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     4. Continued Fractions and Systems of Proportion

Any positive number, a, can be expanded as a continued fraction,

a = a0+1/(a1 + 1/(a2 + 1/(a3 + 1/(....))))          (1)

where a1, a2, a3,... are referred to as the indices of a. The continued fraction can be expressed by the notation a = [a0; a1,a2,a3,...]. Rational values of 'a' have a finite number of indices while irrational values are represented by an infinite sequence of indices. Truncating the continued fraction at various levels leads to rational approximations of 'a' known as approximants.

The golden mean satisfies the quadratic equation,

x2 - x -1 = 0          (2a)

which can be rewritten as,

x = 1 + 1/x.          (2b)

Replacing x by its recursive definition in Equation 2b yields the continued fraction expansion,

f = [1; 1,1,1,...] = [1; 1]

where 1 denotes the infinite repetition of the index 1.

Truncating this sequence at different levels leads to the sequence of approximants,

1/1, 2/1, 3/2, 5/3, 8/5, 13/8,...          (3)

the ratio of successive terms of the Fibonacci, F-sequence:

1 1 2 3 5 8 13 21 ...

This sequence has the property : an+1 = an-1 + an. From Equation 2 it follows that f 2 = f + 1. From this it follows that the f -sequence,

... 1/f2  1/f  f   f2  f3   f4 ...

is both a double geometric and a Fibonacci sequence.

The number f is the basis of a one-dimensional model for quasicrystals having "forbidden" five-fold symmetry [Schroeder 1991] as illustrated by the laser diffraction pattern of Figure 5.



4Kb

Figure 5. Laser diffraction pattern of a quasicrystal with five-fold symmetry.

Any solution to the quadratic equation x2 - px - q = 0 for p > 0, is the ratio of successive terms of a generalized Fibonacci sequence in the sense that

an+1 = p an + q an-1.          (4)

If q = 1, the solution to the quadratic is referred to as the p-th positive silver mean, while if q = -1, the solution is said to be the p-th negative silver mean. Any value of q 1 has been referred to as the (p,q)-th bronze mean [Spinadel 1998] . For example, if p = 2 and q = 1, then q = 1 + Ö 2 is the 2-nd positive silver mean corresponding to the continued fraction q = [2; 2]. The number q is sometimes simply called the silver mean because it is second in importance to the golden mean in the study of dynamical systems. Clearly Ö 2 = [1;2] and has the following sequence of approximants:

1/1, 3/2, 7/5, 17/12, 41/29,...          (5)

which leads to the following pair of integer sequence:

1 1 3 7 17 41 99 ... (6a)


0 1 2 5 12 29 70 ...         (6a)

These generalized Fibonacci sequence are known as Pell sequence and were used as the basis of the ancient Roman system of proportions [Kappraff 1996a,b]. They possess the additive property,

an+1 = an-1 + 2an

where lim an+1/an = q, when n ® ¥.

These additive properties are found in the pair of double geometric sequence,

...Ö2 /q    Ö2   Ö2 q   Ö2 q2    Ö2 q 3    Ö2 q 4 ...
               ... 1/
q       1        q        q2         q3          q4 ...          (6b)

Any arithmetical property possessed by the integer sequence 6a is also possessed by the geometric sequence 6b. For example,

3 + 2 = 5          Ö2 + 1 = q
7 + 5 = 12         
q Ö2 + q = q2

2 + 5 = 7          1 + q = q Ö2
5 + 12 = 17         
q + q2 = q2Ö2
       1 + 3 + 1 + 2 = 7         
Ö 2/q + Ö 2 + 1/q + q = q Ö2
3 + 7 + 2 + 5 = 17                  
Ö 2 + q Ö 2 + 1 + q = q 2Ö2

Also each element of the lower sequence of 6b is the arithmetical mean of the two terms of the upper sequence that brace it while each element of the upper sequence is the harmonic mean of the two terms from the lower sequence that brace it. Note that the harmonic mean m of x and y is defined as m = 2xy/(x+y). The integer sequence 6a has the same property in an asymptotic sense.

In the context of quasicrystals, b leads to a one-dimensional model of a quasicrystal with a "forbidden" eight-fold rotational symmetry first described by Wang, Chen, and Kuo [1987].

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