Are pictures useful when studying?

Question

I was given a general hint by a math professor that if you have difficulties to solve a problem then draw a picture. Is that a good advice in general? I mean, to draw a picture I need to figure out the situation on my head. But then I have everything of the situation in my head so pictures won't give me any extra information. And also cases like Banach–Tarski paradox makes me wonder that sometimes pictures might lead astray.

Answer 1

I think it's some of the best advice you can give. So many relationships and ideas can be better expressed graphically and more clearly too. That's not to say that everything should be drawn, nor does it mean that diagrams don't have flaws too. Take all those "proofs" that $\pi=4$ for example. When you consider that (from a learning perspective) linear algebra is pretty much fancy analytic geometry, which in turn is pretty much fancy geometry, you can see how some arguments can be much easier and simpler in simpler systems. Why bother showing that $\sum abc\dots=x$ when you can draw it out and see that $x$ is a square.