A while ago I have had the pleasure to come across those lectures of Topology & Geometry by Dr Tadashi Tokieda (I do recommend watching at least the first lecture, both parts). My question is based primarily on the second part. In few words my question is about Proofs done by picture.
During my studies, I must say, that I was not (am not) very motivated to draw/sketch the problems we faced (face). I think this is related to the fact, that coming from high school students are not used to rigorous proofs/arguments and tend to prove something by giving, although general, but not sufficient example by picture. I remember this happening during the class of geometry, e.g. discussing something only for equilateral triangle and not arbitrary triangle.
I think this process was motivated by our need to imagine the problem in a way we can understand it better. And I am still behind this idea, although I get the feeling, that mathematicians sometimes discourage this more than it is necessary (drawing etc.), because it does not represent the problem in general, or it is not really equivalent - discussing the problem and the picture. Also mathematical papers, books lack the presence of images (again, maybe this is only my feeling - and I also does not consider myself any great reader of math stuff) or have only few. It seems to me, that the pictures may not be there because of the extra work you need to put into it, while providing only non-equivalent intuition.
Example (not really a picture-one)
A few weeks ago we discussed the homotopy between two continous functions $f, g: [0,1] \to X$ such that $f(0) = g(0),\,f(1) = g(1)$ that is they form a loop (see - only intuition, there might be no loop - in case $f \equiv g$ or sth completely different). But I want to show different thing. Lets put $f\neq g$ and form a hole in the interior formed by $f,g$, e.g. $f,g$ create a loop around the origin in $\mathbb{R}^2$ and we take the origin away, being left with $\mathbb{R}^2 \ \{(0,0)\}$. The problem is, that there is no way to find a homotopy from $f$ to $g$ because of the hole.
I've got the idea of a bit real life example being somewhat this case. Imagine a jacket with zipper. Let $f$ and $g$ be each side of the zipper. The only adjustment we need to make is close the zipper on both ends (normally you would start from one end, and the other is still open). Then ask your friend to stick a stick between the zipper and try to fasten it. No matter how you try, the stick prevents you from being able to do it (fun observation the nothing = hole is suddenly very solid object and the space is nothing).
This is not really one of the examples I would like you to show, but nothing better came to my mind right now (maybe aside from Pythagorean theorem).
Question
If you are still reading, yet have no idea, what is my question, here it is:
Can you give an example of a theorem (maybe provide some additional background if you think it is necessary) from any branch of Mathematics (not only geometry) where a good picture is a sufficient proof (not that the picture would be the only thing used in the proof) of the theorem? Or that the truthness of the theorem is almost obvious from the theorem?
I am aware that the fact, that mathematicians created the tools, which allow us to prove something without relying on an unreliable pictures is great step. I just get the feeling whether we haven't sometimes stepped a bit further than it is necessary.
Mods I understand this, aside from others, can be opinion based, and I am prepared that this might be flagged and closed as off-topic, still I want to give it a try.
