Daishin Ueyama

The science of patterns.

The mysteries of self-organisation revealed by mathematics and simulation

Various patterns around us

There are many different patterns in nature. A visit to a zoo or aquarium will reveal a wide variety of epidermal patterns on living creatures. If you look closely at the patterns of zebras and giraffes, you will see that each individual has a slightly different pattern. The heart, an important organ that sustains our life, is electrically controlled, but when the electricity spirals out over the surface of the heart, it causes a dangerous condition called ventricular fibrillation, which can lead to death. Wind ripples form in the desert unseen, and looking up at the sky, different forms of clouds can be observed depending on the season. The cacti in the room are unique in shape, and if you look closely you will notice that their thorns are arranged in spirals. Look up at the trees and you will be amazed by the finely formed branches and the clever arrangement of the leaves, which cover the sky and keep the light out. These 'patterns' vary from biological to physical phenomena, but how exactly are they formed?

Using mathematics and computers to explore the mysteries of pattern formation

My research uses mathematics and computers to investigate the formation mechanisms of various patterns found in nature. Specifically, I derive mathematical models (mainly partial differential equations) of the target pattern formation phenomena and use computers to find solutions to these equations (called simulations). If the mathematical model is appropriate for the phenomenon under study, the equations can be investigated in detail using mathematics and computers to understand how the pattern is formed. Simulation of the partial differential equations obtained in this way can also provide information on how patterns are formed over time, such as stripes, spirals, snowflake-like patterns and patterns that divide and multiply like cell division over time. Although you may have the impression that the solution to an equation is a number, these complex patterns are the solution to an equation. I use simulation techniques and a mathematical theory called bifurcation theory to investigate the origin of the solutions of these equations, which show complex behaviour, and to gain an understanding of the actual phenomena. These findings are expected to be useful, for example, in controlling undesirable patterns (such as spiral waves appearing on the heart).

Linking self-organising mechanisms to applications

The characteristic feature of pattern formation in nature is that the final, regular and beautiful pattern is created on its own without any detailed instructions. This phenomenon of autonomous formation of structures is called self-organisation, and almost all pattern formations in nature, whether biological or non-biological, are the result of self-organisation. Self-organisation, which produces regular patterns without manual intervention, is the ultimate in manufacturing, and living organisms make good use of such self-organisation to create patterns and shapes. My research not only aims to understand such self-organisation mathematically, but also to link it to engineering applications. For example, by inputting the shape of Tokyo Bay into a computer and simulating the equations for pattern formation within the shape of Tokyo Bay, a pattern of spots automatically appears and is arranged to fit the complex shape of Tokyo Bay. By connecting the positions of these spots together, a regular and uniform collection of triangles (called a triangular mesh) can be obtained. This representation of complex regions as a collection of triangles is essential for precise computer simulations of, for example, the ocean currents in Tokyo Bay. Living organisms are a treasure trove of self-organisation, so I would like to continue my research to gain useful knowledge by using mathematics and computer-based analysis to penetrate into mysterious phenomena.