Quest for Universality of
Mathematical Structure in Nature

Topology is a mathematical field that treats invariant property of spaces under continuous transformation and neglects other detail structure. At the topological view point, we may understand various phenomena in the range from the Universe to elementally particles, and from virus to human's activity by the universal mathematical structure without dependence of kind of matters or energy scale. We investigate various phenomena to prove the universality hidden in nature's mathematical structures, and quest the reason of existence of the universality itself in nature.

Hierarchical Honeycomb Structures

The fact that simple progression laws describes some phenomena or structures in nature impresses people. We discovered the universal hierarchical honeycomb structure following by a new progression law An+1=9An-2 in the two-dimensional electron system. The electrons align on the triangle lattice are forced hierarchical attractive force due to topology of k-space. The force attracts proximate electrons and makes a molecular structure, and then it attracts proximate molecules and make a super molecular structure. The progression law is the same with the packing problem in the gaps opened when two dimensional spheres are attracted each other. We are trying to apply the universal hierarchical structure to the problem of emergence of hierarchical structures in the Universe, namely galaxy-cluster galaxy-super galaxy structure.
Reference: T. Toshima Doctoral thesis (2006)

Topology-Change Method and Topological Rigidity

The nontrivial topology of real spaces affects order-parameter fields such as crystals, magnetism, and charge-density waves) on the spaces. If topological phase were added in the phase of the order parameter r(x)eif(x) like the wave function of the path integral expression, the new property of rigidity, named topological rigidity, would emerged. To prove hypothesis experimentally, we change topology of topological crystals by the topology-change method. We cut the crystals and observe the change of their shapes. As the results, we discovered that the cut-rings becomes trochoidal curves. This form can not be explained by elastic energy models, and suggests necessary of new concept in crystallography, such as topological rigidity.

FigureFCutting experiment of ring-shaped crystals
ReferenceFT. Matsuura Doctoral Thesis(2007)

Knot-like Vortices

From galaxy to tornado, vortices are observed every scale. However, Their creation-annihilation mechanism has been still unclear due to their nonlinear and nonequilibrium nature. We investigated the role of topology in creation of vortices. The laser is radiated to a metal surface and excite vortices of metallic fluid. Then the trail of the vortices are observed by electron microscopy. As the results, we discovered the vortices with topologically classified knot structures (helicity) are excited. We trying to explore the universal property of the creation-annihilation mechanism by the investigation of relation between the helicity of vortices and the exciting mechanism.

ReferenceFT. Miyazawa Bachelor's Thesis(2007)

Discovery of Polytope Crystals

Recently we discovered polytope crystals of MX2 (shown in figure). These crystals are topologically same with bulk crystals because the forms can be identified under continuous transformation (topologically homeomorphic). Topological classification neglects cusps and negative curvatures. However, the straight line connected between two points A and B can be drawn on the region of crystal but the line connected between C and D points cannot. Making cusps are analogous with adding singularity points in networks and changing of Voronoi diagram. At the viewpoint, bulk crystals and the polytope crystals are not topologically homeomorphic. We sublimate topological classification based on invariant property under continuous transformation to new method involved information of global forms (relation of vertices and branches, and changing of Euler number) and further investigate universality of mathematical structure hidden behind more complex phenomena.

FigureFPolytope crystals

Figure: the line between point A and B is on the hexagon,
however, line between pint C and D is not on the hexagram.

Networks as the world

Recently large scale network analysis can be performed due to development of computer's performance. As the results, it has been clarified that an order exists in the anomalous networks found in human society and nature. By the discovery of concept of small-world and scale-free, which characterize network property, the research of networks has exploded across many fields, such as food chain in nature, gene networks, chemical reaction of proteins, World Wide Web, telephone networks, power transmission networks, collaboration networks of researchers, actor's networks, roots of words, relation of friends, and relation of economic trade. When network analysis is applied to the anomalous phenomena, many information must be neglected. We explore universal structure in networks from topologically-invariable information like sequence and forms. This approach is expected to be a strong tool for resolve the real entangled phenomena.

Figure: Network of a community currency flow
(the second research at Tomamae-cho by N. Kichiji)