This site contains: mathematics courses and book; covers: image analysis, data analysis, and discrete modelling; provides: image analysis software. Created and run by Peter Saveliev.

From Intelligent Perception

What it means to puncture a hole in a surface is clear. But the meaning of a tunnel isn't as obvious.

For example, the hole in a doughnut is visible but the fact there are two in a tire might escape some people.

How many tunnels are there in a porous material?

Suppose we drill 3 perpendicular holes through a cube:

How many tunnels here?

There are so many ways to enter and exit these structures!

We can even try to list the seemingly reasonable answers:

• 1, because they are all connected;
• 3, by the number of times it was drilled;
• 6, by the number of entrances;
• 5 + 4 + 3 + 2 + 1 = 15, by the number of ways one can enter and exit the structure.

As a related question, in the next image, how many tunnels do these wireframes have?

To answer the question topologically, we "flatten" them:

Then we see that there are only 3 holes in the pyramid and 5 in the cube.

The issues becomes even more complicated if you want to describe a tunnel in a surface without the benefit of a bird's-eye view, from the inside the surface. The issue becomes real when we need to understand the topology of our universe:

See Homology as an equivalence relation.