Originally posted by AstroDave More interestingly, however, it seems to me that another crop mode would be very nice to have: a "square" picture mode, on the order of 4900x4900 pixels. The width of this square would be about what Pentax is offering, and the height would fit just right into the full frame height of 4912 pixels.
Perhaps Pentax figures we are all so used to the standard (from 35mm film days!) 1.5:1 width to height ratio for images, that we don't want anything else. However, assuming that a normal APS-C lens produces uniform quality imaging in all directions (i.e. same height as width (actually round), even though we usually throw away the extra height resolution), the extra height pixels should have just as good image quality as the outer width pixels.
This would put you outside the APS-C image circle, don't just think width and height but consider the diagonal of your rectangles. 4900x4900 pixels on a k-1 is roughly a 24mm by 24mm square, which importantly has a diagonal of about 33.94mm. APS-C is 16x24mm rectangle, which is a diagonal of about 28.8mm, and that's as wide as an APS-C lens needs to cover. Add in shake reduction or composition adjust, and you might be moving well into a poor area of an aps-c lens.
Of course, many APS-C designated lenses would cover your proposed square just fine but I think Pentax likes to avoid built-in doohickies that might lead us into the danger zone..