That’s a wonderful compendium of graphical proofs. Math Overflow has become such an important mathematical resource: you could imagine this thread being a pretty good short book.
Graphical reasoning can be transformative, when it’s possible. In the field I worked on as a PhD student, string diagrams are amazingly powerful (http://ncatlab.org/nlab/show/string+diagram).
(I introduced the string diagram notation into my dissertation specifically for the sake of one particular proof that would have been incomprehensible expressed any other way.)
I love visual proofs, but the problem with them is that they rarely work without some explanation (and the ones that seem to work without some explanation are often those that re-express theorems you already sort of understand). What the visual proof does is eliminate excess words so that you have multiple streams of information.
For instance, I have published a cool visual proof of the amortised analysis of array resizing,[0] but even in animated form it would take at least a little bit of explanation for someone who didn't already know the basic proof to understand what's going on with it. The elegance of it is that I only need a few words to explain what the visuals mean.
This is cool, although I thought it was going to be something different.
One of my biggest pet peeves with ADHD trying to work on a Physics/Math degree was the excessive wordiness of books. I generally have a photographic and methodic memory, and so many proofs have tons and tons of words in them.
Oddly enough, I got along best with the older russian professors. I talked with one of my professors about this. I guess most of them had been taught in a much less wordy matter, and taught with even less words because English wasn't their primary language.
There's a few books that have been around forever, and they have a completely different level of readability between the editions. The Churchill/Brown book of Complex Variables and Applications has added something like 100+ pages of wordiness to the book since the 60's edition.
