## 1 Introduction
The edge ab in G(R\({}_{j-1}\), R\({}_{j}\)) is therefore an isotropic edge, and the vertices \(a\) and \(b\) are also isotropic. This means that the isotropic edges and vertices form a defined layer that has no confluences and breaks in its structure. The layers located between R\({}_{j-1}\) and R\({}_{j}\) that exhibit breaks and confluences (Figure 2) are considered anisotropic layers, or anisotropic edges and vertices in terms of G(N). Figure 2a shows a layered anisotropic pattern crossed by five transects. The corresponding five-partite graph (G(5)) is shown in Figure 2b. Graph G(5) is composed of bipartite graphs G(R\({}_{1}\), R\({}_{2}\)), G(R\({}_{2}\), R\({}_{3}\)), G(R\({}_{3}\), R\({}_{4}\)), and G(R\({}_{4}\), R\({}_{5}\)). The isotropic edges and vertices (shown in blue) and the anisotropic edges and vertices (shown in red) for bipartite graphs G(R\({}_{j-1}\), R\({}_{j}\)) are presented in Figure 2c. The parameter of the amount of disorder in the layer structure (DStr) was introduced to describe the deviation of an anisotropic pattern from its isotropic analog [4]. DStr is  
Figure 2: Isotropic and anisotropic layers as building blocks of layered patterns. (**a**) From the anisotropic layered pattern to graph. (**b**) Corresponding graph of an anisotropic layered pattern. (**c**) Isotropic (**blue**) and anisotropic (**red**) edges and vertices of the 5-partite graph. proportional to the number of anisotropic vertices and edges. If DStr = 0, then the pattern consists of layers without breaks and confluences. This means that the pattern consists of only isotropic layers (Figure 1b). A DStr of zero thus indicates the highest level of order in a layered pattern and a DStr of 1 indicates the highest level of disorder in the structure of a layered pattern, i.e., G(N) is only comprised of anisotropic layers. It is necessary to specify that special point patterns [8] are beyond the scope of the present work. There are an infinite number of both natural and manmade patterns which visually do not consist of layers. Figure 3 shows patterns of various categories, including those found in nature (Figure 3a), medicine (Figure 3b), the geosciences (Figure 3c), materials (Figure 3d,e), and art (Figure 3f). We call such categories of patterns arbitrary. In this paper, we focus on binary arbitrary patterns and grayscale patterns with a high level of edge gradient. Such a focus allows us to use a relatively simple procedure to convert these patterns into a binary format. The present paper continues our previous work [4; 6; 7]. The title of this paper states our goal: to justify that binary arbitrary patterns and layered patterns, both natural and manmade, are composed of isotropic (IL) and anisotropic (AL) layers. To reach this goal, this paper will do the following:  
* show that AL and IL are the building blocks of binary arbitrary patterns (meaning that the EM could be used to formalize the morphological properties of arbitrary patterns);
* illustrate the possibility of reconstructing the original arbitrary pattern from its EM; and
* introduce three parameters for layers: disorder of layer size (DSize), pattern complexity (PCom), and disorder of layer structure (DStr). We justify the usefulness of our proposed metric by showing that the DStr, DSize, and PCom parameters can be used to quantify the morphology of binary patterns, reveal changes in these patterns' size and structure, and act as shape characteristics. Our proposed method can be applied to both arbitrary patterns, and convex and concave polygons of any configuration, even if the pattern or polygon consists of straight lines alone. Figure 3: Examples of arbitrary patterns. (**a**) Mars’ surface; Credit: NASA [9]. (**b**) Melanoma; Credit: Wikipedia [10]. (**c**) Contour map; Credit: [11]. (**d**) Microstructure of graphene; Credit: [12]. (**e**) Microstructure of bronze: Credit: Wikipedia [13]. (**f**) Artwork; Credit. Wikipedia. Hokusai [14].