The image above shows two viewing angles of the same three dimensional object. I recently posted a problem with this image, and the question asked how many holes it had. Not a single person guessed that it could have \(3\) holes, which has led me to create this note.
Firstly, How do you define a hole, and how is one made?
My initial thought was that it is a piece of area in an object that has been completely "punched out", so that you can pass through the entire object without passing through any solid matter.
If this were the case, then a hole in the ground would mean that you have not made a hole in the Earth, as it does not pass completely through it. Rather, you have transformed the Earth so some area has been moved to a difference space.
Secondly, Does an object maintain the amount of holes it has even after undergoing a transformation?
For the answer of the original holes problem, I posted this gif which shows that the object would have 3 holes if it was transformed. Is this a valid proof to show that the unchanged object has 3 holes?
Take a copper pipe, for example. It is a hollow cylinder with \(2\) openings at either end. However if we flattened it into a disc, there would only be one area that is completely "punched out", which means one hole by this definition.
For the green 3D object in question, if you were to fill in the three holes you would see that both versions of the object, before and after the transformation, have no completely "punched out" areas. The initial version looks like a teacup, and the flattened version looks like a pancake. So does that means a teacup has no holes, despite the fact its curved shape would suggest that the object has a big hole in the middle for the liquid to sit?
Thirdly, Can holes be created or destroyed?
Do holes have to be conserved? If so, how many holes would the object have?
Are there any other questions to consider?
I would love to hear your thoughts on this topic.