Seeds of Mathematics in Everyday Life

My research field is a branch of geometry within mathematics called "topology." Here, I would like to introduce an example from the realm of topology where a seemingly familiar object was the "seed" of a major mathematical theory.

Please compare the left and right images in Figure 1. On the left is a soccer ball, and on the right is a polyhedron made by combining regular pentagons and regular hexagons. Looking at it this way, you can see that the pattern on a soccer ball is a "tiling of a sphere" modeled on the polyhedron on the right. Tiling of a sphere can also be done using other polyhedra as models. A typical example of such polyhedra is the "regular polyhedron" shown in Figure 2. Now, regarding this kind of spherical tiling,

(Number of vertices) - (Number of edges) + (Number of faces)

if we examine this quantity, we find that it is always 2 for any tiling.

However, if we consider the same quantity not for a sphere but for a tiling on a curved surface like the surface of a donut, called a "torus" as shown in Figure 3, the result is 0, not 2 (try to imagine a tiling of a torus and count it). This indicates that the sphere and the torus are fundamentally different surfaces, a discovery attributed to L. Euler (1707–1783). This discovery then developed into a major theory called "homology theory."

The "four-color theorem" and "knot theory" are other examples of how more familiar objects have developed into major fields of mathematics.

The "four-color problem" began when mapmakers in Europe several hundred years ago stated from experience that "four colors are sufficient to color any map." It was solved in 1976 and is now a major theorem known as the "four-color theorem."

"Knot theory" began as an attempt to mathematically formulate the differences between various knots, such as those in ropes and neckties. Mathematics was lurking even in ropes and neckties. This "knot theory" has now developed into a major theory that is also related to DNA.

Perhaps there are still many "seeds of mathematics" all around us that no one has yet noticed.