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
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.
FigureFCutting experiment of ring-shaped crystals
ReferenceFT. Matsuura Doctoral Thesis(2007)
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.
ReferenceFT. 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: 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)