## 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) 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/(....)))) 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