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Focus on Mathematics

Shin-Ichiro Ei, Professor: Patterns and Mathematics

1.Patterns in the Natural World

There is a wide variety of patterns appearing in the natural world, but how can we understand them in a theoretical manner? First of all, what we describe as “patterns” extend from the very large, such as those that represent assemblages of stars and galaxies, to the very small, such as those that depict the distribution of atoms and electrons. These two extremes are each associated with a theory, cosmology in the first case and quantum theory in the second. Here, I would like to focus on patterns that we can see with our own eyes in things that make up the world around us. Some examples are the patterns of snow crystals and resin crystals, the patterns seen in a flame, the markings on the coat of a zebra, panther, and other animals, and the spiral patterns that can be seen in some chemical reactions.

Figure 1: Snow crystals (left) and resin crystals (right) (from [1])

Figure 2: Flame (from [1])

Figure 3: Coat patterns of various animals

Figure 4: Spiral pattern in a chemical reaction (provided by Inomoto, Kai Research Laboratory, Kyushu University)


In addition to these, there are many other kinds of patterns in the world around us.


So we can ask ourselves: “Just how are such patterns generated?” Could they really be explained by theory? Here, I would like to introduce one approach to this problem.


Our aim is to theoretically analyze the formation of patterns, but the various types of pattern phenomena that I just introduced have different mechanisms. For example, the formation of snow crystals and resin crystals is a problem related to material properties, while the formation of patterns on the coats of zebras, panthers, and other animals is a problem belonging to biology. Thus, it is practically impossible to lump these diverse types of patterns together and discuss them as one, and even if we could, we would probably end up talking in very general terms. I will therefore zero-in on one specific theme in the following discussion.


But before I get too specific, I will first clear up how it is that we recognize patterns in the first place. As reflected in the few examples that I presented above, each pattern can be expressed as two regions in different states and the boundary between them. In the case of a snow crystal, we can see a beautiful hexagonal pattern, which turns out to be the pattern formed by the boundary between ice and water. In the case of animal-coat patterns, a pattern is formed by the boundary between an area with much pigment and one with little, which is characteristic of animals. Let’s take a closer look at the area in the vicinity of such a boundary. In this area, we have two different states that are adjacent to each other with the boundary sandwiched between them. Now, as you cross that boundary, there is going to be a place in which it is hard to say which of the two states you are in. For a snow crystal, it is hard to decide on either an ice or water state in the area around the boundary, and for an animal’s coat, pigment may be bountiful or scare near the boundary. If such an in-between state should be somewhat broad in extent, that pattern will no longer be easily recognizable. Thus, for a pattern to be clearly recognizable, we can say that this in-between state distributed about the boundary must be very narrow, that is, the boundary must be well formed. We can summarize this requirement for pattern recognition in the following way: “Given two regions corresponding to two different states and a sufficiently narrow boundary between those regions, the pattern is recognized as the shape of that boundary.”


2.Theoretical Handling of Patterns

We learned in the previous section that patterns can be handled by analyzing the shapes formed by boundaries. How, then, should we deal with boundaries themselves? We just established that a boundary, if it has a width, must be sufficiently narrow for pattern recognition. This being the case, there’s no reason why we can’t assume an ideal width of zero as one method of dealing with boundaries. If we now think of these patterns as being on a plane, the boundaries making up these patterns take on the form of curves. We can also consider a method that considers boundary width to be non-zero while being sufficiently narrow. Such a method would be very significant from a theoretical perspective, but here, we will restrict ourselves to the method that deals only with ideal boundaries of width zero. Now, with our method decided, I present the simplest of several phenomena whose theoretical analysis has nearly been completed.


We assume very small magnetic-like elements that cover a plane uniformly with no gaps between them. Each element is oriented either straight up or straight down as a stable state. At the same time, the fact that these elements cover the plane in a dense, side-by-side manner means that their magnetic fields will mutually interact, and we consider that any one element will be affected by the states of surrounding elements. Furthermore, as one of the simplest ways of being affected, we assume that an element will attempt to orient itself in the same way as the elements in its vicinity. Here, we use the expression “magnetic-like” since, in the case of actual magnetic materials, a pair of adjacent elements in an up/down configuration would be the most stable of states making the above assumption a bit strange. However, an up/down pair as a stable state will make our discussion quite complicated, so we stick to our original assumption. In short, we assume elements with properties at odds with those of actual magnetic materials, and for this reason, we describe them as being “magnetic-like.”


Figure 5: Elements covering a plane

Now, to simplify our discussion even further, we assume that an element pointing upward will appear black and one pointing downward will appear white. Thus, if we were to look down on the plane, what kind of pattern would the black and white regions form?


To begin with, let’s assume for each element that it is easier for it to orient itself in one direction (up or down) than the other. For example, we can assume that it is clearly easier for an element to point upward. In this case, we can predict that, regardless of the intermediate process that each element follows, all of the elements will point upward in the end and the entire plane will simply appear black. We therefore choose to assume that a single element if unaffected by mutual interaction with other elements would be able to point in one direction just as easily as it could the other.


Figure 6: Initial pattern

Under this assumption, we consider the initial state (pattern) shown in the figure.


From here on, how might curve Γ between the black and white regions change over time? By adding a few assumptions to the basic conditions described above, research since the 1980s has established that the movement of Γ can be described as follows.


Figure 7: Movement of Γ

Result 1 “Curve Γ moves according to the following equation:

(2.1) V= –Κ


Here, V denotes velocity in the direction of the outside normal and K is the curvature of Γ.”



This movement as expressed by Eq. (2.1) is called “curvature flow.”


Figure 8: Curvature κ

Let me explain this equation of motion. First, since V is velocity in the direction outward from Γ, we must decide beforehand what is inside and outside of Γ. Here, we decide that the black region is on the inside of curve Γ. Accordingly, V signifies the speed at which Γ is moving in the direction perpendicular to itself from the black region that it encloses toward the white region.


Let’s now examine curvature K. Given a point P on Γ, we draw the largest circle that we can, tangent to curve Γ (maximum inscribed circle such that any larger radius would force part of the circle to protrude from Γ). This circle is uniquely determined. We let r denote its radius and set k = 1/r. The curvature of Γ at point P is therefore defined as follows: If the maximum inscribed circle can be drawn inside Γ, then Κ = k, and if outside Γ, then Κ = - k. For example, if curve Γ is a straight line in the vicinity of point P, an infinitely large inscribed circle can be drawn so that r = ∞, which means that κ = 0.


Let me give a straightforward explanation of the geometric meaning of curvature. Curvature is a quantity that indicates the extent to which curve Γ is bending. As the bending becomes sharper toward the inside, the radius of the inscribed circle becomes smaller and curvature κ takes on a larger positive value. Conversely, as the bending of curve Γ becomes sharper toward the outside, the inscribed circle becomes smaller on the outside of Γ and curvature κ takes on a larger negative value.


Thus, referring to Eq. (2.1) describing the movement of Γ, κ > 0 at a point that’s bulging outward so that V = –κ < 0. At such a point, Γ is moving toward the inside. Conversely, at a point that’s bulging inward, V = –κ > 0 so that Γ is moving toward the outside.


Figure 10: Movement of curve Γ

What all this means is that Γ is moving so that the concave-convex characteristics of its shape disappear. In other words, Γ is gradually approaching the shape of a circle. And once Γ finally becomes a circle, curvature κ will be positive at all points on Γ (with the inside of the circle being the black region) so that Γ will start to contract and eventually disappear. In this way, the initial relationship between the black and white regions can tell us which of the two colors will finally occupy the entire plane.


Something else of interest is known about the movement of Γ from Eq. (2.1). That is, the arc length of a curve that moves according to that equation will surely shorten with time. Thus, if a region should have a calabash shape such as shown in the figure, movement will stop at locations where arc length is minimum and the curve will come to a standstill.


Figure 11: Movement of Γ for a calabash-shaped region

Curvature-flow type of movement as expressed by Eq. (2.1) is known to appear in many places and situations. For example, if two types of organisms are in a mutually competitive relationship and their biological powers of survival are essentially equal, the boundary between the two regions that they each occupy will move according to this equation of curvature flow.


The various types of patterns that I introduced at the beginning of this article, such as those formed by the boundary between ice and water and the boundary between pigments on the coats of animals, are due to completely different types of phenomena and mechanisms. In any case, however, we have found that pattern movement can be described by an equation related to the curvature of the boundary, and that a pattern approximating actual phenomena can be reproduced by computer simulation. This method for investigating the pattern of something in the real world, by concentrating only on its outline, deriving the equation that governs the movement of that outline, and then theoretically analyzing the pattern, has only recently become possible. Looking forward, we can expect this research to solve many heretofore insolvable problems through association with geometric quantities.

Reference: P. Pelce, Dynamics of Curved Fronts, Perspectives in Physics, Academic Press, 1988.