As I already wrote once, I like strange theories and paradoxes, but I believe that there is always some underlaying logic. For example look at this image:

Do you see the reason for this? There is nothing wrong with this picture. Moreover this is exactly what you will get if you will make your own triangle.

The explanation is very simple. The area of all of the smaller triangles is equal, in both pictures, to the original one. But if you will look carefully you will see that that the slope, of the second figure, is different.

Update: This wasn't clear to some of the readers, so I wrote a more detailed explanation.

Such things are the reason why in math everything must be proven. We cannot just trust our intuition. It was relatively easy to spot the reason in this case, but it is not always easy. Geometry we can at least visualize, and even make pictures. When the question is totally abstract it is much more difficult to understand it - especially if our intuition is wrong about it.

A few days ago I overheard two students talking about their linear algebra exam. There was a question in the exam to find a basis for some vector space. The student found it, but didn't prove it is indeed a basis. He didn't get any points for the question naturally. And he was angry about it...

But without proof there is nothing. Even if the answer is correct, without proof it is worthless. Such approach is the only way in which paradoxes can be solved and mistakes can be avoided

Simplicial Sets vs. Simplicial Complexes

2 days ago

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