ELEMENTARY FIGURES.

Fig 6
Fig. 6

Fig. 7

Fig. 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 q = 360/Nº, 2p/N radians. The lower the order, the simpler the elementary figures. As the sum of the inner angles is equal to p x (n-2), n being the number of sides or angles, for the triangle ( n=3):

a + b + g = 180,
k x q + k'x q + k" x q = p,
(k + k' + k") x q = p,
(k + k' + k") x 2p/N = p
and we obtain the interesting formula:

k + k'+ k" = N/2
Since 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"=2
two 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"' = N
In general, the number:

s x k[i] = (n-2) x N/2
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),   ...


Last Updated: 15 November, 1999