first three rationals of the hierarchy. The relative widths of the plateaus are ordered according to the terms of the Farey sequence with elements within each row having approximately the same width. 3. Gray Code, Divisibility, and the Towers of Hanoi Notice that the strings of O's and l's (bit strings) in the third column of Table 1 are labeled as Gray code. Gray code is a system of representing integers as bit strings such that from integer to integer only a single bit changes in its representation unlike binary in which more than one bit can change from one integer to the next, e.g., 7 = 111 whereas 8= 1000.