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Focus on Mathematics

Homework of Two Thousand Years Ago: Hiroyuki Ochiai, Professor

Up to high school, we learn that two lines on a plane will either intersect at one point or be parallel to each other (Figure 1). This fact can also be understood intuitively. It is also true that, given a point that does not lie on a line, there is only one line which passes through that point and is parallel to the given line (Figure 2).


Let’s now consider a somewhat strange world. This world consists of the interior of a fixed circular disk as would exist in an ordinary world. The border (circumference) of this disk is a circle. Let’s call the interior of the fixed disk “this world” and its exterior “that world.” In addition, there is a large snake lying along the border of “this world” (Figure 3). This snake is very terrifying, and while it is possible to approach it, it cannot be touched. As a result, the people who live in “this world” cannot see or visit “that world.”


Now, a “point” in “this world” is the same as a point in an ordinary world, but a “line” in “this world” is defined to be either an arc or a straight line that intersects the snake at right angles (Figure 4). With this established, we can discuss the geometry based on this concept of “points” and “lines” in “this world.” For example, given any two points, there is only one line that passes through those two points (Figure 5). A figure enclosed by three “lines” is called a triangle and we consider the interior angles of such a triangle.


Is the property of “parallelism” the same in both an ordinary world and “this world.” Unfortunately, this is not the case. Given a line and a point lying outside that line, the number of lines that do no intersect with a given line passing through that point is more than one—in fact, the number is infinite (Figure 6). In short, “this world” is not the same as an ordinary world.


However, we can consider a rotation about a point and a translation movement in some direction in “this world”, as on an ordinary plane (Figure 7). We can then consider problems, for example, what happens making a rotation, a translation in a certain direction, and then making a reverse rotation. We can move a figure in “this world,” so we can discuss the congruence between triangles and another figure.


These observations may appear to be trivial and just for play, but in actuality, they were born of extensive and profound studies by many mathematicians of Euclid’s “5th Axiom” appearing in “Elements,” his famous treatise written more than 2000 years ago. In modern mathematics, the geometry introduced here is referred to as “non-Euclidean geometry.” The specialized term for “this world” is “Poincare disk,” which appears in many areas of mathematics such as differential geometry, complex function theory, number theory, and knot theory. It is known that a “statistical manifold” consisting of all normal distributions has the structure of “this world”. In addition, generalized functions called “Sato’s hyperfunctions” live on this large snake, that is, on the border between “this world” and “that world.” A rotation or a translation in “this world” shown in Figure 7 can also be expressed by a two-dimensional matrix, which can lead to group theory and harmonic analysis. And to give one more example, Lorentz space, which lies at the basis of Einstein’s theory of relativity, has a hyperbolic structure like that of “this world” different from that of an ordinary world.

Just as an unbounded world can extend over a chess board, an unbounded world can develop from a small geometric object like the Poincare disk.