# A Model of Islamic Rectilinear
Interlaced Lattices

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

## LATTICE ORDERS and DISTANCES

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, 0º, 60º, 120º, 180º, 240º and 300º. If we consider a triangle, it must be equilateral, as shown before, and
the angles will be 60º, 60º, 60º. 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, 45º, 90º, 135º, 180º, 225º, 270º and 315º. If we consider a
triangle with angles 45º-45º-90º, 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 **Ö**2, the new distance on the diagonal will be
**Ö**2 x **Ö**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 **Ö**2, will provide us with
another combination of them: they form what is known as a Vector Space, with 1 and **Ö**2 as a Base of the Space:

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

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Orders and Distances - part 2

05-June-95