What are the dangers of visual exposition of mathematics? I've heard several times (such as this one)  that it's dangerous to learn/prove/teach mathematics through images. I've also read somewhere that showing mathematics through images helps one's intuition because we understand better through images, due to our long date use of the vision sense. I can conceive that this information should be half-true - there might be things that can be taught with images and things that can not. From here, I have two doubts:


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*What can and what can't be learned/proved/taught with images and why?

*Using mathematics this way isn't the same as geometrical thinking? 


I'm also open to books/articles on this topic.

One topic where pictures are very useful is in that of van Kampen diagrams in combinatorial group theory, to show one relation is a consequence of others. Here is a diagram 

which shows (as I leave you to work out exactly!) that the relation $x^7$ is a consequence of the relations 
$$r= x^2yxy^3,\;\; s= y^2xyx^3.$$
The $2$-dim picture is very important, and from this one can work out a linear expression of $x^7$ in terms of $r,s$: this is discussed around p. 72 of the book Nonabelian algebraic topology, from which the above (non original) diagram is borrowed. A web search on "van Kampen diagrams" gives even more elaborate examples, and more explanation. 
The assumption that mathematics is truly kosher only when written on a line fails to live up to what is going on in higher dimensional algebra, where completely valid arguments often need to put in more than one dimension. One then gets the usual problem of how one writes down what is essentially a $3$-dimensional argument on a piece of paper! 
