## 1 Introduction  
Anisotropy is a common structural feature of various systems formed in nature and beyond. The commonly accepted definition of anisotropy is different properties in different directions. For instance, in materials science, materials have different mechanical properties depending on the direction of their measurements [1]. In seismic geology, the velocity of elastic waves depends on the direction of those waves' propagation [2]. In addition, \"morphogenesis in plants depends critically on directional (anisotropic) growth\" [3]. We will consider anisotropic features of layered systems, which are to some extent different from the broadly used definitions of isotropy and anisotropy. The patterns found in fish scales are one example of a layered growth system that has an anisotropy of structure. The incremental growth of fish scales results in the formation of layers across a 2-D plane. One growth layer describes a cycle of fish scale growth at an instance of time \(\mathrm{t_{i}}\), and the thickness of the layer is the measure of the growth rate at ti. These layers are distinguished by numerous breaks and confluences due to the fact that the growth rate of fish scales varies in different directions. That is, the growth layers exhibit an anisotropic structure due to the anisotropic growth of the fish scales. Cross sections of lamellar bones, corals, and trees similarly exhibit patterns of anisotropic growth layers formed by living systems where a layer is formed at an instant of time ti. Layered anisotropic structures can be found in both natural patterns and manmade objects, including satellite images of the surface of Mars, the human aorta, alloys, and many others [4], even though their layered structures were formed differently than those of the growth layers found in living systems. The anisotropy of the physical, mechanical, or electrical properties of an object under study implies that the property's measurements are carried out along N transects in N different directions. The anisotropy of a structure made up of layered patterns implies that measurements are carried out along N parallel transects, which are oriented in one direction only (Figure 1a). If the layers represent growth increments, then the direction of the numbered layers coincides with the direction of pattern formation. Thus, the set of transects R1,..., Rj,..., RN represents the coordinate system used for the measurement of patterns with anisotropic characteristics. A layered pattern with an isotropic structure implies that there are no breaks and confluences in the structure of the layers (Figure 1b). An empirical model (EM) of 2-D layered patterns was proposed with the aim of providing a tool to quantify the morphological features of anisotropic layered objects [6,7]. The EM has three components: an N-partite graph G(N), a Boolean function (BF) to describe the 2-D structure of the layers, and table T\({}_{\text{M,N}}\), which comprises the thickness of the layers along transects R1,..., RN; EM = {BF, G(N), T\({}_{\text{M,N}}\)}. Transects R1,..., RN that are used for the construction of the EM are straight lines that are always distributed evenly across a 2-D  
Figure 1: Two categories of layered pattern structures. (**a**) Human bone as a pattern with a layered anisotropic structure. (**b**) Mars’ surface in orange and green as a pattern with a layered isotropic structure. Credit: NASA [5]. layered pattern and are perpendicular to the direction of pattern formation or numbering of layers. The N-partite graph G(N) consists of a sequence of bipartite graphs, G(R\({}_{1}\), R\({}_{2}\)),..., G(R\({}_{j-1}\), R\({}_{j}\)),..., G(R\({}_{N-1}\), R\({}_{N}\)), where G(R\({}_{j-1}\), R\({}_{j}\)) is a bipartite graph that describes the structure of a layered pattern situated between nearby transects R\({}_{j-1}\) and R\({}_{j}\). An isotropic layer here would imply that the vertex a\(\in\)R\({}_{j-1}\) connects only with the vertex b\(\in\)R\({}_{j}\), and b\(\in\)R\({}_{j}\) connects only with a\(\in\)R\({}_{j-1}\).