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There's a book called "Proofs without words". Fun to have a glance. (https://ia801405.us.archive.org/24/items/proofs-without-word...) It also has a sequel.

I've never really been a fan of proofs without words; they've always felt way too slippery to me, for lack of a better term. A well worded proof with nice explanatory diagrams hits the spot for me instead.

Here’s a proof with just a few words that got published in a serious math journal: https://fermatslibrary.com/s/shortest-paper-ever-published-i...

Nice. There's an entire book like this for geometric statements. Every picture is a fact, proofs are supplied by the reader:

https://users.mccme.ru/akopyan/papers/EnGeoFigures.pdf

Caution: proofs of some of the statements in it are difficult.

Anyone who enjoys this should read David Bessis’s Mathematica.

Another great site is https://theoremoftheday.org/ with a neat one-pager overview of each theorem

"The sum of the first $n$ positive integers is ${n+1 \choose 2}$" is beautiful! for anyone lacking the background to get it, the right hand side is "(n + 1) choose 2", the number of ways of selecting 2 elements out of a set of (n + 1). and if you look at the picture, selecting any two balls in the bottom row uniquely identifies a ball in the triangle, and vice versa (selecting a ball in the triangle picks a unique pair of balls in the bottom row). so the sum of all the balls in the first n rows is indeed the number of ways of choosing two balls from the bottom row!

There is also this youtube channel called 'Mathematical Visual Proofs' on similar theme: https://www.youtube.com/@MathVisualProofs

I made a game out of creating proofs without words: https://brianberns.github.io/Tactix/

I know a nice proof of volume of tetrahedron being 1/3 of the corresponding paralellepiped. You split it into smaller tetrahedra by midpoints and count them.

Also there is a nice visual proof that in an equilateral triangle, for every point in it, the sum of distances from all the sides is constant.

I’m not a huge fan of these, but this time I noticed that the best ones feel a lot like naturality arguments. As in, moving structural bits in a way that makes it clear that we’re not touching anything that ought to be universally quantifiable.

I still don’t love this sort of thing being presented as “proof”, but I thought that idea is interesting. Is there a way to formalize naturality into technical diagrams? Probably!

See also O. Byrne, "The First Six Books of the Elements of Euclid, in Which Coloured Diagrams and Symbols are Used Instead of Letters for the Greater Ease of Learners", https://www.c82.net/euclid/ (reproduction in CSS by Nicholas Rougeux)

and also how to lie with visual proofs: https://www.youtube.com/watch?v=VYQVlVoWoPY