Fig. 6Fig. 6 |
Among all possible EE, the RIGE tradition shows a preference for the symmetrical ones; for instance, in fig. 6 and 7, all the 10 EE are symmetrical, 8 with only 1 axis, 1 with 2, and 1, the star, with 8 axes, the maximum number for a 8-lattice. Note that those numbers are all divisors of N. See something more of this subject in the 'Musical Relationship' paragraph. The EE receives names according to its appearance: stars, barrels, candles, almonds, streets, diamonds, are names used in old times and nowadays [Jawad al-Janab, 1982., Nuere, 1989. Pavón Maldonado, 1981, 1990. Prieto y Vives, 1977.]. This attribution of things names to abstract designs is not the smaller pleasure which RIGE design offers, the possibilities being limited only by the intuition of the practitioner: new forms can appear after months of freqcuentation of a form. See some of these EE in the illustration above>.
Geometrically, the EE are polygons whose angles are multiple of the Basic Angle
and we obtain the interesting formula:
k + k'+ k" = N/2Since k, k,' k" are integers, N must be even, which justifies the parity of the orders formerly considered, at least for triangles. Now, if we have an 8-lattice, N/2=4, and only one possibility remains:
k = k'= 1, k"=2two 45º angles and one 90º, that is, an isosceles right angled triangle, as intuition would tell us. For a 10-lattice, two more combinations are possible, both isosceles: 1-1-3 and 1-2-2. More quadrilaterals are possible. The formula now becomes:
k + k'+ k" + k"' = NIn general, the number:
must be an integer. In the following table, some possible polygons - only the symmetric and convex ones - are listed, accompanied by the values of N and n. |

N a/b exact n k[i] polygons ------------------------------------------------------------------------ 6 60 1 1 3 3 1-1-1 4 9 1-22-3 6 12 6*2 ------------------------------------------------------------------------ 8 45 1.414 3 4 1-1-2 4 8 4*2, 1-1-1-5, 1-2-2-3 6 16 11-22-33, 2*(2-3-3), 22-3333, 11-333-5 8 24 6*3, 4*(1-5) 16 56 8*(1-6), 8*(2-5) ------------------------------------------------------------------------ 10 36 1.618 3 5 1-2-2, 1-1-3 4 10 22-33, 2-3-2-3, 111-7, 1-333 6 20 3333-44, 8 30 12222-777, ... 10 40 10*4, ... 20 90 10*(1-8), ...