Why no octagonal IC packages?

I don't think you'd get any advantage. If you look at a square and an octagon with the same perimeter, the octagon does not look great. On top of the that, you normally lose a bit of space at the corners (compare PLCC and TQFP packages) so I think you'd lose a lot with double the corners.

I think it would make sense from a PCB designer point of view, however production wise it is not the best idea.

Comparing a square and octagonal TQFP package with the same area, shows that the ammount of pins stays equal without any problems. And surely, routing a PCB could be a lot easier with this!

But, to provide another viewpoint:

The production of octagonal dies is less efficient than for square dies. So if the chip designer wants to make his chip as small as possible, this is not going to help... The die inside the package would still be square to make production cost effective.

And of course, square dies are easily sawn by vertical and horizontal lines, whereas ocotagonal shapes need another cutting method...

Traditionally new packages are developed for better manufacturing yields or higher density. So if there would be any advantage, it would have to be in that primarily. I don't think that manufacturing yield is a reason.. still the same pin technology, pin pitch etc. Density is pins/area -> so a given package should have more pins on it to make sense.

If you consider a square and an octagon with the same overall width/height (e.g. the octagon fits inside the square), you could calculate the perimeter of both packages. I assume the perimeter as a direct indicator of how many pins can fit on a package.

For a rectangle the perimeter is 2*w + 2*h. For an octagon the perimeter is 8 times the length of a side. The length of a side for an octagon is 0.4142*w. (source: wikipedia).

Substitute x=y=1cm, you get 4*1=4cm perimeter for a square and 8*0.4142*1=3.3136cm for an octagonal. That's 17.16% loss in perimeter length, or 17.16% less pins on the same width/height package.

That makes sense, because you're basically "cutting a corner" 4 times. So I can understand why these packages don't exist.