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     5. The Root 3 System of Proportions


If p = 2 and q = 2 in Equation 4, then the bronze mean, y = 1 + Ö 3, is a solution corresponding to y = [2;1,2]. A proportional system based on Ö 3 is derived from the continued fraction expansion, Ö 3 = [1;1,2] with the following sequence of approximants:

1/1 2/1 5/3 7/4 19/11 26/15 71/41 97/56...          (7)

leading to the double integer sequence,

1 2 5 7 19 26 71 97 ...
                  1 1 3 4 11 15 41 56 ...          (8a)

These sequence share aspects of both the Fibonacci and Pell sequence in that,

1 + 2x2 = 5, 2 + 5 = 7, 5 + 2x7 = 19, 7 + 19 = 26, etc.
1 + 2x1 = 3, 1 + 3 = 4, 3 + 2x4 = 11, 4 + 11 = 15, etc.

They also have other additive properties such as,

2 + 1 = 3, 7 + 4 = 11, 26 + 15 = 41, etc.
3 + 4 = 7, 11 + 15 = 26, 41 + 56 = 97, etc.

The corresponding additive properties of sequence 8a are also found in the pair of double geometric sequence,

... Ö3/a   Ö3   Ö3a   Ö3b   Ö3a b   Ö3b 2   Ö3a b2   Ö3b 3   Ö3a b3 ...
              ...1/a      1       a       b       ab       b2       ab2           b3       ab 3...          (8b)

where a = y /2 and b = y 2/2. Analogous to sequence 8a, sequence 8b has the following additive properties:

1 + 2a = b, a + b = a b, a + Ö 3a = b, 1 + a = Ö 3a, 1 + 2Ö 3b = b 2          (9)

It should be noted that b = 2 + Ö3 is the 4-th negative silver mean, i.e., it satisfies Equation 4a for p = 4 and q = -1 and can be expressed as the following continued fraction with negative entries,

b = 4 - 1/(4 - 1/(4 - 1/(4 - 1/(....))))

The approximants of this sequence : 4/1, 15/4, 56/15, ... are found in the terms of sequence 8a and 8b related to powers of b. The proportion b is the basis for generating quasicrystals with the forbidden twelve-fold symmetry [Chen 1988].

A second pair of double geometric sequence and their accompanying integer sequence shed light on this system of proportions,

...Ö3/y   Ö3   Ö3y   Ö3y2   Ö3y3   Ö3y4 ...
              ...1/y       1     y       y2       y3        y4 ...          (10a)

and

1 1 4 10 28 76 208 ...
                 0 1 2 6 16 44 140 ...          (10b)

Each of sequence 10a and 10b are Generalized Fibonacci sequence with the additive property,

an+1 = 2 (an-1 + an)          (11)

Also the ratio of successive terms in each sequence of sequence 10b has the property lim an+1/an = y and lim bn+1/bn = y while lim bn/an = Ö 3 when n ® ¥. Whereas each term from the lower sequence divides the two terms from the upper sequence that brace it in a ratio of 1:2, e.g., 2(16 - 10) = 28 - 16, each term from the upper sequence is the harmonic means of the two terms from the lower sequence that brace it in an asymptotic sense, and divides this interval in a ratio which asymptotically approaches a = 1.366..., e.g., (140-76)/(76-44) = 11/8 = 1.375 where 76 is approximately the harmonic mean of 44 and 140. The same property holds exactly for sequence 10a. As an example of the additive properties of the root 3 system, consider edge AB of the twelve pointed star shown in Figure 1c. Taking the edge of the dodecagon to be 1 unit and using Equation 9, the length of the longest diagonal AB is,

AB = 1 + 1/y + 1 + 1/y + 1 = a + 1 + a = b

and, using Equation 11, the diagonals intersect in the ratio,

1/(1/y + 1 + 1/y + 1) = 1/y


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