3.2.9  Mirror curves in art

Let us consider mirror curves occurring in art of different cultures, distant in space and time, and try to discover some common construction approaches used by them for the continuation of mirror curves. We will try to compare mirror curves from Tamil art, Tchokwe sand drawings and Celtic art, try to discover the common properties of the constructions used, and establish some hierarchy with regard to their complexity. As the final result, we will describe a kind of stratified algorithmic approach used by these cultures for the construction of knotwork designs and compare it with similar approaches used in knot theory. 

Every culture probably in the beginning used plates - rectangular square grids RG[a,b] with the sides a, b. Plates have been recognized as the basis of all knotworks by the antiquarian J. Romilly Allen whose twenty years' work is summarized in the book Celtic Art in Pagan and Cristian Times (1904). The initial number of mirror curves for plates without internal mirrors is k = GCD(a,b) (GCD- greatest common divisor), so a single curve is obtained iff a, b are mutually prime numbers. From the knot theory point of view every single-curve plate, turned into an alternating knot by introducing the relation "over-under", represents a Lissajous knot (Bogle, Hearst, Jones and Stoilov, 1994). The first infinite series of them obtained for an arbitrary a (a ³ 3) and b = 2 consist of rational KLs of the form 3 1 3, 3 1 2 1 3, 3 1 2 1 2 1 3, 3 1 2...2 1 3. Naturally, for every odd b we obtain a knot, and for every even b a 2-component link. For those KLs, the number of different projections is: 1, 4, 13, 68, 346,..., respectively, but in knotworks only one of them- the most symmetrical, is used for each a. These projections can be found and drawn by using the LinKnot function MaxSymmProjAltKL. For a = 3 we have the projection 3 1 3, for a = 4 the projection (((1,(3,1),1),1),1,1,1), for a = 5 the projection (((1,((1,(3,1),1),1),1),1),1,1,1), for a = 6 the projection (((1,((1,((1,(3,1),1),1),1),1),1),1),1,1,1) etc.. The sequence that gives the number of projections was not in the Encyclopedia of Integer Sequences; in fact, it is possible to obtain a large number of new infinite sequences defined by numbers of different projections of specific classes of KLs. For an arbitrary a (a ³ 3) and b = 3 we have polyhedral KLs. The first of them for a = 3 is the 3-component link 8*2:2:2:2, the second obtained for a = 4 is the knot 1312*:2 0:::2 0.2.2 0, etc.

Any monolinear mirror curve placed in some polyomino without internal mirrors is a plate design