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RigeLawsFrom several of our own studies and papers, specially a model of islamic rectilinear interlaced_ attices and lacework_analysis we derive what we call Rige Lws, that is, the set of rules that constitutes the model or style of these figures.

FORMAL RULES of Rige

We formally characterize the Rectilinear Lace by means of the following restrictions or rules for the Rige forms:

R0. A Rige is developed in two dimensions, however, reliefs and holes are used to mark the designs.

R1. Only figures limited by straight segments are considered. Curvilinear design can be used to fill elementary polygonal figures, to avoid rigidity,

drynessandtoo geometricalforms. But they do not contradict the basic rectilinear criterion, which remains.R2. All these segments are parallel to a few directions in the plane, derived from the division of the compáss (360 degrees) by an integer N. This integer is called the Order, and the N directions, the main directions. Direction 0 will be the Reference, and 2=360º/N will be called the Basic Angle. Violations of this rule are exceptional, and will be considered in the

Metabolêsection below. They do not, however, affect this very basic law.R3. All these segments are situated on specific straight lines, parallel to the main directions. The distances between those lines are always equal to a very small number of quantities, 1, 2 or 3, according to the order. These distances are not arbitrary, but are determined by this order and by the angles - all multiples of 2 - between the main directions.

R4. The set of the N families of parallel straights is called the Lattice or Net of the form, briefly the N-lattice. All the segments belong to the Net; all the figures belong to the Net; the Net can be considered as the union or set of many possible Rige forms of a given order.

R5. The segments form closed polygonal figures, convex or not, symmetrical or not. Those without segments between them will be the Elementary Elements -EE- (elements in the lowest level), which, by juxtaposition, give larger figures.

R6. Their sizes are not too different: a general aspect of constant density of lines can be observed (and measured), recalling a grid, or a

tissue.R7. At least one point can be found which is a symmetry centre for all, or a part, of the form. The order of this central symmetry is equal to the order of the net to which the form belongs.

R8. The segments which limits the figures are united in polygonal lines with a width, as frames or sheets, which are never interrupted: whether they are closed or whether they finish in the external limits of the figure (actual figures are always limited, while nets are infinite by nature).

R9. The width of the frame is constant for the same form.

R10. When frames intersect, only two do so at the same point. The angle is non‑zero (and of course, equal to the angle between two main directions). Frames never bend at the intersection point, or, equivalently, opposite angles are equal.

R11. The intersection is usually represented (from Arabic countries to the West) as in three dimensions: one frame covers the other.

R12. When one frame (f1) covers another (f2) at a given intersection, at the following intersection it will be covered in turn by a third (f3) (which could be the second, f2).

R13. The form can actually be constructed by means of physical frames, bent and interlaced according to the former rules. Simple figures (EE) appear as empty spaces between frames.

R14. When the widths of frames are minima, ++++++ they fill the entire plane, with the disappearance of elementary figures. The aspect is now a polygonal chess board (damero), used in wood‑work, musical instrument decoration, etc.

From the second paper:

12. Lattices and forms will be as simple and economical as possible, reducing the number of different elements ( i.e. not reducible to others by geometrical transformations) to a minimum.

13. A segment can be slightly slanted, bent or displaced to comply with several lattices.

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