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:
leading to the double integer sequence,
1 2 5 7 19 26 71 97 ...
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.
They also have other additive properties such as,
2 + 1 = 3, 7 + 4 = 11, 26 + 15 = 41, 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
b^{2}
Ö3b
^{3}
Ö3a
b^{3} ...
where a = y /2 and b = y ^{2}/2. Analogous to sequence 8a, sequence 8b has the following additive properties: 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,
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
Ö3y^{2}
Ö3y^{3}
Ö3y^{4} ...
and
1 1 4 10 28 76 208 ...
Each of sequence 10a and 10b are Generalized Fibonacci sequence with the additive property, a_{n+1} = 2 (a_{n-1} + a_{n}) (11) Also the ratio of successive terms in each sequence of sequence 10b has the property lim a_{n+1}/a_{n} = y and lim b_{n+1}/b_{n} = y while lim b_{n}/a_{n} = 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 |