A Model of Islamic Rectilinear Interlaced Lattices

Fig.1 An 8 Lattice with some EE figures

Fig.1. An 8-Lattice with some EE Figures.


In theory, the order of a lattice could be any integer, but constructive, psychological, aesthetic and mathematical reasons limit its values to small numbers with many divisors. Typical cases are 8, 10, 16, 20, 24 and 32. The constructive reasons are (or were) the difficulty of making wood, plaster, ceramic pieces with extremely fine angles, and with too many possibilities: in a 36-lattice there may appear angles of 10, 20, 30, ... 350 degrees, 35 different values. The tools would have to have been very exact, which was not easy in medieval times. The psychological reasons relay on the difficulty in perceiving so many different angles and distances. This leads us to aesthetics: the sense of unity is lost as in an intricate and baroque relief. Some mathematical reasons can also be found, probably related to the former: the higher the order, the more polygons are possible and the lattice will need many different distances, complicating again the form for designer, maker and observer. Let us expand further this subject of distances between the straights of a family.

If we do not impose any symmetrical rule, any distance is possible. But if we do, as Tradition demands, once we choose one, the gyration for central symmetry will create 'alias' in another radius, thus creating new distances as a necessity. Let us see some cases from the simplest to some more complicated ones.

ORDER 6. This is a very simple case. The angles are the 6 multiples of q = 360/6 = 60 that is, 0, 60, 120, 180, 240 and 300. If we consider a triangle, it must be equilateral, as shown before, and the angles will be 60, 60, 60. If we choose any distance on the horizontal as a side, the same distance appears by rotating it on the oblique side, for equilateral. Thus we have only one distance necessary in the case N=6. We can use more, of course, but the order does not impose it, one is enough.

ORDER 8. A simple, but already interesting case. The angles are the 7 multiples of q = 360/8 = 45 , that is, 45, 90, 135, 180, 225, 270 and 315. If we consider a triangle with angles 45-45-90, once we gyrate the hypotenuse on the horizontal side, we obtain a new distance as a consequence of the first one. The ratio between them is the Square Root of 2. So the lattice order implies a specific distance ratio. See this ratio in Fig.1

Now the problem is to know how many different distances we need. If we raise another perpendicular to the horizontal at point 2, the new distance on the diagonal will be 2 x 2 = 2. But this distance is 2 times 1, that is, a second distance equal to the first, 1: we do not need another distance in this case. In general, any gyration of a linear combination of these two distances, 1 and 2, will provide us with another combination of them: they form what is known as a Vector Space, with 1 and 2 as a Base of the Space:

(a + b(2) x 2 = 2b + a(2) = b' + a(2)

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