## 4 Scattering by Coplanar Multilevel Ordered Strip System
### Some Novel Classes of Cantor Functions  
In this subsection, we will consider novel types of irregular Cantor functions, which enable us to create new original and strong coplanar multilevel grating, consisting of flat strips with parallel edges. The transverse cross-section of such coplanar grating structure represents an appropriate number of collinear intervals in two-dimensional space, ordering in original and mathematical strong sense.  
The basic fractal perfect set, the Smith-Cantor set, has numerous modifications to generalize a series of one-dimensional perfect sets with variable fractal dimension. The simplest modification uses triadic principle of fractal perfect set creation [5; 12]. For that modification, a straight-line segment (the initiator) is divided into three specific parts: two equal sub-segments and an interval or gap between them. On that interval, the Cantor function is equal to zero and we have the main step (Figure 2a). On two gaps of the second construction stage for Cantor function values, we put \(\pm 0.5\). So, new two equal steps are added to graph of Cantor function and so on.  
Figure 2: (a,b) The cross-sections of not classic strips systems.  
Figure 2a shows a stage of the Cantor function creation procedure constructed on the base of the fourth stage of a triadic perfect set [7; 13]. One can see seven intervals corresponding to eight segments of the fourth stage of the triadic perfect set. That Cantor function constructive procedure deals with Smith-Cantor perfect set, as the base for domain of that function definition. Due to the simplest generalization of the Smith-Cantor set, the class of triadic perfect sets with variable fractal dimensions can construct a lot of novel terrible Cantor functions.  
The second generalization of basic Smith-Cantor set uses rather unwieldy principle of perfect set creation to obtain five-parts classes of perfect sets with variable fractal dimensions. For that class a straight-line segment (the initiator) is divided into five specific parts: three equal sub-segments and two intervals or gaps [9]. On these two intervals for Cantor function values we put \(\pm d\) (Figure 2b). So, we have two basic equal steps (third and sixth in Figure 2b) for graph of Cantor function: \(x=\pm b+at\), \(z=\pm d\). One can see three geometrical parameters, which have to be multiplied by the wave number in correct modeling of scattering process. These parameters determine pre-fractal construction process and stages of Cantor function's construction as well.  
The lower part of Figure 2b shows the second stage of a perfect set creation: nine equal segments. Then we correctly defined Cantor function on six unequal gaps (intervals) of the second construction stage of perfect set creation. For instance, on the intervals number 4 and number 5 for Cantor function values, we put \(\pm d/3\) and so on. It will be noted that Cantor function's \"steps\" are intervals, unlike stage of a perfect set's creation is a set of segments. In Figure 2b, intervals (in general not equal) of domain of Cantor function's definition are between nine equal segments of the second stage of a perfect set's creation. Due to the generalization of the Smith-Cantor set, a five-part class of perfect sets with variable fractal dimensions can be constructed, as well as a lot of novel irregular Cantor functions. It is important that such Cantor functions depend on three or four dimensionless geometrical parameters.  
In the next subsection, we will consider the simplest novel types of non-classical coplanar gratings, dealing with irregular Cantor functions, in order to obtain explicit solutions of the plane wave scattering problem.