Existence of a Polycrystal Filled with an Arbitrary Finite Number of Self-Similar Crystals

Abstract

We discuss a sufficient condition for a space to be filled with an arbitrary finite number of self-similar spaces using a topological concept.It is a great issue in the fields of materials science and geology, which are closely related to crystallography, that a space can be filled with an arbitrary finite number of grains, each of which is characterized as self-similar. Mathematical methods for the crystallography have long been developed mainly by the group theory, because it is quite useful to discuss symmetry and periodicity in the crystal. However, it is not necessary for the above problem to consider a periodicity of the internal elements in each grain using a concept of space group. We have studied the mathematical structures of self-similar condensedmatters using a point set topology [1]. To our knowledge, such fundamental topological approach has not been applied for the study on the structures of aggregates of grains. Here we can also discuss the tiling problems according to a same procedure for the current problem on aggregate of grains by replacing an aggregate of grains with a tiling and a grain with a tile, respectively. The unit of the discussion is now single crystal or grain, whose characters can be defined topologically. Therefore, it is also possible to discuss the structure of aggregates of the noncrystalline or amorphous grains using a topological method.Although structures of self-similar materials are often discussed using a fractal dimension in fractal science, such discussions are done only within a real space as it can be seen. In the present report, we employ somewhat indirect method of observation of the structures of material. That is, the structures are expressed indirectly through the mathematical observations of the formation of a set of equivalence classes. (see, for example, a literature by Fernández [2], concerning the use of quotient space in statistical physics) The concept of the equivalence class is familiar in the diffraction crystallography as a reciprocal lattice point [3]. Group of the lattice plane is gathered as a concept of equivalence class, and then the geometrical structure in a real space can be determined by diffraction patterns in a reciprocal space. The structures of materials are observed here mainly from the viewpoint of this equivalence class, that is, a decomposition space, which is similar idea as that for the diffraction analysis. From a viewpoint of topology, any pattern can be obtained from the space which is characterized, in principle, as 0-dim, perfect, compact T 2 space. In the followings, we discuss this sufficient condition for a space to be filled with the self-similar spaces from the viewpoint of topological aspects.Let us start the mathematical observations of the structure of material for the case shown in Fig. 1. This figure shows the structure of aggregates of grains (X), where each X i corresponds to each grain. Let (X, τ ) = ({0, 1} Λ , τ Λ 0 ), CardΛ ≻ ℵ 0 (aleph zero) be the Λ−product space of ({0, 1}, τ 0 ) where τ 0 is a discrete topol...

Universal topological representation of geometric patterns

“…Recently, the authors proposed the mathematical sufficient condition for an issue in material science and geology that a polycrystal can be filled with an arbitrarily finite number of self-similar crystals [9]. According to the issue, if a geometric structure of polycrystal is characterized by a topological space X which has a specific topological structure, namely, X is a 0-dim [14], perfect, compact Hausdorff-space, then the mathematical procedure of construction of an arbitrarily finite number of self-similar crystals filling with the polycrystal is ensured.…”

“…In several topological methods, we have been successfully studied the mathematical structures of condensed matters by using a fundamental topological approach, that is, a point set topology [7,8,9]. By means of this topological method an universal geometric structure of condensed matters which is independent of each detailed nature of structure of them has been investigated qualitatively.…”

“…Note that the method is mathematically based on a technique of algebraic topology. By using a fundamental topological approach, that is, a point set topology, we have successfully studied the mathematical structures of geometric patterns of matters [9][10][11]. By means of this mathematical method, TSM have been investigated independently of the detailed properties of each matter.…”

Topological characterization of network structures of aggregates of atoms

“…To investigate geometric structures of condensed matters, mathematical methods using topology have been applied to condensed matter physics, for example, topological defects [8] and quasicrystals [9]. In several topological methods, the mathematical structure of aggregates has been successfully studied using more fundamental mathematical approach, that is, a point set topology [10,11]. In particular, we have focused on a somewhat indirect method of observation of the material structure.…”