This article proposes a topological method that extracts hierarchical structures of various amorphous solids. The method is based on the persistence diagram (PD), a mathematical tool for capturing shapes of multiscale data. The input to the PDs is given by an atomic configuration and the output is expressed as 2D histograms. Then, specific distributions such as curves and islands in the PDs identify meaningful shape characteristics of the atomic configuration. Although the method can be applied to a wide variety of disordered systems, it is applied here to silica glass, the Lennard-Jones system, and Cu-Zr metallic glass as standard examples of continuous random network and random packing structures. In silica glass, the method classified the atomic rings as short-range and medium-range orders and unveiled hierarchical ring structures among them. These detailed geometric characterizations clarified a real space origin of the first sharp diffraction peak and also indicated that PDs contain information on elastic response. Even in the Lennard-Jones system and Cu-Zr metallic glass, the hierarchical structures in the atomic configurations were derived in a similar way using PDs, although the glass structures and properties substantially differ from silica glass. These results suggest that the PDs provide a unified method that extracts greater depth of geometric information in amorphous solids than conventional methods.T he atomic configurations of amorphous solids are difficult to characterize. Because they have no periodicity as found in crystalline solids, only local structures have been analyzed in detail. Although short-range order (SRO) defined by the nearest neighbor is thoroughly studied, it is not sufficient to fully understand the atomic structures of amorphous solids. Therefore, medium-range order (MRO) has been discussed to properly characterize amorphous solids (1-3). Many experimental and simulation studies (4-7) have suggested signatures of MRO such as a first sharp diffraction peak (FSDP) in the structure factor of the continuous random network structure, and a split second peak in the radial distribution function of the random packing structure. However, in contrast to SRO, the geometric interpretation of MRO and the hierarchical structures among different ranges are not yet clear.Among the available methods, the distributions of bond angle and dihedral angle are often used to identify the geometry beyond the scale of SRO. They cannot, however, provide a complete description of MRO because they only deal with the atomic configuration up to the third nearest neighbors. Alternatively, ring statistics are also applied as a conventional combinatorial topological method (2, 8, 9). However, this method is applicable only for the continuous random network or crystalline structures, and furthermore it cannot describe length scale. Therefore, methodologies that precisely characterize hierarchical structures beyond SRO and are applicable to a wide variety of amorphous solids are highly desired.In recent years, ...

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To evaluate performance in a complex survival task, we studied the morphology of the Physarum plasmodium transportation network when presented with multiple separate food sources. The plasmodium comprises a network of tubular elements through which chemical nutrient, intracellular signals and the viscous body are transported and circulated. When three separate food sources were presented, located at the vertices of a triangle, the tubular network connected them via a short pathway, which was often analogous to the mathematically shortest route known as Steiner's minimum tree (SMT). The other common network shape had high fault tolerance against accidental disconnection of the tubes and was known as cycle (CYC). Pattern selection appeared to be a bistable system involving SMT and CYC. When more than three food sources were presented, the network pattern tended to be a patchwork of SMT and CYC. We therefore concluded that the plasmodium tube network is a well designed and intelligent system.

When two food sources are presented to the slime mold Physarum in the dark, a thick tube for absorbing nutrients is formed that connects the food sources through the shortest route. When the lightavoiding organism is partially illuminated, however, the tube connecting the food sources follows a different route. Defining risk as the experimentally measurable rate of light-avoiding movement, the minimum-risk path is exhibited by the organism, determined by integrating along the path. A model for an adaptive-tube network is presented that is in good agreement with the experimental observations. Introduction.-The plasmodium of Physarum polycephalum is an amoebalike organism with a body made up of a tubular network through which nutrients, signals, and body mass are transported. Studies of this organism have shown that it is able to determine the shortest path through a maze as well as ''solve'' other geometric puzzles [1][2][3]. In a maze, a starved organism forms a tube that connects food sources (FS) placed at the two exits of the maze via the shortest path, while nearly the entire protoplasm of the amoeba gathers over the two FS. The organism meets its physiological requirements in adopting this shape by absorbing nutrients from the FS as rapidly as possible while maintaining sufficient connectivity to permit intracellular communication. Such behavior in a primitive organism of this kind may offer insights into the evolutionary origins of biological information processing.Here we give the plasmodium a new type of task involving optimization behavior. Two separate FS are presented to the organism, which is illuminated by an inhomogeneous light field. Because the plasmodium is photophobic, tubes connecting the FS do not follow the simple shortest paths but form according to the illumination inhomogeneity. We report on the behavior of the organism under these conditions and discuss its physiological significance. We also propose a mathematical model for the cell dynamics and present a computational algorithm for its problem solving.Organism and methods.-The plasmodium of Physarum polycephalum, which regenerated from the sclerotia in ca. one-half day in the dark (25 C), was used in the experiments. A plastic film was placed onto a 1% agar gel, leaving a rectangular area (1 2 cm 2 ) of the gel uncovered. A few pieces (0:5 1 cm 2 ) of the regenerated plasmodium were placed in the rectangular area, and the preparation was placed in the dark for a few hours. The

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We report that various geometric patterns can be formed upon mechanical deformation of hexagonal micro polymer mesh. The patterning of micromesh can be applied to the fabrication of micropatterned soft-materials for cell culturing. A microporous film was prepared from a viscoelastic polymer, poly(ε-caprolactone). The film was a hexagonal mesh of 4 μm diameter. Plastic deformation of the film was caused by loading tensile force in one direction. Geometrical patterns such as elongated hexagons, rectangles, squares, and triangles were found in the stretched microporous film. These four types of deformation were reproduced by computer simulations using a viscoelastic network of hexagonally connected viscoelastic bonds. On the stretched hexagonal mesh, cardiac myocytes formed fibrous tissue where cells were aligned along the direction of the long axis of micropores. The hierarchical structure of blood vessels could be modeled by the coculture of endothelial cells and smooth muscle cells using a stretched honeycomb film as a micropatterned substrate.