Why not a 21.6mm x 21.6mm sensor?

Leaving aside the dubious claim that 4:3 is "the ideal ratio for projecting on a screen", 4:3 does not use the most of the image circle of other aspect ratios (that honor falls to 1:1).  In any case, I'm talking about a sensor that circumscribes the image circle, not a sensor that is inscribed in the image circle, so all aspect ratios can be used with 100% efficiency.