Phi x Phi =(1 + Square Root 5)/2 x (1 + Square Root 5)/2 =(1 + 5 + 2(Square Root 5)) / 4 =(3 + Square Root 5) /2 = 1 + Phitherefore
(a + b x Phi) x Phi = a x Phi + b(Phi x Phi) = (a + b) + b x Phi = a' + b x Phiwhich means that also the other height is a linear combination of the 1 and Phi, and, again in this 10-space, the pair 1, Phi is a base vector. For instance, in Fig.2, all the distances are equal to 1, Phi, 1 - Phi or any other linear combination.
Beautiful and well-known relations can be found for this number Phi; see several developments of this subject in [Coxeter,1973. Ghica, 1977. Rademacher & Toeplitz,1970.].
GENERAL CASE. As we have seen in the former cases, any order which engenders in other radii distances which can be expressed as a linear combination of two, will accept this pair as a base, and any linear combination of them, or what is equal, the distance between any two straights of the same series, will form a legal or permitted triangle (with vertices on some nodes of the lattice), using some others combinations of the base vectors.
If we try all the linear combinations of distances which determine a valid triangle in an N-lattice, we will arrive at the expression:
(ms x n - m x ns) x (sin(Alpha) x mp - sin(Beta) x m) - (mp x n - m x np) x (sin(Alpha) x ms - sin(Gamma) x m) =0where Alpha, Beta and Gamma are the angles; (m,n), (ms,ns), (mp,np) are the coefficients of the linear combination of the two unknown base vectors for the distances between straights of the three series whose intersection generate this triangle. By trying different combinations with small values (otherwise the job will last a infinite time) we obtain two-vector bases whose ratios allow legal triangles, as we have shown individually for the 6, 8, and 10 cases.
As any elemental figure (EE) of a RIGE can be decomposed into triangles, the found basis also allows legal RIGE, and therefore constitute answers (not all, but some) to the question raised about the distances between straight series. In Table 1 we can see some of these answers. In it are also shown several possible polygons in each lattice, some of which were shown in Fig.1 and 2. The angles are expressed one by one or, when possible, as a multiple, for regular convex and star polygons. In Table 2 are listed base vector ratios for greater values of N.
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