## 4 Discussion  
### Method: Pros and Cons  
The central point of the present paper is the justification that binary arbitrary patterns (and those converted to binary patterns) consist of building blocks that are made up of isotropic and anisotropic layers. In terms of G(N), these building blocks are made up of isotropic and anisotropic edges/vertices (Figure 2). It is therefore concluded that the EM is applicable for the formalization of the morphology of layered, as well as arbitrary patterns.  
The parameters DStr, DSize, and PCom were introduced in order to describe the morphological characteristics of arbitrary patterns. Figure 8a shows the positions of sixteen arbitrary patterns in the DStr vs. DSize space. It follows from Figure 8a that the similarity of the DStr and DSize does not guarantee the similarity of their corresponding patterns. For instance, the difference between the DStr and DSize of a graphene microstructure and fish skin patters are minimal, which is contrary to the visual impression that these patterns have nothing in common. The same is true for butterfly-2 and plant cells (Figure 8a). Nevertheless, the DStr and DSize could be used as the measure of similarity if the natural patterns being analyzed and the problem statement under consideration allow us to do so. Our experiments with Alef justify the potential possibility of quantifying the similarity between patterns in the DStr and DSize space (Figures 18 and 19).  
The usefulness of DStr, DSize, and PCom are also demonstrated via a morphological analysis of: (a) a 2-D honeycomb pattern, in order to pinpoint cells with different level of size anomaly (Figure 11e); (b) artwork (Figures 15 and 16), in order to show that the DStr,  
Figure 19: The influence of basic graphical elements on Alef’s complexity. (**a**) Basic graphical elements. (**b**) Shape of Alef as the function of basic graphical elements. (**c**) Alef’s complexity. Credit: Drawing by Dr. [PERSON], courtesy of Carta Jerusalem [51].  
DSize, and PCom are able to reveal anomalies in the morphological description of hand-drawn geometrical figures that can be used as an MSC of the artwork; (c) environmental patterns, such as contour maps, river meanders, and shape of island coastlines, in order to justify that the DStr, DSize, and PCom could be used to reveal changes in the size and structure of patterns under study.  
The robustness tests for DStr, DSize and PCom are not performed in the present work because the focus of our study is binary patterns and patterns with low levels of fuzziness around the edges. The robustness test for the DStr parameter was performed in our previous work [4], resulting in the conclusion that the first step in applying our proposed method to solving any particular problem has to be further study of the relationship between DStr, DSize and pattern binarization.  
Let us consider the limitations of the proposed method. Many limitations are still unknown because the EM of layered and arbitrary patterns is a relatively new object of study. Thus, we list the most obvious limitations that follow from the pattern analyses presented in the Results section:  
* The EM = [BF, G(N), T\({}_{\rm M,N}\)] does not account for the material properties of layers.
* The EM = [BF, G(N), T\({}_{\rm M,N}\)] does not account for the thickness of lines used for the pattern construction.
* The algorithms used to construct the EM in the present work are based on a pattern in the binary raster (pixel) format only.
* The EM cannot be constructed for patterns made up of only one-pixel dots.
* The problem of arbitrary pattern normalization is outside the scope of this work.
* There is no answer in the present work to the question \"Are identical morphological characteristics possible for different binary arbitrary patterns for any angle of its rotation?\"