even at 1x such configurations will have little discernible difference between it and pixels
2x size pixels and a screen pattern (could be any of the shapes you describe) it's fruitless, interesting for a restriction for your art but nothing more, I'm sure alot of the "scientists" and technicians that have been making and developing displays for decades came to a similar conclusion, it's not efficient or economical because it would likely just be harder to manufacture with little to no discernible difference between it and the current methods,
besides you miss the point to some degree, pixel's are the digital picture element used to display visual data, monitors don't have "pixels" they are dot matrices and essentially each element is basically circular in nature anyway, and with cathode ray methods they were circular rays of light clustered closely together anyway as far as I recall.
and in all likely hood anything more complex than a 4 sided pixels are going to have more overhead than is worthwhile, and as you are doing it now you are relying on square pixels to form your proposed patterns which is a bizarre paradox within itself