Chess without squares?!

I recently ran into an interesting post on Reddit, on the subreddit /r/dataisbeautiful. It's an image that shows the pattern of "78 million chess moves, from 1 million top-level games". The image looks like this:

There are some interesting comments on the Reddit thread, so if you're interested check it out. For instance, one comment points out that bishops & knights seem to move towards the center and very rarely towards the sides of the board-- a "large data" way of learning how to play chess like the pros, in a way.

But what I found striking with this image is that, of course, the pieces' movements are bound by the squares on the board (giving the image the jagged, linear, square-y appearance that it has, with a ton of untouched white space). "What if there were no squares?", asked John Lennon (I think). In other words, is it possible to imagine (no pun intended) a chess game where pieces move freely on the board, no longer restricted by the squares?

In truth, I don't know how we could figure out the legality of moves without the neatness of the squares. (Perhaps it would need to be more "physics-based", with a maximum distance [read: speed?] that can be travelled by the pieces at each move, which itself could depend on their previous momentum.) But those technicalities notwithstanding, it's fun to imagine...