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     6. A Dynamical Systems Approach to Proportions


Consider a rectangle of proportions 1: x subdivided according to the law of repetition of ratios into a unit of the same ratio and a gnomon (see Figure 6a). In Figure 6b a square placed into this rectangle produces a pair of proportions 1/x and 1-1/x = (x-1)/x = 1/y where 1/x + 1/y = 1. If x and y are irrational numbers then we can apply Beatty's Theorem.


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Figure 6. a) Law of repetition of ratios; b) production of ratios 1/x and 1/y.

Theorem (Beatty's):

If x and y are irrational numbers such that 1/x + 1/y = 1 (Beatty pairs), then the set {[nx], [ny]} for n = 1,2,3,..., where [ ] denotes "the integer part of", equals the set of natural numbers with no repeats.

The following Beatty pairs correspond to the golden mean, root 2, and root 3 systems of proportion, respectively: 1) x = f, y = f2; 2) x = Ö2, y = qÖ 2; and 3) x = Ö3, y = a Ö3. Table 1 illustrates Beatty's Theorem for the Ö3 system.

n   [nÖ3]   [naÖ3]

     1         1         2        
     2         3         4        
     3         5         7        
     4         6         9        
     5         8         11       
     6         10        14       
     .         .         .        
     .         .         .        

If the numbers in column 2 are assigned a 1 while the numbers in column 3 are assigned a 0, the numbers in Table 1 are listed in order as

1 0 1 0 1 1 0 1 0 1 0 1 1 0 1 0 1 0 1 ...          (12a)

This sequence is known in the theory of dynamical systems as the symbolic dynamics. Notice that it contains all the information concerning the approximating sequence to Ö 3 given by sequence 8a. For example 1/2 of the first two numbers in sequence 12a are 1's, 3/5 of the first five numbers are 1's, 11/19 of the first 19 numbers are 1's, etc. In a similar manner the symbolic dynamics for the f system and the Ö 2 system are:

f:            1 0 1 1 0 1 0 1 1 0 1 1 0 1 0 1 1 ...          (12b)
Ö2:          1 1 0 1 1 0 1 1 1 0 1 1 0 1 1 1 0 ...          (12c)

These also replicate their approximating sequence 3 and 5.

The symbolic dynamics can also be generated by drawing a line with slope x/y on a coordinate system with vertical and horizontal lines drawn at integer values of the coordinates. Where this line cuts a vertical coordinate line assign a 1; where it crosses a horizontal coordinate line assign a 0. The symbolic dynamics for the f, Ö2, and Ö3 systems is illustrated in this manner in Figure 7. The ratio, 1/x, can be thought of as the probability of getting a 0 in the above sequence while 1/y is the probability of getting a 1, in which case the slope of the line represents the odds of getting a 0.


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Figure.7a. Square lattice and straight line with slope x/y = 1/f generates the approximating sequence (10110...) to f. The lower straight line has slope x/y = 1/q and generates another self-similar approximating binary sequence to Ö2.




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Figure 7b. Square lattice and straight line with slope x/y = 1/a generates the Ö3 approximating sequence.


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