I wonder why [Apostol's proof of the irrationality of $\sqrt{2}$][1] - which is as visual as a proof can be (in my opinion) - has not been mentioned: One can literally *see* at a glance that it proves what it's supposed to prove: the **impossibility of a isosceles triangle with integer side length** (by infinite descent):

[![enter image description here][2]][2]

Note that it's not a proof completely without words. It helps a lot to read the comments of the author:

> Each line segment in the diagram has integer length, and the three
> segments with double tick marks have equal lengths. (Two of them are
> tangents to the circle from the same point.) Therefore the smaller
> isosceles right triangle with hypotenuse on the horizontal base also
> has integer sides.

But through own thinking one could come up with this by oneself (having in mind what's to be proved).


  [1]: http://hipatia.dma.ulpgc.es/profesores/personal/aph/ficheros/resolver/ficheros/crp/2root_2000.pdf
  [2]: https://i.sstatic.net/vogIH.png