When you ask a mathematician about holes, prepare for a mind-bending journey that challenges everything you thought you knew about openings, tunnels, and topological spaces.

Experts in mathematics have long grappled with a seemingly simple question: What exactly constitutes a 'hole'? According to Katie Steckles, a lecturer in mathematics at Manchester Metropolitan University, mathematicians define a hole as something that goes completely through a shape, much like the opening in a doughnut that allows you to peer directly through its center.

This definition might sound straightforward, but it quickly becomes complex when you start examining everyday objects. Take a straw, for instance. Ask a group of people how many holes a straw has, and you'll receive a variety of answers ranging from zero to two. Mathematical communicator James Arthur suggests an imaginative approach to resolving this puzzle: mentally transforming the straw into play dough and slowly squishing it until it resembles a doughnut.

In the realm of topology - a branch of mathematics that focuses on fundamental spatial connections rather than precise shapes - objects are classified by their holes. A golf ball, baseball, and Frisbee are essentially identical from this perspective, as they could theoretically be manipulated without creating or closing holes. However, a bagel, doughnut, or basketball hoop represent an entirely different topological category because of their central opening.

The nuanced understanding of holes extends beyond simple geometric considerations. When engineers constructed the Channel Tunnel connecting the United Kingdom and France, they began with two separate digging projects. Once these tunnels connected, the entire structure transformed from two distinct openings into a singular 'through hole' - much like a tube with openings at both ends.

Steckles emphasizes that what many might casually consider a hole - like a depression in sand at the beach - doesn't truly qualify in mathematical terms. A genuine hole, according to topological definitions, must completely traverse an object, allowing something to pass entirely through it.

This mathematical perspective offers a fascinating lens through which we can examine the world, transforming mundane objects into complex topological puzzles. By reimagining everyday items as malleable, interconnected shapes, mathematicians reveal the elegant, abstract beauty underlying our physical reality.