Some of them seem completely arbitrary. Others vaguely relate to the law involved - e.g. #6 where the circles share a common boundary.

I prefer the rules in my copy of Design Elements - A Graphic Style Manual. Its rule #2 make a comment that is apt here: communicate don't decorate.

the shapes are the raison d'etre for the article.

"How many UX concepts can I attempt to illustrate with basic shapes and shading?"

I'll hazard a guess: the site is a portfolio piece, meant to show off both the author's ability to make artistic design, and their familiarity with some theory behind UX.

If it were meant to convey information first, it would have looked like this: https://news.ycombinator.com/item?id=24031163.

I think they were just meant to make the design more interesting :)

you can scroll down to get the actual content. but get this, while that site is about UX, its UX is so bad you didn't figure that out right away.