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,

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

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,

and

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

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,

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