Topology: Seeing the 'Structure' Beyond Shape

Are All Golf Ball Dimples Hexagons? Exactly 12 Pentagons Are Required
Subway Maps: "Ignore Actual Distance Between Stations, Show Only Connectivity"

Topology is a branch of mathematics that deals with shapes. However, it is quite different from the 'precise' shapes we commonly know. Mathematics that studies shapes can be broadly divided into geometry, which measures 'lengths and angles' with precision, and topology, which focuses on the essential structure of 'connectivity.'

While geometry is a discipline aimed at calculating (or measuring) precise lengths, angles, and areas, topology is concerned with properties that remain unchanged even when an object is continuously deformed (such as stretched, compressed, or twisted, as long as it is not torn). For example, the essential structure of space, such as the number of holes or the way parts are connected, is the focus of topology.

For instance, even if a paper cup is crushed or stretched, as long as it is not torn, it is still a cup 'without a hole.' Topology studies how shapes can be continuously transformed. From this perspective, a coffee mug and a donut may look different at first glance, but since they both have 'one hole,' they share the same topological property and are considered the same object in topology.

The best example of topological thinking in everyday life is the 'subway map.'

Choi Sooyoung, a professor in the Department of Mathematics at Ajou University, explained, "A subway map ignores the actual distances between stations and shows only the connections between lines and transfer points?this 'connectivity' is a topological perspective." She added, "Topology is a discipline that seeks to see the essential structure beyond form, even in complex systems."

Professor Choi added, "When you try to understand the structure beyond the form, you sometimes arrive at very unexpected results. Topology is characterized by explaining what properties arise when you look at the overall shape of an object, why those properties are possible, or why they are impossible."

Not all dimples on a golf ball are hexagonal. There are always 12 pentagons. Photo by Pixabay

Topologists do not focus solely on the hexagonal shape of the dimples when looking at a golf ball. While it may seem that all the dimples on the surface of a golf ball are hexagonal, in reality, there must always be exactly 12 pentagonal faces for a perfect sphere to be formed. This is a structural constraint that is inevitable in order to maintain the topological shape of a sphere, and topology reveals such unavoidable characteristics.

Just as urban planners try to improve traffic flow in a complex city road network by installing traffic signals in strategic locations, a topological perspective allows us to understand the overall connectivity of the road system. With this understanding, it becomes possible to identify exactly where 'bottleneck' points will inevitably occur and why it is fundamentally difficult to resolve congestion at those points, no matter how much effort is made. Through topology, we can see through such structural limitations that are otherwise impossible to predict.

This content was produced with the assistance of AI translation services.